qm_a0704

2007 4 5 c○
1 1
1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 1 22
2.1 1 . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 1 40
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.6 . . . . . . . . . . . . . . . . . . 52
4 57
4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 74
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

ii
6 116
6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.6 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7 133
7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8 WKB () 148
8.1 WKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.3 . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9 160
9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
10 175
10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
11 209
11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

iii
11.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
11.7 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
12 221
12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12.3 : 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
12.4 : 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13 229
13.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
13.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
14 237
14.1 ( Levi–Civita ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
14.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
14.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
14.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
15 248
15.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
15.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
15.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
15.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
15.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
15.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
15.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
15.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
15.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
284
()
J. J. , ()
,
,
1, 2, 3 ()
1, 2 ()
()
I, II ()

1 1
1
1.1
m¨r = F 2
, ,
,
, , 1
r = (x, y, z) , r
, 3 x, y, z f(x, y, z) f(r)
✓ 1
✏
= h/2π ( h ) p ,
ˆp = − i∇
ˆp x = − i ∂
∂x , ˆp y = − i ∂ ∂y , ˆp z = − i ∂ ∂z
, ˆ ,
✒
2
ˆp x xf(r) = −i ∂
(
∂x xf(r) = −i f(r) + x ∂f(r) )
, x ˆp x f(r) = −ix ∂f(r)
∂x
∂x
( )
x ˆp x − ˆp x x f(r) = if(r)
✑
f(r) , f(r) ,
x ˆp x − ˆp x x = i
, 2
Â, ˆB [ Â , ˆB ] = Â ˆB − ˆBÂ [ Â , ˆB ]
[ x , ˆp x ] = i x 1 = x, x 2 = y, x 3 = z
[ x i , ˆp j ] = i δ ij , [ x i , x j ] = 0 , [ ˆp i , ˆp j ] = 0 (1.1)
✓ 2
✏
V (r) m ,
i ∂ ψ(r, t) = Ĥψ(r, t) ,
∂t Ĥ
= ˆp2
2
+ V (r) = −
2m 2m ∇2 + V (r) (1.2)
ψ(r, t) V (r) t
, r = (x, y, z) d 3 r = dx dy dz
|ψ(r, t)| 2 d 3 r
(1.2)
✒
✑

1 2
H ,
ψ(r, t) (1.2) , , Ĥψ(r, t) , (1.2)
ψ(r, t) = 0 , ψ(r, t)
,
,
ψ(r, t) , ψ(r, t)
,
(1.2) ψ 1 () , C , ψ(r, t) (1.2) ,
C ψ(r, t) (1.2) , ψ(r, t) C ψ(r, t)
, ψ(r, t) C ψ(r, t) ,
, ψ 1 (r, t), ψ 2 (r, t), · · · (1.2) ,
ψ(r, t) = ∑ i
C i ψ i (r, t) ,
C i =
(1.2)
ψ(r, t) = C 1 ψ 1 (r, t) + C 2 ψ 2 (r, t)
,
|ψ| 2 = |C 1 ψ 1 | 2 + |C 2 ψ 2 | 2 + 2Re (C ∗ 1 C 2 ψ ∗ 1ψ 2 )
, |C 1 ψ 1 | 2 + |C 2 ψ 2 | 2 3 C 1 ψ 1 C 2 ψ 2
,
2 ,
∫
P (t) = d 3 r |ψ(r, t)| 2 (1.3)
2
∂ψ(r, t)
∂t
V ,
∂ψ ∗ (r, t)
∂t
=
(
−
2im ∇2 + 1 )
i V (r) ψ(r, t)
=
(
2im ∇2 − 1
i V (r) )
ψ ∗ (r, t)
, ρ(r, t) = |ψ(r, t)| 2
∂ρ(r, t)
∂t
= ∂ψ∗ ∂ψ
ψ + ψ∗
∂t ∂t = (
)
ψ(r, t)∇ 2 ψ ∗ (r, t) − ψ ∗ (r, t) ∇ 2 ψ(r, t)
2im
∇·(ψ∇ψ ∗ ) = (∇ψ)·∇ψ ∗ + ψ∇ 2 ψ ∗
∂ρ(r, t)
∂t
= ∇·(
)
ψ(r, t)∇ψ ∗ (r, t) − ψ ∗ (r, t) ∇ψ(r, t)
2im

1 3
j(r, t) =
(
)
ψ ∗ (r, t)∇ψ(r, t) − ψ(r, t) ∇ψ ∗ (r, t) = (
)
2im
m Im ψ ∗ (r, t)∇ψ(r, t)
∂ρ(r, t)
∂t
(1.4)
+ ∇·j(r, t) = 0 (1.5)
V
∫
∫
∫
d
d 3 r ρ(r, t) = − d 3 r ∇·j(r, t) = − dS n·j(r, t)
dt
V
V
S V , n S V S
, S ψ(r, t) 0 0 ,
∫
d
d 3 r ρ(r, t) = dP
dt
dt = 0
P
P ψ(r, t)/ √ P (1.2) ψ(r, t)
∫
d 3 r |ψ(r, t)| 2 = 1 (1.6)
S
|ψ(r, t)| 2 d 3 r (1.6)
(1.5) , ρ , j
(1.5) , ρ = |ψ| 2 , (1.4)
j ()
1.1
V (r)
dP
dt = 2 ∫
d 3 r |ψ(r, t)| 2 Im V (r)
V
✓ 3
✏
A(r, p) Â = A(r, ˆp) ψ(r, t) ,
, 〈A ˆ 〉
∫
〈A ˆ 〉 = d 3 r ψ ∗ (r, t) A(r, −i∇) ψ(r, t)
, ψ(r, t) , ,
,
✒
 ,  ψ(r, t)
,
∫
〈A ˆ 〉 ̸= d 3 r A(r, −i∇) ψ ∗ (r, t) ψ(r, t)
  = A(r)
∫
∫
〈A ˆ 〉 = d 3 r A(r) ψ ∗ (r, t) ψ(r, t) = d 3 r A(r) |ψ(r, t)| 2
✑

1 4
, 2 |ψ(r, t)| 2
,
, 1 (Ehrenfest)
〈A ˆ 〉 , Â
i d ∫
〈A ˆ 〉 = d 3 r
(i ∂ψ∗ Âψ + ψ ∗ Â i ∂ψ )
dt
∂t
∂t
(1.2)
i d ∫
〈A ˆ 〉 =
dt
1 2
∫
∫
d 3 2
r (Ĥψ∗)Âψ = −
2m
∫
)
d 3 r
(− (Ĥψ∗)Âψ + ψ∗ÂĤψ ∫
∫
d 3 2
r (Ĥψ∗)Âψ = − d 3 r (∇ 2 ψ ∗
2m
)Âψ + d 3 r V (r)ψ ∗ Âψ
∫
∫
d 3 r ψ ∗ ∇ 2 Âψ + d 3 r V (r)ψ ∗ Âψ = d 3 r ψ ∗ ĤÂψ
i d ∫
〈A ˆ 〉 =
dt
) ∫
d 3 r
(− ψ ∗ ĤÂψ + ψ∗ÂĤψ = d 3 r ψ ∗ [ Â , Ĥ ] ψ (1.7)
 Ĥ , 〈A ˆ 〉
Ĥ , 〈H ˆ 〉 , Â = 1
∫
〈A ˆ 〉 = d 3 r |ψ(r, t)| 2 = P
, P
(Ehrenfest)
(1.7) Â = r , Â = ˆp
d
dt 〈 r 〉 = 1 ∫
d 3 r ψ ∗ [ r , Ĥ ] ψ ,
i
d
dt 〈 ˆp 〉 = 1 ∫
d 3 r ψ ∗ [ ˆp , Ĥ ] ψ
i
[ x , Ĥ ] , Ĥ x ˆp x
Â, ˆB, Ĉ
[ x , Ĥ ] = 1
2m [ x , ˆp2 x ]
[ Â , ˆBĈ ] = Â ˆBĈ − ˆBĈÂ )
(Â = ˆB − ˆBÂ)
Ĉ + ˆB
(ÂĈ − ĈÂ = [ Â , ˆB ] Ĉ + ˆB [ Â , Ĉ ]
[ x , ˆp x ] = i
[ x , Ĥ ] = 1 (
)
[ x , ˆp x ] ˆp x + ˆp x [ x , ˆp x ] = i 2m
m ˆp x

1 5
y , z
[ r , Ĥ ] = i m
ˆp (1.8)
d
dt 〈 r 〉 = 1 ∫
d 3 r ψ ∗ ˆp ψ = 1 m
m 〈 ˆp 〉
(
)
[ ˆp , Ĥ ] = [ ˆp , V (r) ] = − i ∇V (r) − V (r)∇
, ψ ,
(
) ( )
[ ˆp , Ĥ ]ψ = − i ∇V (r)ψ − V (r)∇ψ = − i ∇V (r) ψ (1.9)
∇ V (r) F (r) V (r)
F (r) = − ∇V (r)
∫
d
dt 〈 ˆp 〉 = d 3 r ψ ∗ (F (r)) ψ = 〈F (r)〉
,
d
dt 〈 r 〉 = 1 m 〈 ˆp 〉 ,
d
〈 ˆp 〉 = 〈F (r)〉 (1.10)
dt
〈F (r)〉 = F ( 〈r〉 ) , , ,
〈F (r)〉 ̸= F ( 〈r〉 ) 1 X = 〈 x 〉
F (x) = F (X + x − X) = F (X) + F ′ (X)(x − X) + F ′′ (X)
(x − X) 2 + · · ·
2
∫
〈F (x)〉 = dx F (x) |ψ(x, t)| 2
∫
∫
= F (X) dx |ψ| 2 + F ′ (X) dx (x − X)|ψ| 2 + F ′′ ∫
(X)
dx (x − X) 2 |ψ| 2 + · · ·
2
∫
∫
dx |ψ| 2 = 1 , X = dx x |ψ| 2
m d2
dt 2 〈 x 〉 = F (〈 x 〉) + F ′′ (〈 x 〉)
(
〈 x 2 〉 − 〈 x 〉 2) + · · · (1.11)
2
2
V (x) ∝ x n , n = 0, 1, 2 (1.12)
F (x) 2 0 , 〈 x 〉
, 〈 x 〉 , |ψ(x, t)| 2 x = 〈 x 〉
|ψ(x, t)| 2 , 〈 x 〉
(1.12) ,

1 6
( 〈 x 2 〉 ≈ 〈 x 〉 2 ), F (x) ( F ′′ (〈 x 〉) ≈ 0 ),
1, 2, 3 ,
, ∆x = √ 〈 x 2 〉 − 〈 x 〉 2 , ∆p = √ 〈 ˆp 2 〉 − 〈 ˆp 〉 2 , 11 ,
∆x∆p ≥ /2
, x ,
∫
〈 ˆp 〉 = − i d 3 r ψ ∗ (r, t) ∇ψ(r, t)
,
∫
〈 ˆp 〉 ∗ = i d 3 r ψ(r, t) ∇ψ ∗ (r, t)
,
∫
dx ψ(r, t) ∂ψ∗ (r, t)
[
] x=+∞
∫
= ψ(r, t)ψ ∗ (r, t) − dx ψ ∗ ∂ψ(r, t)
(r, t)
∂x
x=−∞
∂x
∫
= − dx ψ ∗ ∂ψ(r, t)
(r, t)
∂x
∫
〈 ˆp 〉 ∗ = − i d 3 r ψ ∗ (r, t) ∇ψ(r, t) = 〈 ˆp 〉
,
∫
〈 ˆp 〉 = − i d 3 r ψ(r, t) ∇ψ(r, t) = − i 2
∫
d 3 r ∇ψ 2 (r, t) = i ×
〈 ˆp 〉 0
〈 ˆp x 〉 = − i ∫
d 3 r ∂
2 ∂x ψ2 (r, t) = − i ∫
2
, 〈 ˆp 〉 ̸= 0
dy dz [ ψ 2 (r, t) ] x=∞
x=−∞ = 0
(1.4) j
j(r, t) = 1 (
)
ψ ∗ (r, t) ˆp ψ(r, t) − ψ(r, t) ˆp ψ ∗ (r, t)
2m
∫
d 3 r j(r, t) =
〈 ˆp 〉 + 〈 ˆp 〉∗
2m
= 〈 ˆp 〉
m
j(r, t) , v(r, t) j = ρ v
, j v (1.4) j
1.2
, ψ α (r, t) , α
, ψ α (r, t)
|ψ α 〉 | α 〉 | 〉

1 7
| α 〉, | β 〉 〈 α | β 〉
∫
〈 α | β 〉 = d 3 r ψα(r, ∗ t) ψ β (r, t)
〈 α | β 〉 ∗ = 〈 β | α 〉
, 〈 α | α 〉 , u, v | γ 〉 = u | α 〉 + v | β 〉
〈 α ′ | γ 〉 = u 〈 α ′ | α 〉 + v 〈 α ′ | β 〉 , 〈 γ | α ′ 〉 = u ∗ 〈 α | α ′ 〉 + v ∗ 〈 β | α ′ 〉
 ψ α(r, t) Âψ α(r, t)  | α 〉 | β 〉
∫
〈 β | Â | α 〉 = d 3 r ψβ(r, ∗ t) Â ψ α(r, t)
1.3
ψ α (r, t), ψ β (r, t)
(∫
∗ ∫
d 3 r ψα(r, ∗ t) Â ψ β(r, t))
= d 3 r ψβ(r, ∗ t) ˆB ψ α (r, t) , 〈 α | Â | β 〉∗ = 〈 β | ˆB | α 〉
ˆB  ˆB = †
〈 α |  | β 〉∗ = 〈 β | † | α 〉 (1.13)
 , † =  , Â
A ij A † (A † ) ij = A ∗ ji ,
(1.13)
| γ 〉 = Â| β 〉 | µ 〉
〈 γ | µ 〉 = 〈 µ | γ 〉 ∗ = 〈 µ |  | β 〉∗ = 〈 β | † | µ 〉
| µ 〉 = ˆB † | α 〉 ,
〈 γ | ˆB † | α 〉 = 〈 β | † ˆB† | α 〉
〈 α | ˆB | β 〉∗ = 〈 α | ˆB | γ 〉 ∗ = 〈 γ | ˆB † | α 〉 = 〈 β | † ˆB† | α 〉
〈 α | ˆBÂ | β 〉∗ = 〈 β | ( ˆBÂ)† | α 〉
( ˆBÂ)† = † ˆB†
n Â1Â2 · · · Ân
(Â1 Â 2 · · · Ân
) †
=  † n · · · † 2† 1 (1.14)

1 8
,
 ˆB
(Â ˆB) † = ˆB † Â † = ˆBÂ
 ˆB ,  ˆB
|ψ〉 Â ˆB|ψ〉 = ˆBÂ|ψ〉 = |ψ〉, Â ˆB = ˆBÂ = 1 ,
 Â−1 1 Â
Ĉ ˆD ˆD −1 Ĉ −1 = ĈĈ−1 = 1 , ∴ (CD) −1 = ˆD −1 Ĉ −1
, †  = † = 1, † = A −1
  ˆB
(Â ˆB) −1 = ˆB −1 Â −1 = ˆB † Â † = (AB) †
 ˆB  ˆB
Û | α′ 〉 = Û| α 〉
〈 α ′ |  | β 〉 = 〈 β | † | α ′ 〉 ∗ = 〈 β | † Û | α 〉 ∗ = 〈 α | († Û) † | β 〉 = 〈 α | Û †  | β 〉 (1.15)
〈 α ′ | 〈 α ′ | = 〈 α |U † | β ′ 〉 = Û| β 〉
〈 α ′ | β ′ 〉 = 〈 α | Û † Û | β 〉 (1.16)
, Û 〈 α′ | β ′ 〉 = 〈 α | β 〉
 ϕ(r)  ϕ(r) , ϕ(r)
, ϕ a (r)
 ϕ a (r) = a ϕ a (r)
a , ϕ a (r) Â
, a a
 ,  ,
a 1 , a 2 , · · · a i ϕ i (r) :
 ϕ i (r) = a i ϕ i (r) ,  | i 〉 = a i | i 〉 (1.17)
, 〈 i | i 〉 = 1 ϕ i (r)
1. (1.17) ϕ ∗ i 〈 i |Â| i 〉 = a i〈 i | i 〉 = a i
〈 i |Â| i 〉 a i
2. (1.17) ϕ ∗ j 〈 j |Â| i 〉 = a i〈 j | i 〉
〈 i |Â| j 〉 = a i〈 i | j 〉

1 9
 | j 〉 = a j | j 〉 | i 〉
〈 i |Â| j 〉 = a j〈 i | j 〉
a i ≠ a j
(a i − a j ) 〈 i | j 〉 = 0
∫
〈 i | j 〉 = d 3 r ϕ ∗ i ϕ j = 0
ϕ i ϕ j ϕ i
〈 i | j 〉 = δ ij (1.18)
3. , ψ(r) ϕ i (r)
ψ(r) = ∑ c i ϕ i (r) , | ψ 〉 = ∑ c i | i 〉 (1.19)
i
i
c i ϕ ∗ j (r)
〈 j | ψ 〉 = ∑ i
ψ(r)
c i 〈 j | i 〉 = ∑ i
c i δ ij = c j (1.20)
| ψ 〉 = ∑ i
| i 〉〈 i | ψ 〉 (1.21)
ψ(r) = ∑ i
∫
∫
ϕ i (r) d 3 r ′ ϕ ∗ i (r ′ ) ψ(r ′ ) = d 3 r ′ ψ(r ′ ) ∑ i
ϕ i (r) ϕ ∗ i (r ′ )
∑
ϕ i (r) ϕ ∗ i (r ′ ) = δ(r − r ′ ) (1.22)
i
(1.22)
, (1.18), (1.22)
(1.13)
〈ψ|  |ψ〉 = 〈ψ|  |ψ〉∗ = 〈ψ| † |ψ〉
,  = † 3 , ,
✓ 3 ′
✏
 , 1  ϕ i(r) 1
, ϕ i
∫
|c i (t)| 2 , c i (t) = d 3 r ϕ ∗ i (r) ψ(r, t)
 a i , ψ(r, t)
, c i ψ(r, t) (1.19)
✒
✑

1 10
ψ
∫
1 = d 3 r ψ ∗ (r, t) ψ(r, t) = ∑ ij
∫
c ∗ i c j d 3 r ϕ ∗ i (r) ϕ j (r) = ∑ ij
c ∗ i c j δ ij = ∑ i
|c i | 2
, 1 ,
, ∑ |c i | 2 = 1
Â
〈 Â 〉 = ∑ i
a i |c i | 2
 |ψ〉 = ∑ i
c i  | i 〉 = ∑ i
c i a i | i 〉
〈 ψ | A | ψ 〉 = ∑ ij
c ∗ j c i a i 〈 j | i 〉 = ∑ i
a i |c i | 2
, 3 ′ 3
 1 ϕ k(r)
∫
∫
c i = d 3 r ϕ ∗ i (r) ψ(r, t) = d 3 r ϕ ∗ i (r) ϕ k (r) = δ ik
, Â i ≠ k a i 0 ,
a k
1.2
ˆp = − i∇ , 〈 α | ˆp | β 〉 ∗ = 〈 β | ˆp | α 〉
ˆp † = (− i∇) † = (−i) ∗ ∇ † = i ∇ †
∇ † = − ∇ ∇
F (x)
F (Â)
F (x) =
∞∑
f n x n
n=0
∞
F (Â) = ∑
f n  n
n=0
 Â| i 〉 = a i| i 〉 Â
∞ F (Â)| i 〉 = ∑
∞∑
f n  n | i 〉 = f n a n i | i 〉 = F (a i )| i 〉 (1.23)
n=0
F (a i ) F (x)
(1.23) F (Â) |ψ〉
n=0
|ψ〉 = ∑ i
c i | i 〉 ,
c i =

1 11
F (Â)|ψ〉 = ∑ i
c i F (Â) | i 〉 = ∑ i
c i F (a i ) | i 〉 (1.24)
,
e =
 ˆB , eÂ+ ˆB = eÂe ˆB
∞∑
n=0
 n
n!
eÂe − = e −Âe = 1 , ∴ ( eÂ) −1
= e
−Â
 ˆB eÂ+ ˆB ≠ eÂe ˆB (4.44)
(
e  ) †
∑
∞
= (Ân ) †
n=0
n!
=
∞∑ († ) n
n=0
n!
= exp ( A †)
 (iÂ)† = − iÂ
(
e  ) †
= e ,
(
e iÂ) †
= e
−i = ( e iÂ) −1
e , e iÂ
1 ψ(x + c)
 = c d
dx
ψ(x + c) = ψ(x) + c dψ
dx + c2 d 2 ψ
2! dx 2 + · · · + cn d n ψ
n! dx n + · · ·
ψ(x + c) =
∞∑
n=0
ˆp = − i d
dx
(
n!Ân 1
ψ(x) = exp(Â) ψ(x) = exp c d )
ψ(x) (1.25)
dx
e icˆp/ ψ(x) = ψ(x + c) (1.26)
e icˆp/ − c x
1.4
|ψ〉 Â , Â (
)
√
) 2
∆a = 〈ψ|
(Â − a |ψ〉 , a = 〈ψ| Â |ψ〉
a , 〈ψ| a |ψ〉 = a〈ψ|ψ〉 = a
√
) √
√
∆a = 〈ψ|
(Â2 − 2a + a2 |ψ〉 = 〈ψ|Â2 |ψ〉 − a 2 = 〈ψ|Â2 |ψ〉 − 〈ψ|  |ψ〉2

1 12
|ψ〉 Â 〈ψ|Â2 |ψ〉 = 〈ψ| Â |ψ〉2 = a 2 , 〈ψ|Â2 |ψ〉 ̸=
〈ψ| Â |ψ〉2
2 Â ˆB
∆a∆b ≥ 1 2
∣
∣〈ψ|[ Â , ˆB ]|ψ〉
∣ ∣∣ (1.27)
〈ψ|[ Â , ˆB ]|ψ〉 ̸= 0 ,
 ˆB , ,  = x, ˆB = ˆpx
[ x , ˆp x ] = i
∆x∆p x ≥ 2
(1.28)
,
(1.27)
c ψ(r) = ψ α (r) + c ψ β (r)
∫ (
)(
)
〈ψ|ψ〉 = d 3 r ψα(r) ∗ + c ∗ ψβ(r)
∗ ψ α (r) + c ψ β (r)
= 〈 α | α 〉 + |c | 2 〈 β | β 〉 + c 〈 α | β 〉 + c ∗ 〈 β | α 〉 ≥ 0 (1.29)
c , c = − 〈 β | α 〉/〈 β | β 〉 〈 β | α 〉 = 〈 α | β 〉 ∗ (1.29)
〈ψ|ψ〉 = 〈 α | α 〉 −
|〈 β | α 〉|2
〈 β | β 〉
≥ 0 , 〈 α | α 〉〈 β | β 〉 ≥ |〈 β | α 〉| 2 (1.30)
( ) ψ = 0 , ψ α ∝ ψ β
 ′ =  − a , | ψ a 〉 = Â′ | ψ 〉 (1.16)
) 2
〈 ψ a | ψ a 〉 = 〈 ψ |Â′† Â ′ | ψ 〉 = 〈 ψ |(Â′ ) 2 | ψ 〉 = 〈 ψ |
(Â − a | ψ 〉 = (∆a)
2
1 ˆB
(∆b) 2 = 〈 ψ b | ψ b 〉 , | ψ b 〉 = ˆB ′ | ψ 〉 =
,
(
ˆB − b
)
| ψ 〉 , b = 〈 ψ | ˆB | ψ 〉
(∆a) 2 (∆b) 2 = 〈 ψ a | ψ a 〉〈 ψ b | ψ b 〉 ≥ |〈 ψ a | ψ b 〉| 2 ∣
= ∣〈 ψ | Â′ ˆB′ ∣
| ψ 〉
∣ 2
λ
ψ b (r) = λ ψ a (r) (1.31)
Â′ ˆB′ 2
 ′ ˆB′ = i Ĉ + ˆD ,
Ĉ = 1 2i
(Â′ ˆB′ − ˆB ′ Â ′) = 1 2i [ Â , ˆB ] , ˆD = Â ′ ˆB′ + ˆB ′ Â ′
2
Â′ ˆB ′ (Â′ ˆB′ ) † = ˆB ′ Â ′ Ĉ ˆD
,
∣
∣〈 ψ |Â′ ˆB′ | ψ 〉 ∣ 2 = ∣ i 〈 ψ | Ĉ | ψ 〉 + 〈 ψ | ˆD | ψ 〉 ∣ 2 = 〈 ψ | Ĉ | ψ 〉2 + 〈 ψ | ˆD | ψ 〉 2

1 13
(∆a) 2 (∆b) 2 ≥ 〈 ψ | Ĉ | ψ 〉2 + 〈 ψ | ˆD | ψ 〉 2 ≥ 〈 ψ | Ĉ | ψ 〉2 = 1 4
(1.29)
∣
∣〈 ψ |[ Â , ˆB ]| ψ 〉
∣ ∣∣
2
(1.29) (1.31)
〈 ψ | ˆD | ψ 〉 = 1 (
〈 ψ
2
|Â′ ˆB′ | ψ 〉 + 〈 ψ | ˆB
)
′ Â ′ | ψ 〉 = 1 (
〈 ψ a | ψ b 〉 + 〈 ψ a | ψ b 〉 ∗) = 0 (1.32)
2
(1.31)
λ + λ ∗
〈 ψ a | ψ a 〉 = 0
2
(∆a) 2 = 〈ψ a |ψ a 〉 ̸= 0 ν λ = iν
ψ b (r) = iν ψ a (r) ,
(1.32)
〈 ψ a | ψ b 〉 = 1 2
(1.33)
i.e.
( )
)
ˆB − b ψ(r) = iν
(Â − a ψ(r) (1.33)
(
)
〈 ψ a | ψ b 〉 − 〈 ψ b | ψ a 〉 = 1 (Â′
2 〈 ψ | ˆB′ − ˆB ′ Â ′) | ψ 〉 = 1 2 〈 ψ |[ Â , ˆB ]| ψ 〉
ν = − i 〈 ψ a | ψ b 〉
〈 ψ a | ψ a 〉 = − i 〈 ψ |[ Â , ˆB ]| ψ 〉
= − i 〈 ψ |[ Â , ˆB ]| ψ 〉
2〈 ψ a | ψ a 〉
2 (∆a) 2 (1.34)
1.3
ψ(x)
1 x ˆp ∆x∆p = /2
ψ(x) =
(
1
(2π(∆x) 2 ) exp − (x − x 0) 2 )
1/4 4(∆x) 2 + ik 0 x
, x 0 = 〈ψ| x |ψ〉 , k 0 = 〈ψ| ˆp |ψ〉
(1.35)
1.5
∂ψ(r, t)
i = Ĥψ(r, t) (1.36)
∂t
ψ(r, t) = f(t)ϕ(r)
i df(t)
= 1
f(t) dt ϕ(r)Ĥϕ(r)
t , r ,
E f
df(t)
dt
= E i f(t)

1 14
f(t) = f 0 e −iEt/ f 0 ϕ(r)
ψ(r, t) = e −iEt/ ϕ(r) (1.37)
Ĥϕ(r) = Eϕ(r) (1.38)
ϕ(r)
Ĥ , E
(1.37) , t = 0 Ĥ ϕ , e−iEt/
, t = 0 , (1.37) ψ(r, t)
|ψ(r, t)| 2 = |ϕ(r)| 2 , Â
, 〈Â 〉
∫ (
∗
∫
〈Â 〉 = d 3 r e ϕ(r)) −iEt/ Âe −iEt/ ϕ(r) = d 3 r ϕ ∗ (r)Âϕ(r)
(1.7) , [ Â , Ĥ ] ≠ 0
, Â Ĥ Â
(1.38) , ,
Ĥϕ n (r) = E n ϕ n (r) , Ĥ | n 〉 = E n | n 〉
(1.18) c n c n e −iEnt/ ϕ n (r) (1.36)
, , (1.36) , (1.36)
ψ(r, t) = ∑ n
c n e −iE nt/ ϕ n (r) , |ψ(t)〉 = ∑ n
c n e −iE nt/ | n 〉 (1.39)
c n (1.39) ϕ ∗ j (r) , (1.18)
〈 j |ψ(t)〉 = ∑ n
c n e −iE nt/ δ nj = c j e −iE jt/
t = 0
∫
c n = 〈 n |ψ(0)〉 = d 3 r ϕ ∗ n(r) ψ(r, 0) (1.40)
, t = 0 ψ(r, 0) c n , ψ(r, t)
, ψ(r, 0) Ĥ ϕ k c n = 〈 n | k 〉 = δ nk
ψ(r, t) = e −iEkt/ ϕ k (r)
(1.39)
|ψ(r, t)| 2 = ∑ c ∗ nc n ′e i(En−E n ′ )t/ ϕ ∗ n(r)ϕ n ′(r)
nn ′
,
ψ(r, t) = c 1 e −E 1t/ ϕ 1 (r) + c 2 e −E 2t/ ϕ 2 (r)

1 15
(
)
|ψ(r, t)| 2 = |c 1 ϕ 1 (r)| 2 + |c 2 ϕ 2 (r)| 2 + 2Re c ∗ 1c 2 e i(E 1−E 2 )t/ ϕ ∗ 1(r)ϕ 2 (r)
, |E 1 − E 2 |/ ,
∫
d 3 r |ψ(r, t)| 2 = ∑ ∫
c ∗ nc n ′e i(En−E n ′ )t/ d 3 r ϕ ∗ n(r)ϕ n ′(r) = ∑ c ∗ nc n ′e i(En−E n ′ )t/ δ nn ′ = ∑
nn ′ nn ′ n
|c n | 2
 〈ψ(t)|  |ψ(t)〉 = ∑ nn ′ c ∗ nc n ′e i(En−E n ′ )t/ 〈 n |  | n′ 〉 (1.41)
, , c n = δ nk
〈ψ(t)| Â |ψ(t)〉 = 〈 k | Â | k 〉 = 〈ψ(0)| Â |ψ(0)〉
, 〈 n | Ĥ | n′ 〉 = E n 〈 n | n ′ 〉 =
E n δ nn ′ ,
〈ψ(t)| Ĥ |ψ(t)〉 = ∑ n
|c n | 2 E n
( ) t E n
|〈 n |ψ(t)〉| 2 = |c n e −iEnt/ | 2 = |c n | 2
e −iĤt/ e −iĤt/ | n 〉 = e −iEnt/ | n 〉 (1.39)
|ψ(t)〉 = ∑ n
c n e −iĤt/ | n 〉
e −iĤt/ n
|ψ(t)〉 = e −iĤt/ ∑ n
c n | n 〉 = e −iĤt/ |ψ(0)〉 (1.42)
(1.26) , ,
(1.42) (1.36)
, (1.15) , (1.41)
〈ψ(t)| Â |ψ(t)〉 = 〈ψ(0)| eiĤt/ Â e −iĤt/ |ψ(0)〉 (1.43)
1.4
ψ(r, t) Ĥ ϕ n(r)
ψ(r, t) = ∑ n
f n (t) ϕ n (r)
(1.36) f n (t) i df n
dt = E nf n (1.39)

1 16
1.5
Ĥ
〈 n |[ Â , Ĥ ]| n′ 〉 = (E n ′ − E n ) 〈 n |Â | n′ 〉
(1.41) (1.7) , (1.43) (1.7)
1.6
〈 n |[ Â , Ĥ ]| n 〉 = 0 Â = r· ˆp
1
2m 〈 n | ˆp2 | n 〉 = 1 ˆp2
〈 n | r·(∇V ) | n 〉 , Ĥ = + V (r) (1.44)
2 2m
1.6
ˆp = − i∇
∇ exp(ik·r) = ik exp(ik·r) , ˆp exp(ik·r) = k exp(ik·r)
exp(ik·r) k N k
ϕ k (r) = N k exp(ik·r)
, Ĥ = ˆp2 /(2m)
Ĥϕ k (r) = E k ϕ k (r) , E k = 2 k 2
ϕ k (r) Ĥ E k |ϕ k (r)| r
∫
∫
d 3 r |ϕ k (r)| 2 = |N k | 2 d 3 r =
, r V
V → ∞ |N k | 2 V = 1 ,
ϕ k (r) = 1 √
V
exp(ik·r)
2m
N k = (2π) −3/2
∫ ( )
〈 k | k ′ 〉 = NkN ∗ k ′ d 3 r exp i (k ′ − k)·r = (2π) 3 |N k | 2 δ(k − k ′ ) = δ(k − k ′ )
,
,
∂ψ(r, t)
i = − 2
∂t 2m ∇2 ψ(r, t)
, (1.39), (1.40) ϕ i (r) ϕ k (r) , k i
∫
∫
ψ(r, t) = d 3 k c(k) ϕ k (r)e −iωkt 1
= d 3 k c(k) exp (ik·r − iω
(2π) 3/2 k t) ,
ω k = E k
= k2
2m

1 17
, ,
(1.40) | i 〉 | k 〉
∫
1
〈 k |ψ(0)〉 = d 3 r exp (− ik·r) ψ(r, 0)
(2π) 3/2
= 1 ∫
∫
(2π) 3 d 3 r exp (− ik·r) d 3 k ′ c(k ′ ) exp (ik ′·r)
∫ ∫ d
= d 3 k ′ c(k ′ 3 r
( ) ∫
)
(2π) 3 exp i (k ′ − k)·r = d 3 k ′ c(k ′ ) δ(k − k ′ ) = c(k)
1.7
ψ(r, t) ψ(r, t) = N exp(ik·r − iEt/)
1. ρ j j = ρv v
k k/m ρ v
j = ρ v
2. ∆x ∆p x
1.8
∫
d 3 r
ϕ(k, t) = exp(− ik·r) ψ(r, t)
(2π)
3/2
∫
1. ψ(r, t) d 3 k |ϕ(k, t)| 2 = 1
2. ∫
∫
d 3 r ψ ∗ (r, t) ˆp ψ(r, t) = d 3 k k |ϕ(k, t)| 2
3.
k |ϕ(k, t)| 2
3 ′
∫
d 3 (
r
(2π) 3/2 xn exp(−ik·r) ψ(r, t) = i ∂ ) n
ϕ(k, t)
∂k x
∫
d 3 r
(2π) 3/2 V (r) exp(−ik·r) ψ(r, t) = V (i∇ k)ϕ(k, t)
∇ k k
4. p = k
∂ϕ(p, t)
i = p2
∂t 2m ϕ(p, t) + V (ˆr) ϕ(p, t) , ˆr = i∇ p (1.45)
r ,
(1.1)
1.7
r t exp (ik·r − iωt) ω > 0 k ω = k 2 /(2m)
t , θ = k·r − ωt 2 r 1 , r 2
k·r 1 − ωt = k·r 2 − ωt , ∴ k·(r 1 − r 2 ) = 0

1 18
k r 1 − r 2 , t , r k
, exp (ik·r − iωt)
, k k t ∆t , k
∆r
k·r − ωt = k·
(r + k k ∆r )
− ω (t + ∆t) , k = | k |
v p = ∆r
∆t = ω k
v p = ω/k v p
v p = k/(2m) , k/m
2
∂ψ(r, t)
(S) i = − 2
∂t 2m ∇2 ψ(r, t) , (M) ∇ 2 ψ(r, t) − 1 ∂ 2 ψ(r, t)
c 2 ∂t 2 = 0 (1.46)
(S) , (M)
ψ(r, t) = exp (ik·r − iωt)
(S) ω = k2 , (M) ω = ck (1.47)
2m
exp (ik·r − iωt) (S) , k/(2m) k
, , (M) , c
(1.47) ω k ω k
(1.46) ψ 1 , a(k) ,
∫
ψ(r, t) =
( )
d 3 k a(k) exp ik·r − iω k t
(1.46) 1 k x = k
ψ(x, t) =
∫ ∞
−∞
( )
dk a(k) exp ikx − iω k t
(1.48)
,
a(k) = a 0 e −α2 (k−k 0 ) 2 , α > 0 (1.49)
α , k k 0 a(k) e −α2 (k−k 0 ) 2
= e −1
k |k − k 0 | = 1/α , (1.48) k 0 − 1/α < k < k 0 + 1/α
α → ∞
a(k) =
{
a0 , k = k 0
0 , k ≠ k 0
α → 0 a(k) = a 0 (1.46) , θ = kx − ω k t k 2
θ(k) = θ(k 0 ) + dθ
dk ∣ (k − k 0 ) + 1
k=k0
2
d 2 θ
dk 2 ∣
∣∣∣k=k0
(k − k 0 ) 2
= k 0 x − ω 0 t + (x − v g t) (k − k 0 ) − γt(k − k 0 ) 2

1 19
ω 0 = ω k0 ,
v g = dω k
dk
(1.48) k − k 0 k
ψ(x, t) = a 0 e ik 0x−iω 0 t
∫ ∞
−∞
∣ , γ = 1
k=k0
2
(
dk exp − ( α 2 + iγt ) )
k 2 + i (x − v g t) k
= a 0 exp
(ik 0 x − iω 0 t − (x − v gt) 2 ) ∫ (
∞
4(α 2 dk exp
+ iγt) −∞
(14.22) , z 1 , z 2 Re z 1 > 0
∫ ∞
dx exp ( −z 1 (x − z 2 ) 2) √ π
=
z 1
|ψ(x, t)| 2
−∞
d 2 ω k
dk 2 ∣
∣∣∣k=k0
− (α 2 + iγt)
√
( π
ψ(x, t) = a 0
α 2 + iγt exp ik 0 x − iω 0 t − (x − v gt) 2 )
4(α 2 + iγt)
(
|ψ(x, t)| 2 = a2 0 π
αD exp − (x − v gt) 2 )
2D 2 , D(t) =
√
α 2 +
(
k − i(x − v ) ) 2
gt)
2(α 2 + iγt)
(1.50)
( ) 2 γt
(1.51)
α
v g ψ(x, t) k = k 0 ,
v g v g
(S)
dω k
dk = k m ≠ ω k
k = k
2m , (M) dω k
dk = c = ω k
k
, (S) ,
|ψ(x, t)| x = v g t D ,
(M) γ = 0 D(t) = α ,
(= ) , (S) γ = /(2m) ≠ 0
, D(t) , ω k ∝ k ,
ω k /k k ,
, ω k k
v g = dω k /dk
, (1.51) ψ(x, t)
∫ ∞
∫
dx |ψ(x, t)| 2 = a2 0 π ∞
dx exp
(− (x − v gt) 2 )
αD
2D 2 = a2 0 π √
√ α
2πD2 = 1 , ∴ a 0 =
αD
π √ 2π
−∞
−∞
√
(
1 α
ψ(x, t) = √
2π α 2 + iγt exp ik 0 x − iω 0 t − (x − v gt) 2 )
4(α 2 + iγt)
|ψ(x, t)| 2 1
= √ exp
(− (x − v gt) 2 )
2π D 2D 2
(1.52)
(1.53)
t = 0 (1.35) x 0 = 0 , ∆x = α
, , v g = k 0 /m

1 20
, ,
t 〈 · · · 〉
〈 x 〉 =
〈 x 2 〉 =
∫ ∞
−∞
∫ ∞
−∞
∫
dx x |ψ(x, t)| 2 1 ∞
= √ dx (x + v g t) e −x2 /(2D 2) = v g t
2π D
−∞
∫
dx x 2 |ψ(x, t)| 2 1 ∞
= √ dx (x + v g t) 2 e −x2 /(2D 2 )
2π D
−∞
∫
1 ∞
= √ dx ( x 2 + (v g t) 2) e −x2 /(2D 2) = D 2 + (v g t) 2
2π D
−∞
(∆x) 2 = 〈 x 2 〉 − 〈 x 〉 2 = D 2 ,
(
∂
∂x ψ(x, t) = ik 0 −
x − v )
gt
2(α 2 ψ(x, t)
+ iγt)
〈 ˆp 〉 =
∫ ∞
(
)
x − v g t
dx k 0 + i
2(α 2 |ψ(x, t)| 2 = k 0 +
i ∫ ∞
√ dx x /(2D 2 )
e−x2
+ iγt)
2π D 2(α 2 + iγt)
−∞
= k 0 = mv g
〈 ˆp 〉 = m d〈 x 〉/dt ,
∫ ∞
〈 ˆp 2 〉 = − 2 dx ψ ∗ (x, t) ∂2
ψ(x, t)
∂x2 −∞
∫ ∞
= 2 dx ∂ψ∗ (x, t) ∂ψ(x, t)
−∞ ∂x ∂x
∫ ∞
= 2 dx
∣ ik 0 −
x − v 2
gt
−∞ 2(α 2 + iγt) ∣ |ψ(x, t)| 2
∫ ∞
(
= 2 dx k0 2 + k 0(x − v g t)γt
α 2 D 2 + (x − v gt) 2 )
4α 2 D 2 |ψ(x, t)| 2 = 2 k0 2 + 2
4α 2
−∞
, (∆p) 2 = 〈 ˆp 2 〉 − 〈 ˆp 〉 2 = 2 /(4α 2 )
√
∆x∆p = D
2α = ( ) 2 γt
1 +
2 α 2
t ≠ 0 α ∆p γt
α = ∆p
m t
( ) 2 ∆p
(∆x) 2 = D 2 = α 2 +
m t t = 0
t = t 0
∆x ,
∆p/m
(1.53) |ψ(x, t)| 2 t 0
, 〈 x 〉 〈 p 〉 ,
∆x ∆p , 〈 x 〉
−∞
t = 2t 0
t = 3t 0
0 v g t 0 2v g t 0 3v g t 0 x
〈 p 〉 ,

1 21
1.9
(1.48)
(1.48)
ψ(x, t) = 1
2π
∫ ∞
−∞
a(k) = 1
2π
∫ ∞
−∞
dx e −ikx ψ(x, 0)
∫ ∞
dx ′ dk e ik(x−x′ )−iω k t ψ(x ′ , 0) =
−∞
G(x, t) = 1
2π
∫ ∞
−∞
∫ ∞
−∞
dk e ikx−iω kt
dx ′ G(x − x ′ , t) ψ(x ′ , 0) (1.54)
(1.55)
G(x, t) 1 (14.21)
√ ( )
m
G(x, t) =
2πit exp i mx2
(1.56)
2t
(1.52) t = 0
ψ(x, 0) =
(1.54) (1.52)
G(x, t) (1.55)
(1.56) t → 0
)
1
√√ exp
(ik 0 x − x2
2π α 4α 2
G(x, 0) = 1
2π
lim
t→0
√
∫ ∞
−∞
dk e ikx = δ(x)
( )
m
2πit exp i mx2
2t
= δ(x)
1
δ(x) = √ 1 e −x2 /ε
lim √ π ε→0 ε

2 1 22
2 1
2.1 1
1
− 2
d 2
ψ(x) + V (x)ψ(x) = Eψ(x) (2.1)
2m dx2
ψ i (x) 1
c 1 ψ 1 (x) + c 2 ψ 2 (x) + · · · + c n ψ n (x) = 0 =⇒ c 1 = c 2 = · · · = c n = 0
1 ,
1 (2.1) 2 ψ 1 (x)
ψ 2 (x) :
− 2
d 2
2m dx 2 ψ 1(x) + V (x)ψ 1 (x) = Eψ 1 (x) ,
− 2
d 2
2m dx 2 ψ 2(x) + V (x)ψ 2 (x) = Eψ 2 (x)
ψ 2 ψ 1
ψ 2 (x) d2
dx 2 ψ 1(x) − ψ 1 (x) d2
dx 2 ψ 2(x) = 0
ψ 2 (x) d2
dx 2 ψ 1(x) − ψ 1 (x) d2
dx 2 ψ 2(x) = d (
ψ 2 (x) d
dx dx ψ 1(x) − ψ 1 (x) d )
dx ψ 2(x)
ψ 2 (x) d
dx ψ 1(x) − ψ 1 (x) d
dx ψ 2(x) = (2.2)
x → ± ∞ ψ 1 (x) = ψ 2 (x) = 0 = 0
ψ 2 (x) d
dx ψ 1(x) − ψ 1 (x) d
dx ψ 2(x) = 0 , d
dx
ψ 1 (x)
ψ 2 (x) = 0
ψ 1 (x) = × ψ 2 (x) ψ 1 ψ 2
x → ± ∞ 0 , ( V = 0 ), e ikx
e −ikx E = 2 k 2 /(2m) ,
2 1
(2.1) ψ(x) ψ(x) = ψ R (x) + iψ I (x) V (x)
(
− 2
2m
d 2
)
dx 2 + V (x) − E
ψ R (x) = 0 ,
(− 2
d 2
)
2m dx 2 + V (x) − E ψ I (x) = 0
, C ψ I (x) = C ψ R (x)
ψ(x) = (1 + iC)ψ R (x)
ψ R (x) 1 + iC ψ R (x)

2 1 23
, V (x) = V (−x) ψ(x)
(2.1) x −x
V (x) = V (−x)
− 2
d 2
ψ(−x) + V (−x)ψ(−x) = Eψ(−x)
2m dx2 − 2
d 2
ψ(−x) + V (x)ψ(−x) = Eψ(−x) (2.3)
2m dx2 ψ(−x) ψ(x) E , , c
ψ(−x) = c ψ(x)
ψ(x) = c ψ(−x) = c 2 ψ(x) c 2 = 1
, ψ(x)
ψ(−x) = ± ψ(x)
x = a V (x) , a − ε ≤ x ≤ a + ε
dψ ′ (x)
dx
= 2m ( )
2 V (x) − E ψ(x) ,
ψ ′ (x) = dψ(x)
dx
ψ ′ (a + ε) − ψ ′ (a − ε) =
∫ a+ε
a−ε
dx F (x) ,
F (x) = 2m ( )
2 V (x) − E ψ(x) (2.4)
V (x) x = a F (x) , a − ε ≤ x ≤ a + ε
F min ≤ F (x) ≤ F max
F min , F max
2εF min ≤ ψ ′ (a + ε) − ψ ′ (a − ε) ≤ 2εF max
ε → + 0
ψ ′ (a + 0) − ψ ′ (a − 0) = 0
, ψ ′ (x) x = a
∫ a+ε
a−ε
dx ψ ′ (x) = ψ(a + ε) − ψ(a − ε) ,
∫ a+ε
a−ε
dx ψ ′ (x) = 2εψ ′ (a) + · · ·
ε→0
−−−−→ 0
ψ(x) x = a
V 1 (x) V (x) = v 0 δ(x − a) + V 1 (x) v 0 δ(x − a)
⎧
⎨ v 0
V ε (x) = 2ε , |x − a| < ε
⎩ 0 , |x − a| > ε

2 1 24
ε → + 0 ψ(x) ψ ′ (x) (2.4)
ψ ′ (a + ε) − ψ ′ (a − ε) = 2mv 0
2
2 ε → + 0 0
ψ(a) + 2m
2
ψ ′ (x) x = a x ≠ a
∫ a+ε
a−ε
dx
( )
V 1 (x) − E ψ(x)
ψ ′ (a + 0) − ψ ′ (a − 0) = 2mv 0
2 ψ(a) (2.5)
− 2 d 2 ψ
2m dx 2 + V 1(x)ψ = E ψ
x > a x < a x = a , (2.5)
(2.5) ψ x = a ,
− 2 d 2 ψ
(
)
2m dx 2 + v 0 δ(x − a) + V 1 (x) ψ = E ψ
ψ(a) ≠ 0 d 2 ψ/dx 2 x = a ( ψ(a) = 0 δ(x − a)
V 1 (x) ) ψ(x) 1
ψ(x) =
{
ψ − (x) , x < a
ψ + (x) , x > a , ψ(x) = θ(a − x)ψ −(x) + θ(x − a)ψ + (x)
θ(x)
θ(x) =
{
1 , x > 0
0 , x < 0
dθ(x)/dx = δ(x) ψ + (a) = ψ − (a)
ψ ′ (x) = − δ(x − a)ψ − (x) + θ(a − x)ψ −(x) ′ + δ(x − a)ψ + (x) + θ(x − a)ψ +(x)
′
(
)
= δ(x − a) ψ + (a) − ψ − (a) + θ(a − x)ψ −(x) ′ + θ(x − a)ψ +(x)
′
= θ(a − x)ψ ′ −(x) + θ(x − a)ψ ′ +(x)
(
ψ ′′ (x) = δ(x − a)
)
ψ +(a) ′ − ψ −(a)
′ + θ(a − x)ψ −(x) ′′ + θ(x − a)ψ +(x)
′′
(
) )
δ(x − a)
(− 2
ψ +(a) ′ − ψ −(a)
′ + v 0 ψ(a)
2m
+ θ(a − x)
(− 2
2m
d 2
dx 2 + V 1(x) − E
)
ψ − + θ(x − a)
(− 2
2m
x = a δ(x − a) 0
(
)
− 2
ψ ′
2m
+(a) − ψ −(a)
′ + v 0 ψ(a) = 0
d 2
dx 2 + V 1(x) − E
)
ψ + = 0

2 1 25
(2.5)
(2.1) ψ ∗ (x)
∫
∫
∫
NE = − 2
dx ψ ∗ (x) d2 ψ(x)
2m
dx 2 + dx V (x) |ψ(x)| 2 , N = dx |ψ(x)| 2
1
∫
− 2
dx ψ ∗ (x) d2 ψ(x)
2m
dx 2
∫
∫
= 2
dx dψ∗ dψ(x)
2m dx dx
= 2
dx
2m
∣
dψ(x)
dx
∣
2
> 0
E > 1 ∫
dx V (x) |ψ(x)| 2
N
V (x) ≥ V min
E > 1 ∫
N
dx V (x) |ψ(x)| 2 ≥ V min
N
∫
dx |ψ(x)| 2 = V min
2.2
−a
O
a
a, V 0 V (x)
{
0 , |x| > a
V (x) =
− V 0 , |x| < a
(2.6)
|x| = a ,
|x| = a
, F (x) = − dV/dx x = − a F > 0, x = a
F < 0 , 0 , ,
,
−V 0
• ( , )
• ,
1
(− 2
2m
d 2 )
dx 2 + V (x)
ψ(x) = E ψ(x) (2.7)
V (x) (2.6) − V 0 < E < 0
√
√
α = − 2ma2 E
2ma2 (E + V 0 )
2 , β =
2 (2.8)
, (2.7)
|x| < a
|x| > a
d2 ψ(x)
dx 2
d2 ψ(x)
dx 2
= − β2
a 2 ψ(x)
= α2
a 2 ψ(x) (2.9)

2 1 26
V (x) = V (−x) , (2.9) x ≥ 0
x ≥ 0 (2.9)
A, B, C, D
⎧
⎨ A sin βx βx
+ B cos
ψ(x) = a a ,
⎩
Ce −αx/a + De αx/a ,
0 ≤ x < a
x > a
x → ∞ ψ(x) D = 0
ψ ′ (0) = 0 A = 0 x = a ψ(x), ψ ′ (x)
B cos β = Ce −α ,
βB sin β = αCe −α
2
,
α = β tan β (2.10)
α 2 = β 2 tan 2 β , tan β > 0
v 0 =
α 2 + β 2 = v 2 0 cos2 β = β 2 /v 2 0
√
2ma2 V 0
2 (2.11)
| cos β| = β v 0
, tan β > 0 (2.12)
C = Be α cos β
⎧
⎨ B cos βx
ψ(x) = a ,
⎩
B cos β e −α(x−a)/a ,
0 ≤ x < a
x > a
(2.13)
B ψ(x)
∫ a
(
1 +
0
dx cos 2 βx
a = a 2
cos 2 β
∫ ∞
a
)
sin 2β
= a (
1 + α )
2β 2 v0
2
dx e −2α(x−a)/a = a
2α cos2 β = a 2
β 2
αv 2 0
(2.14)
(2.15)
∫ ∞
−∞
(
dx |ψ(x)| 2 = a|B| 2 1 + α v0
2
) (
+ β2
αv0
2 = a|B| 2 1 + 1 )
√
1
= 1 , ∴ B =
α
a
α
1 + α
ψ(0) = 0 B = 0 x = a
A sin β = Ce −α ,
βA cos β = − αCe −α
α = − β cot β (2.16)

2 1 27
(2.12)
| sin β | = β v 0
, tan β < 0 (2.17)
⎧
⎨ A sin βx
ψ(x) = a ,
⎩
A sin β e −α(x−a)/a ,
0 ≤ x < a
x > a
√
1 α
, A =
a 1 + α
(2.18)
(2.12), (2.17) | cos β | | sin β | β/v 0 β
β/v 0
v 0 = 5 , (2.12), (2.17) 2 , ◦
β/v 0 = 1 α 2 = v 2 0 − β 2 = 0 , E = 0 ,
(n − 1)π
2
< v 0 ≤ nπ 2
, n = 1, 2, 3, · · ·
n n β β k ( k = 0, 1, 2, · · · )
, k β k (2.12) , (2.17)
1
β 0 β 1 π β 2 β 3 2π β
(2.12), (2.17) β k
E k = −
( )
2
2ma 2 α2 k =
2
2ma 2 βk 2 − v0
2
, v 0 E 0 < E 1 < · · · ,
E 0 , E 1 , E 2 , · · · 1 , 2
, · · · V 0 E k
− α 2 = 2ma 2 E/ 2
−10
−20
−30
−40
0 π 5 2π v 0

2 1 28
V 0 → ∞ E k → − ∞ , E k + V 0
2 , β/v 0 β , (2.12), (2.17) β k = (k + 1)π/2
E k + V 0 =
2
2ma 2 β2 k = 2
2m
( ) 2 (k + 1)π
, k = 0, 1, 2, · · · (2.19)
, α → ∞ , (2.13), (2.18) |x| > a ψ(x) = 0 ,
|x| < a
⎧
⎪⎨
ψ(x) =
⎪⎩
v ,
E E = mv 2 /2 + V (x)
m
2 v2 = E − V (x) ≥ 0
(2.6) , −V 0 < E < 0 ,
|x| < a ,
,
,
|x| > a ψ(x) 0
2a
1 (k + 1)πx
√ cos , k = 0, 2, 4, · · ·
a 2a
1 (k + 1)πx
√ sin , k = 1, 3, 5, · · ·
a 2a
a|ψ(x)| 2
0.8
0.6
0.4
0.2
v 0 = 3
v 0 = 1
v 0 = 0.5
(2.20)
0 a x
,
v 0 = 0.5, 1, 3 (2.13)
|x| > a P ψ(x) , (2.15)
P = 2
∫ ∞
a
dx |ψ(x)| 2 = |B| 2 aβ2
αv 2 0
P V 0
( ) 1 ( )
1
v 0
≤ π/2 |x| > a ψ(x) ∝
e −α|x|/a E ≈ 0, ( α ≈ 0 )
, x
, |x| > a v 0
E ( α ), P
v 0 → ∞ α → ∞ , β → (k + 1)π/2
P → 0 ,
= 1 β 2
1 + α v0
2 = 1 E + V 0
1 + α V 0
1.0
0.0
1 2 3 4 v 0
2.1
V (x) =
{
∞ ,
|x| > a
0 , |x| < a

2 1 29
( ) 2
1. E E > 2 1
2m 2a
2. (2.19) (2.20)
2.2
v 0 → 0
1. E k k = 0 1 E 0 = − 2
2ma 2 v4 0
2. V 0 = u 0 /(2a) a → 0 v 0 = √ mau 0 / 2 → 0
E 0 = −
2
2ma 2 v4 0 = − mu2 0
2 2 (2.21)
{ ∫ 0 , |x| > a
∞
V (x) =
− u 0 /(2a) , |x| < a , dx V (x) = − u 0
−∞
a → 0 V (x) = − u 0 δ(x)
(− 2 d 2 )
2m dx 2 − u 0δ(x) ψ(x) = Eψ(x)
, E (2.21) x = 0
(2.5)
2
E , 2
x → − ∞ ψ(x) → 0 , x → + ∞ ψ(x) → 0
ψ(x) , x → ∞ ψ(x) → 0 ,
x → − ∞ ψ(x) → 0 , ψ(x) , ,
x → − ∞ ψ(x) → 0 1 x → − ∞ ψ → 0
, (2.9) A, C, D, F , G
⎧
Ae αx/a ,
⎪⎨
ψ(x) = C sin βx βx
+ D cos
a a ,
⎪⎩
F e αx/a + Ge −αx/a ,
x = − a ψ, dψ/dx
x < − a
x < |a|
x > a
(2.22)
Ae −α = − C sin β + D cos β , αAe −α = β (C cos β + D sin β) (2.23)
x = a
( )
C α
A = e−α β cos β − sin β ,
( )
D α
A = e−α β sin β + cos β
(2.24)
C sin β + D cos β = F e α + Ge −α , β (C cos β − D sin β) = α ( F e α − Ge −α) (2.25)

2 1 30
F
((sin
A = e−α
β + β ) ( C
2
α cos β A + cos β − β ) ) D
α sin β A
(
= e −2α sin β + β ) ( )
α
α cos β β cos β − sin β
G
A = eα 2
(2.26)
((sin β − β ) ( C
α cos β A + cos β + β ) ) D
α sin β = α2 + β 2
cos β sin β (2.27)
A αβ
, − V 0 < E < 0 E x → − ∞ ψ(x) → 0
, E . F ≠ 0 x → ∞
ψ(x) F = 0 (2.26)
(sin β + β α cos β ) ( α
β cos β − sin β )
= 0
(2.10), (2.16) E ,
E
(2.10) (2.22) α/β = tan β (2.24)
( )
C α
A = e−α β cos β − sin β = e −α( )
tan β cos β − sin β = 0
α = β tan β (2.27)
, (2.22)
( )
D α
A = e−α β sin β + cos β = e −α( )
tan β sin β + cos β = e−α
cos β
G
A = tan2 β + 1
cos β sin β = 1
tan β
⎧
⎪⎨ Ae −α|x|/a , |x| > a
ψ(x) =
⎪⎩ A e−α βx
cos
cos β a , |x| < a (2.28)
ψ(x) = ψ(−x) x (2.13) x ≥ 0
, (2.16) , tan β = − β/α
C
)
A = e−α( − cot β cos β − sin β = − e−α
sin β
D
)
A = e−α( − cot β sin β + cos β = 0
G
A = − cot2 β + 1
cos β sin β = − 1
cot β
⎧
⎪⎨
ψ(x) =
⎪⎩
Ae αx/a ,
− A e−α βx
sin
sin β a ,
− Ae −αx/a ,
x < − a
|x| < a
x > a
(2.29)
x (2.18) x ≥ 0

2 1 31
(2.24), (2.26), (2.27) (2.22) , v 0 = 5
β = β 0 (2.28) β β 0
(2.22) β/β 0 β β 0
, x → ∞ , 1 1
1 , 2
ψ(x)
0.99 0.9999
ψ(x)
−a
O
a
x
−a
O
a
x
1.01 1.0001
2.3
V (x) v 0 , a
V (x) = − 2 v
(
)
0
δ(x − a) + δ(x + a)
2m
(2.30)
1. E
av 0 q = − log |q − 1| , q = 2 v 0
√
− 2mE
2 (2.31)
2. (2.31) q q
av 0 ≤ 1 1 , av 0 > 1 2 ,
3. a → 0 a → ∞ (2.21) u 0 = 2 v 0 /(2m)
2.3
, m mg ( x )
x = 0
(− 2 d 2 )
{
2m dx 2 + V (x) ∞ , x < 0
ψ(x) = E ψ(x) , V (x) =
mgx , x > 0
x ≤ 0 ψ(x) = 0, x → ∞ ψ(x) → 0 x > 0
(− d2
dq 2 + 2m2 g
2 α 3 q )
ψ(q) = 2mE
2 ψ(q) , q = αx
α2 (2.32)

2 1 32
α = ( 2m 2 g/ 2) 1/3
( d
2
dq 2 − (q − ε) )
ψ(q) = 0 ,
ε = 2mE
2 α 2 = E ( 2
mg 2 2 ) 1/3
(2.33)
x = q − ε (15.100) , (Airy)
ψ(q) = CAi(q − ε) + DBi(q − ε)
(15.103) q → ∞ Bi(q − ε) D = 0
, q = 0 ψ(0) = CAi(− ε) = 0 , 277
a n Ai(x) ε = − a n , E
( mg 2 2 ) 1/3 ( mg 2 2 ) 1/3
E n = ε
= − a n (2.34)
2
2
0 > a 1 > a 2 > · · · 0 < E 1 < E 2 < · · · (15.102) a n
2
3 |a n| 3/2 + π ≈ nπ , n = 1, 2, 3, · · · (2.35)
4
E n ≈
(
9π 2 mg 2 2) 1/3
2
(
n − 1 4) 2/3
(2.36)
ψ n (q) = CAi(q + a n ) Ai(q) − a n > 0 x
Ai(z) z > l , ( l ∼ 3 ) Ai 2 (z) ≈ 0 ψ(q) = CAi(q + a n )
, q ≥ 0 , q + a n l
q = αx =
( 2m 2 ) 1/3 ( ) 1/3
g
2
2 x , a n = −
mg 2 2 E n
0 ≤ x E ( )
n
2 1/3
mg + ∆ , ∆ = 2m 2 l
g
, E = mv 2 /2 + mgx , mv 2 /2 = E − mgx ≥ 0
0 ≤ x ≤ E/mg
() , , ∆
( )
∆
2 1/3
E n /mg = l
2m 2 g E n /mg =
, n E n /mg ∆ , n E n /mg ≫ ∆ ,
[ x , x + dx ] P cl (x) dx
dt
P cl (x) ∝ dt
dx = 1 v = √ m
2
l
− a n
(E − mgx)−1/2
P cl (x) = mg 1
√ ,
2E 1 − mgx/E
∫ E/mg
0
P cl (x) dx = 1 (2.37)

2 1 33
277
z = q + a n =
( 2m 2 ) 1/3 (
g
2 x − E )
n
−1 , x E (
n
1 + 1 )
mg
mg a n
ψ(q) = CAi(q + a n ) ≈ C sin ( 2
3 |q + a n| 3/2 + π/4 )
√ π |q + an | 1/4
,
C 2
π | q + a n | −1/2 = C2
π
1
√
− an (1 − mgx/E n )
n |ψ n | 2 , 1/ √ 1 − mgx/E n
, n
n = 2 n = 30 |ψ n | 2 P cl
5
|ψn| 2 × En/mg
2
1
n = 2
|ψn| 2 × En/mg
4
3
2
1
n = 30
0 5
q
0 5 10 15 20 25 30
q
mgx , ,
, (2.34) ( Ultra cold neutron )
• Nature 415 (2002) 297
http://www.nature.com/nature/journal/v415/n6869/full/415267a.html
http://www.nature.com/nature/journal/v415/n6869/full/415297a.html
• Physical Review D67 (2003) 102002
http://prola.aps.org/abstract/PRD/v67/i10/e102002
(2.36) E 1 ≈ 1.4 × 10 −12 eV ( 0.1 eV
)
, ,
(1.52) ,
V (x) (2.32) 3.4
2.4
V (x) = mg|x| , E
( mg 2 2 ) 1/3 [ ( 3π
E ≈
k + 1 2/3
, k = 0, 1, 2, · · ·
2 4 2)]
k = 2n − 1 (2.36)

2 1 34
2.4
d 2 ψ
dx 2 + 2m ( )
2 E − V (x) ψ(x) = 0
, E
ψ(x) λ ,
q = x λ ,
,
ε = 2mλ2 E
2 , U(q) = 2mλ2
2 V (x)
d 2 ψ(q)
( )
dq 2 + ε − U(q) ψ(q) = 0 (2.38)
, 2 3 ,
(2.38) ( 131 )
q min ≤ q ≤ q max ,
0 ε ,
ψ(q) , , 1 ε
, ,
1 ( , [] (
) 1992 )ε
ψ − (q, ε) : q = q min q
ψ + (q, ε) : q = q max q
ψ − (q, ε) ψ + (q, ε)
q = q c , ( q min < q c < q max ) ψ − (q c , ε) ≠ ψ + (q c , ε)
q = q c ,
φ − (q, ε) = ψ −(q, ε)
ψ − (q c , ε) , φ +(q, ε) = ψ +(q, ε)
ψ + (q c , ε)
φ − (q c , ε) = φ + (q c , ε) = 1 q = q c , ε
, φ ′ (q) q = q c
φ ′ −(q c , ε) − φ ′ +(q c , ε) =
W (ε)
ψ − (q c , ε)ψ + (q c , ε)
W (ε) = ψ ′ −(q c , ε) ψ + (q c , ε) − ψ − (q c , ε) ψ ′ +(q c , ε)
, W (ε) = 0 , (2.2) W (ε) q c
(2.38) Y 1 (q) = ψ(q) , Y 2 (q) = ψ ′ (q) 1 2
dY 1
dq = Y 2 ,
dY 2
dq = (
U(q) − ε ) Y 1

2 1 35
, ( Runge–Kutta ) , (2.38)
( Numerov ) q min ≤ q ≤ q max N q
q k = q min + ∆q k , k = 0, 1, · · · , N q , ∆q = q max − q min
N q
q = q k f(q) f k = f(q k )
f k±1 = f(q k ± ∆q) = f k ± ∆qf k ′ + (∆q)2 f ′′
2
q = q k 2
(2.38)
k ± (∆q)3
3!
f ′′′
k
+ (∆q)4 f k ′′′′ + · · ·
4!
f k+1 − 2f k + f k−1 = (∆q) 2 f ′′
k + (∆q)4
12 f ′′′′
k + O ( (∆q) 6) (2.39)
ψ ′′′′
k
f ′′
k = f k+1 − 2f k + f k−1
h 2 + O((∆q) 2 ) (2.40)
= d2 F (q)ψ(q)
dq 2 ∣
∣∣∣q=qk
, F (q) = U(q) − ε
, 2 f(q) = F (q)ψ(q) (2.40)
ψ ′′′′
k
= F k+1ψ k+1 − 2F k ψ k + F k−1 ψ k−1
(∆q) 2 + O((∆q) 2 )
ψ ′′
k = F kψ k (2.39)
ψ k+1 − 2ψ k + ψ k−1 = (∆q) 2 ψ ′′
k + (∆q)4
12 ψ′′′′ k + O((∆q) 6 )
= (∆q) 2 F k ψ k + (∆q)2
12
D q = (∆q) 2 /12
)
)
(1 − D q F k+1 ψ k+1 − 2
(1 + 5D q F k ψ k +
k k − 1
(F k+1 ψ k+1 − 2F k ψ k + F k−1 ψ k−1
)
+ O((∆q) 6 )
(1 − D q F k−1
)
ψ k−1 = O((∆q) 6 ) (2.41)
ψ k ≈ 2 (1 + 5D qF k−1 ) ψ k−1 − (1 − D q F k−2 ) ψ k−2
1 − D q F k
, k = 2, 3, · · · (2.42)
ψ 0 ψ 1 k = 2, 3, · · · ψ 2 , ψ 3 , · · · ψ −
, ψ Nq , ψ Nq−1
ψ k ≈ 2 (1 + 5D qF k+1 ) ψ k+1 − (1 − D q F k+2 ) ψ k+2
1 − D q F k
, k = N q − 2, N q − 3, · · · (2.43)
k ψ Nq−2, ψ Nq−3, · · · ψ +
, W (ε) ψ ′ (q) , (2.40)
ψ ′ k = ψ k+1 − ψ k−1
2∆q
+ O((∆q) 2 )
ψ ′ k = 8(ψ k+1 − ψ k−1 ) − ψ k+2 + ψ k−2
12∆q
+ O((∆q) 4 ) (2.44)

2 1 36
(2.44)
ψ ± (q) ,
ψ 0 = ψ − (q min ) , ψ 1 = ψ − (q min + ∆q) , ψ Nq −1 = ψ + (q max − ∆q) , ψ Nq = ψ + (q max )
• − ∞ < q < ∞ ψ(q) q→±∞
−−−−−→ 0
q min , q max ψ(q min ) ≈ 0, ψ(q max ) ≈ 0 ( ,
)
ψ 0 = ψ Nq = 0 , ψ 1 = ψ Nq −1 = ≠ 0 (2.45)
,
q → ± ∞ U(q) → 0 ψ ′′ (q) = − ε ψ(q)
ψ(q) → e ±√ −ε q
ψ 0 = e √ −ε q min
,
ψ 1 = e √ −ε (q min+∆q)
(2.46)
ψ Nq = e −√ −ε q max
, ψ Nq −1 = e −√ −ε (q max −∆q)
(2.47)
q c q c = 0
• 0 ≤ q < ∞ ψ(0) = 0 , ψ(q) q→∞
−−−−→ 0
q min = 0 (2.45) q = q max (2.47) x ≈ λ , q c ≈ 1
W (ε) , W (ε) = 0
ε 2 ε min < ε < ε max
, N e
ε i = ε min + i ∆ε , i = 0, 1, · · · , N e , ∆ε = ε max − ε min
N e
W (ε) , W (ε i )W (ε i+1 ) < 0 ε i < ε < ε i+1 W (ε) = 0
ε ε i ε i+1 (
)N e , ε min < ε < ε max
,
W (ε)
Y (ε) = φ ′ −q c , ε) − φ ′ +q c , ε) = ψ′ −(q c , ε)
ψ − (q c , ε) − ψ′ +(q c , ε)
ψ + (q c , ε) = 0
, q c ψ + (q c , ε c ) = 0 ψ − (q c , ε c ) = 0
ε = ε c Y (ε c ) ε ε c , ψ(q c , ε)
ψ ′ (q c , ε) , Y (ε) ,
, ,

2 1 37
, q c = 0 Y (ε) = 0
, q c
C
•
qmin = q min , qmax = q max , dq = ∆q , nq = N q , nc = q c − q min
∆q
wf[k] = ψ k , pot[k] = U(q k )
main , W (ε)
wf pot N q +1 U(q k )
, nc
k = nc q k q c
• double wronski( double ee )
ε W (ε)
(2.45) for (2.42)
wf[k] = ψ − k ψ − ′ (2.44) k = nc + 2
for (2.43) wf[k] = ψ + k for
ψ − ψ + k ≤ nc − 1 wf[k] = ψ − k , k ≥ nc
wf[k] = ψ + k , k ≤ nc − 1
wf[k] ⇐ C wf[k] ,
• double get_eigen( double e1, double e2 )
C = ψ + nc
= wf[nc]
ψ − nc wf1
2 ε = e1, e2 , W (ε) = 0
|1 − e1/e2| < = 10 −6
#include
#include
double potential( double q );
double wronski( double ee );
double get_eigen( double e1, double e2 );
double wf[ ... ], pot[ ... ], qmin, qmax, dq;
int nc, nq;
int main()
{
double emin, emax, de, ee, w1, w2, qc;
int i, ne;
qmin = ... ; qmax = ... ; qc = ... ; dq = ... ;
nq=(qmax-qmin)/dq;
nc=(qc-qmin)/dq;
qmax=qmin+nq*dq; qc=qmin+nc*dq;
for(i=0; i

2 1 38
}
ee=emin+de*i;
w2=wronski( ee );
if( w1*w2

2 1 39
}
if( w1*w3 > 0 ){
w1=w3; e1=e3;
} else {
w2=w3; e2=e3;
}
}
return e1;
2.5 ( U 0 )
1. U(q) = q 2 2. U(q) = U 0
(
e −2q − 2e −q) 3. U(q) = −
U 0
(cosh q) 2
, ε , q min , q max
q cl , ( U(q cl ) − ε = 0 ) 2 , U 0 ≈ 20 , q max , − q min ≈ 4 ,
∆q ≈ 0.1 ψ k = wf[k]
S =
∫ qmax
q min
( ψ
2
ψ 2 0 + ψN 2
(q) dq ≈ ∆q
q
2
ψ k / √ S ψ k
N q −1
∑
+ ψk
2
k=1
(2.38) n
1. ε n = 2n + 1 , n = 0, 1, 2, · · ·
(
2. ε n = − n + 1 2 − √ ) 2
U 0 , 0 ≤ n < √ U 0 − 1 2
3. ε n = −
(n + 1 2 − √
U 0 + 1 4
) 2
, 0 ≤ n <
√
)
U 0 + 1 4 − 1 2
1. 2. ( Morse )
2. , 3. 1 () 87
ε
ε 0 = − 15.7779 , ε 1 = − 8.8336
ε 2 = − 3.8893 , ε 3 = − 0.9450
n = 2 , 3 q max
,
ε = − 14.8, ε = − 13.8
q c = 0 ψ ′
, ε ε 0
1
0
−15.7779
−8.8336 −3.8839 −0.5924
U 0 = 20
q min = −1.6
q max = 4.0
∆q = 0.05
0 2 4
q

3 1 40
3 1
3.1
1 V (x) ,
,
, ,
(− 2 d 2 )
2m dx 2 + V (x) ψ(x) = E ψ(x)
, ψ(x) x→±∞
−−−−−→ 0 ,
, (1.4) ()
(1.5) 1
ρ(x, t) = |ψ(x, t)| 2
j(x, t) =
2im
∂ρ(x, t)
∂t
+
∂j(x, t)
∂x
= 0 (3.1)
(
)
ψ ∗ ∂ψ(x, t)
(x, t) − ψ(x, t) ∂ψ∗ (x, t)
= (
∂x
∂x m Im ψ ∗ (x, t)
D = [x 1 , x 2 ]
∫
d
x2
dt P 12(t) = j(x 1 , t) − j(x 2 , t) , P 12 (t) = dx ρ(x, t)
x 1
)
∂ψ(x, t)
∂x
D P 12 , D x = x 1 D j(x 1 , t)
x = x 2 D j(x 2 , t) , j(x, t)
x ψ(x, t) = e −iEt/ ψ(x) , ρ(x, t), j(x, t)
x → − ∞ x x → − ∞
ψ(x)
ψ(x) = +
x → + ∞
ψ(x) = x
x → ± ∞ V (x)
V (x) x→−∞
−−−−−→ V − ,
V (x) x→+∞
−−−−→ V +
, E V − = 0
x → − ∞ ,
− 2 d 2 ψ
2m dx 2 = Eψ

3 1 41
E < 0 κ = √ −2mE/ 2 ψ ′′ = κ 2 ψ A, B
ψ(x) = Ae κx + Be −κx
x → − ∞ ψ B = 0 ψ(x) = Ae κx → 0 ,
, E > 0
√
2mE
ψ(x) = Ae ikx + Be −ikx , k =
ψ(x) ψ I (x) ψ R (x)
2
ψ I (x) x→−∞
−−−−−→ Ae ikx ,
ψ R (x) x→−∞
−−−−−→ Be −ikx
, x → − ∞
J I = (
m Im ψI ∗ (x) dψ )∣
I(x) ∣∣∣x=−∞
, J R = (
dx
m Im ψR(x) ∗ dψ )∣
R(x) ∣∣∣x=−∞
dx
J I = k m |A|2 ,
J R = − k m |B|2
, ψ I |ψ I (x)| 2
x→−∞
−−−−−→ |A| 2 , ψ I (x)
|A| 2 , k/m x ,
ψ R (x → −∞) k/m x ψ I (x) , ψ R (x)
E > 0
x → ∞
− 2 d 2 ψ
2m dx 2 = (E − V +) ψ
E < V + ( E > 0 V + > 0 )
√
ψ(x) = Ce −κ′x , κ ′ 2m(V+ − E)
=
x → ∞ ψ(x) → 0 x → ∞ , E > V +
ψ(x) = Ce ik′x + De −ik′x , k ′ =
2
√
2m(E − V+ )
x → ∞ x
D = 0 ,
⎧
Ae ⎪⎨
ikx + Be −ikx , x → − ∞
ψ(x) → Ce −κ′x , x → ∞ , E < V + , ( V + > 0 )
(3.2)
⎪⎩
Ce ik′x , x → ∞ , E > V +
E ,
E > V − = 0 E
(3.2) |ψ(x)| x → − ∞ 0
∫ ∞
−∞
dx |ψ(x)| 2
2

3 1 42
, ,
,
x → ∞
J T = (
m Im ψ ∗ (x) dψ(x) )∣
∣∣∣x=∞
dx
, R T
∣ R =
J R ∣∣∣
∣ , T =
J I
1 ,
∣
∣
J T ∣∣∣
J I
, (3.2)
⎧
⎪⎨ 0 , E < V + , ( V + > 0 )
J T = k ⎪⎩
′
m |C|2 , E > V +
R =
B
∣ A ∣
2
⎧
⎪⎨ 0 , E < V +
, T =
k ⎪⎩
′ 2
C
k ∣ A ∣ , E > V +
A, B, C ,
ψ(x, t) = e −iEt/ ψ(x) , ρ(x, t) j(x, t) , (3.1)
dj(x)/dx = 0 j(x) = x → −∞
j(x) = m Im [ (Ae ikx + Be −ikx) ∗
d (
Ae ikx + Be −ikx)]
dx
= ( [ik
m Im |A| 2 − |B| 2) − 2k Im ( AB ∗ e 2ikx)] = k m
(
|A| 2 − |B| 2) (3.3)
, x → ∞ j(x) = J T
0 < E < V + |A| 2 − |B| 2 = 0 , ∴ R = 1 , T = 0
(
V + < E k |A| 2 − |B| 2) = k ′ |C| 2 , ∴ R + T = 1
R + T = 1
3.2
V (x) V 0
⎧
⎨ 0 , x < 0
V (x) =
⎩ V 0 , x > 0
(3.4)
V (x)
V 0
E E > 0
x < 0
O
x
ψ(x) = Ae ikx + Be −ikx , k =
√
2mE
2 (3.5)

3 1 43
x > 0
E V 0
E > V 0
(3.6) (3.2)
− 2 d 2 ψ
2m dx 2 = (E − V 0) ψ (3.6)
ψ(x) = Ce ik′x , k ′ =
x = 0 ψ(x) ψ ′ (x) (3.5), (3.7)
√
2m(E − V0 )
2 (3.7)
A + B = C ,
k(A − B) = k ′ C
R T
R =
B
∣ A ∣
T = k′
k
2
B
A = k − k′
k + k ′ ,
=
C
A = 2k
k + k ′ (3.8)
( ) k − k
′ 2
(√ √ ) 2
E − E − V0
k + k ′ = √ √ (3.9)
E + E − V0
2 C
∣ A ∣ = 4kk′
(k + k ′ ) 2 = 4 √ E(E − V 0 )
( √E √ ) 2
(3.10)
+ E − V0
R + T = 1
E > V 0 , ,
V 0 > 0 z = E/V 0 − 1 > 0
R =
(√ z + 1 −
√ z
√ z + 1 +
√ z
) 2
= (√ z + 1 − √ z ) 4
=
⎧
⎨
⎩
1 − 4 √ z + · · · , z ≈ 0
1
16z 2 + · · · , z ≫ 1
E ≈ V 0 R ≈ 1 , , E ≫ V 0 R ≈ 0
E < V 0
(3.6) (3.2)
ψ(x) = Ce −κ′x , κ ′ =
x = 0 A + B = C , ik (A − B) = − κ ′ C
√
2m(V0 − E)
2 (3.11)
B
A = k − iκ′
k + iκ ′ ,
C
A = 2k
∣ ∣ ∣∣∣
k + iκ ′ , R = k − iκ ′ ∣∣∣
2
k + iκ ′ = 1 (3.12)
x > 0 j(x) = 0 T = 0 R +T = 1 , x < 0
j(x) |A| = |B| (3.3) j(x) = 0 , (3.11) (3.12)
(3.7), (3.8)
k ′ =
√
− 2m(V 0 − E)
2 = iκ ′

3 1 44
, ,
, E − V (x) ≥ 0 x , E < V 0 x > 0
, x > 0
ψ(x) = Ce −κ′x = A
2k
k + iκ ′ e−κ′x ≠ 0 (3.13)
, x > 0
V 0 < 0 E > 0 > V 0 , R T (3.9), (3.10)
, T = 1, R = 0 T (E)
1.0
0.8
T
0.6
V 0 < 0
V 0 > 0
0.4
0.2
0 1 2
E/|V 0 |
3.3
V 0 , a
⎧
⎨ 0 , |x| > a
V (x) =
⎩ V 0 , |x| < a
V (x)
V 0
|x| > a V = 0
(3.2)
⎧
⎨ Ae ikx + Be −ikx ,
ψ(x) =
⎩ Ce ikx ,
x < − a
x > a
−a
O
a
x
E > V 0
|x| < a
√
d 2 ψ
2m(E −
dx 2 = − k′2 ψ(x) , k ′ V0 )
=
2
F , G
⎧
Ae
⎪⎨
ikx + Be −ikx , x < − a
ψ(x) = F e ik′x + Ge −ik′x , |x| < a
⎪⎩
Ce ikx ,
x > a
(3.14)

3 1 45
x = − a x = a
Ae −ika + Be ika = F e −ik′a + Ge ik′a , k ( Ae −ika − Be ika) ( ) = k ′ F e −ik′a − Ge ik′ a
(3.15)
F e ik′a + Ge −ik′a = Ce ika , ( ) k ′ F e ik′a − Ge −ik′ a
= kCe ika (3.16)
(3.16)
(3.15)
F
A = k′ + k
2k ′ e i(k−k′ )a C A , G
A = k′ − k
2k ′ e i(k+k′ )a C A
(3.17)
R =
B
∣ A ∣
T =
C
∣ A ∣
R + T = 1
2
2
B
A = i ( k ′2 − k 2) sin(2ak ′ )
2kk ′ cos(2ak ′ ) − i (k ′2 + k 2 ) sin(2ak ′ ) e−2iak (3.18)
C
A = 2kk ′
2kk ′ cos(2ak ′ ) − i (k ′2 + k 2 ) sin(2ak ′ ) e−2iak (3.19)
(
k ′2 − k 2) 2
sin 2 (2ak ′ )
=
4k 2 k ′2 cos 2 (2ak ′ ) + (k ′2 + k 2 ) 2 sin 2 (2ak ′ )
(
k ′2 − k 2) 2
sin 2 (2ak ′ )
=
4k 2 k ′2 + (k ′2 − k 2 ) 2 sin 2 (2ak ′ ) = V0 2 sin 2 (2ak ′ )
4E(E − V 0 ) + V0 2 sin2 (2ak ′ )
=
4k 2 k ′2
4k 2 k ′2 + (k ′2 − k 2 ) 2 sin 2 (2ak ′ ) = 4E(E − V 0 )
4E(E − V 0 ) + V0 2 sin2 (2ak ′ )
ε = (2a) 2 2mE
2 , v 0 = (2a) 2 2mV 0
2
R =
v 2 0 sin 2 ( √ ε − v 0 )
4ε(ε − v 0 ) + v 2 0 sin2 ( √ ε − v 0 ) , T = 4ε(ε − v 0 )
4ε(ε − v 0 ) + v 2 0 sin2 ( √ ε − v 0 )
(3.20)
, E > V 0 E ≈ V 0 , ( ε ≈ v 0 )
sin 2 ( √ ε − v 0 ) ≈ ε − v 0
v 2
R ≈
0(ε − v 0 )
4ε(ε − v 0 ) + v0 2(ε − v 0) = v2 0
4ε
, T ≈
+ 4ε v0 2 + 4ε (3.21)
E V 0 T
sin( √ ε − v 0 ) = 0 , √ ε − v 0 = 2ak ′ = nπ , n = 1, 2, 3, · · · (3.22)
E T = 1 ( n = 0 T 0/0
, (3.21) ε = v 0 ) |x| < a
ψ(x) = F e inπx/2a + Ge −inπx/2a
v 2 0

3 1 46
ψ(a) = (−1) n ψ(− a) , x < − a x > a
(−1) n ,
T = 1 ε − v 0 = (nπ) 2 ε − v 0 = (nπ) 2 + x
T =
v 2 0 sin 2 ( √ ε − v 0 )
4ε(ε − v 0 )
=
v 2 0(x/2nπ) 2
4(v 0 + (nπ) 2 )(nπ) 2 + O(x3 )
(
1 + v2 0 sin 2 ( √ ) −1
ε − v 0 )
1
≈
4ε(ε − v 0 ) 1 + x 2 /Γ 2 , Γ = (2nπ)2 √
v0 + (nπ)
v 2 (3.23)
0
, Γ , v 0 , T x = 0 Γ
0 ≤ sin 2 ( √ ε − v 0 ) ≤ 1
T min ≤ T ≤ 1 , T min = 4ε(ε − v 0)
4ε(ε − v 0 ) + v 2 0
( ) 2 v0
= 1 −
2ε − v 0
T 1 T min v 0 T min (ε)
E ≫ V 0 T ≈ 1
0 < E < V 0
|x| < a
√
d 2 ψ
dx 2 = κ′2 ψ(x) , κ ′ 2m(V0 − E)
=
2
E > V 0 k ′ iκ ′ − iκ ′ k ′ → iκ ′
|x| < a
ψ(x) = F e −κ′x + Ge κ′ x
cos(2ak ′ ) → cos(2iaκ ′ ) = cosh(2aκ ′ ) , sin(2ak ′ ) → sin(2iaκ ′ ) = i sinh(2aκ ′ )
(3.18), (3.19)
(3.17)
(
B
κ ′2
A = + k 2) sinh(2aκ ′ )
2ikκ ′ cosh(2aκ ′ ) − (κ ′2 − k 2 ) sinh(2aκ ′ ) e−2iak
C
A = 2ikκ ′
2ikκ ′ cosh(2aκ ′ ) − (κ ′2 − k 2 ) sinh(2aκ ′ ) e−2iak
(3.24)
R =
B
∣ A ∣
T =
C
∣ A ∣
2
2
=
=
F
A = κ′ − ik
2κ ′ e (ik+κ′ )a C A , G
A = κ′ + ik
2κ ′ e (ik−κ′ )a C A
(κ ′2 + k 2 ) 2 sinh 2 (2aκ ′ )
4κ ′2 k 2 + (κ ′2 + k 2 ) 2 sinh 2 (2aκ ′ ) = v0 2 sinh 2 ( √ v 0 − ε )
4ε(v 0 − ε) + v0 2 sinh2 ( √ v 0 − ε )
4κ ′2 k 2
4κ ′2 k 2 + (κ ′2 + k 2 ) 2 sinh 2 (2aκ ′ ) = 4ε(v 0 − ε)
4ε(v 0 − ε) + v 2 0 sinh2 ( √ v 0 − ε )
(3.25)
(3.26)
(3.27)

3 1 47
E < V 0 , |x| < a
, x > a , , E
x > a ,
ε ≈ v 0 , (3.21) ε = v 0
T = 4
1
=
4 + v 0 1 + 2ma 2 V 0 / 2
, , x ≫ 1
sinh x = (e x − e −x )/2 ≈ e x /2 , v 0 − ε ≫ 1
T ≈
4ε(v 0 − ε)
4ε(v 0 − ε) + v0 2 exp (2√ v 0 − ε ) /4 ≈ 16 ε(v 0 − ε)
v0
2
exp ( −2 √ v 0 − ε )
v 0 = 20 (3.20), (3.27) T ε
T min
⎧
⎨ 0 , 0 < E < V 0
T =
⎩ 1 , E > V 0
, T = 1 (3.22) ε ε = 29.87, 59.48, 108.83, · · ·
1.0
0.8
T
0.6
0.4
0.2
0 20 40 60 80 100
ε
T 1 T min , v 0 ε − v 0
v 0 = 2000 T (3.23) , ε = v 0 + (nπ) 2 ,
T ε ≈ v 0 + (nπ) 2 ,
1.0
0.8
T
0.6
0.4
0.2
2000 2020 2040 2060 2080 2100
ε

3 1 48
3.1
(3.3) , (3.14)
)
k
(|A|
⎧⎪ 2 − |B| 2 , x < −a
j(x) = ⎨
m × ( )
k ′ |F | 2 − |G| 2 , |x| < a
⎪ ⎩
k|C| 2 ,
x > a
0 < E < V 0 , |x| < a
2
m κ′ Im (F ∗ G)
, j(x)
3.4
⎧
⎨ 0 , |x| > a
V (x) =
⎩ − V 0 , |x| < a , a
−a
V (x)
−V 0
a
x
V 0
, (3.14) ∼ (3.19)
k =
√
2mE
2 , k ′ =
√
2m(E + V0 )
2
k ′ (3.20) v 0 → − v 0
R =
v0 2 sin 2 ( √ ε + v 0 )
4ε(ε + v 0 ) + v0 2 sin2 ( √ ε + v 0 ) , T = 4ε(ε + v 0 )
4ε(ε + v 0 ) + v0 2 sin2 ( √ ε + v 0 )
(3.28)
n
√ ε + v0 = 2ak ′ = nπ , nπ > √ v 0
ε T = 1
T min ≤ T ≤ 1 ,
( ) 2 v0
T min = 1 −
2ε + v 0
v 0 = 20 T (ε) T min
1.0
0.8
T
0.6
0.4
0.2
0 20 40 60 80 100
ε

3 1 49
, (3.14) ∼
(3.19) E > 0 , − V 0 < E < 0 k = √ 2mE/ 2
, E < 0 κ = √ − 2mE/ 2 k iκ
, (3.14)
B/A, C/A
⎧
Ae ⎪⎨
−κx + Be κx ,
ψ(x) = F e ik′x + Ge −ik′x ,
⎪⎩
Ce −κx ,
x < − a
|x| < a
x > a
(
B
k ′2
A = + κ 2) sin(2ak ′ )
2κk ′ cos(2ak ′ ) − (k ′2 − κ 2 ) sin(2ak ′ ) e2aκ
C
A = 2κk ′
2κk ′ cos(2ak ′ ) − (k ′2 − κ 2 ) sin(2ak ′ ) e2aκ
F
A = k′ + iκ
2k ′ e −(ik′ +κ)a C A , G
A = k′ − iκ
2k ′ e (ik′ −κ)a C ( F
A = A
f = 2κk ′ cos(2ak ′ ) − ( k ′2 − κ 2) sin(2ak ′ )
(2.8) α, β κ = aα , k ′ = aβ
f = 2a 2( αβ ( cos 2 β − sin 2 β ) − ( β 2 − α 2) )
sin β cos β
= 2a 2( )(
)
α cos β − β sin β α sin β + β cos β
) ∗
(3.29)
f = 0 E (2.10), (2.16)
E (3.29) x → ± ∞ ψ(x) → 0
A = 0, B/A → ∞
, B/A E ,
,
3.2
(2.30) , R T
R =
B
∣ A ∣
2
= 1 − T , T =
C
∣ A ∣
2
=
k 2 + v0
2
k 2
(
cos 2ka − v 0
2k sin 2ka ) 2
, B/A, C/A E (2.31)
3.5
, ,
∂ψ(x, t)
i =
(− 2
∂t 2m
∂ 2 )
∂x 2 + V (x)
ψ(x, t) (3.30)

3 1 50
, ,
,
V (x) = 0
A exp(ikx − iω k t) + B exp(−ikx − iω k t) ,
ω k = k2
2m
(3.31)
(3.30) k > 0 B/A k
,
B
A = √ R k e iθ k
R k , E < V 0 , (3.12)
−1 κ′
R k = 1 , θ k = − 2 tan
k , κ′ =
(3.31) A (3.30) ,
ψ(x, t) = ψ I (x, t) + ψ R (x, t)
(3.30)
ψ I (x, t) =
ψ R (x, t) =
∫ ∞
0
∫ ∞
0
dk A(k) exp(ikx − iω k t)
dk B(k) exp(−ikx − iω k t) =
∫ ∞
0
√
2mV0
2 − k 2 (3.32)
dk √ R k A(k) exp(iθ k − ikx − iω k t)
ψ I (x, t) , ψ R (x, t) A(k) k = k 0 ,
k = k 0 , A(k) k = k 0 k − k 0
2
ω k ≈ ω 0 + (k − k 0 )v 0 , ω 0 = ω k0 , v 0 = dω k
dk
kx − ω k t ≈ k(x − v 0 t) + (k 0 v 0 − ω 0 )t
0
∣ = k 0
k=k0
m
(
) ∫ ∞
( )
ψ I (x, t) ≈ exp i(k 0 v 0 − ω 0 )t dk A(k) exp ik(x − v 0 t)
(
)
= exp i(k 0 v 0 − ω 0 )t ϕ(x − v 0 t) (3.33)
ϕ(x) = ψ I (x, 0 ) =
∫ ∞
0
dk A(k) e ikx
θ k ≈ θ 0 + (k − k 0 ) γ , θ 0 = θ k0 , γ = dθ k
dk
∣
∣
k=k0
(
ψ R (x, t) ≈ exp i(θ 0 − γk 0 ) + i(v 0 k 0 − ω 0 )t) ∫ dk √ (
)
R k A(k) exp ik(−x − v 0 t + γ)
(
√
≈ exp i(θ 0 − γk 0 ) + i(v 0 k 0 − ω 0 )t)
R0 ϕ(− x − v 0 t + γ) (3.34)

3 1 51
√ R k k A(k) R k ≈ R 0 = R k0
(3.33) v 0 x , (3.34)
x = −v 0 t + γ , γ (3.32)
γ = dθ k
dk ∣ = 2 √
k=k0
κ ′ , κ ′ 2mV0
0 =
0
2 − k0
2
E 0 < V 0 x > 0 (3.13) x > 0
e −κ′x , e −κ′x ∼ e −1 , x r ∼ 1/κ ′
x ∼ x r γ ∼ 2x r
|ψ(x, t)| 2 = |ψ I (x, t) + ψ R (x, t)| 2
≈ |ϕ(x − v 0 t)| 2 + R 0 |ϕ(−x − v 0 t + γ)| 2
+ 2 √ R 0 Re
(ϕ )
∗ (x − v 0 t) ϕ(−x − v 0 t + γ) e i(θ 0−γk 0 )
, 3 ,
, f(x) ϕ(x) = e ik0x f(x)
(
)
Re ϕ ∗ (x − v 0 t) ϕ(−x − v 0 t + γ) e i(θ 0−γk 0 )
= f(x − v 0 t) f(−x − v 0 t + γ) cos (2k 0 x − θ 0 )
, cos (2k 0 x − θ 0 ) π/k 0 f(x) ,
|ψ(x, t)| 2 π/k 0
, (3.30) ,
, |ψ(x, t)| 2
• ,
,
, (3.33), (3.34) |ϕ(x)|
, (1.50)
, (k − k 0 ) 2

3 1 52
3.6
t = 0 ψ(x, 0) ,
t = t max ψ(x, t) ( , ( ) 7.5 )x
x min ≤ x ≤ x max , ψ(x min , t) = ψ(x max , t) = 0
x min , x max ψ(x, t) 0 λ
(3.30)
q = x λ , τ =
2mλ 2 t
i ∂ ψ(q, τ) = Hψ(q, τ) ,
∂τ H
= − ∂2
∂q 2 + U(q) ,
2mλ2
U(q) =
2 V (x)
, = 1 , 2m = 1
∆τ = τ max /N τ , τ = τ n = n∆τ, ( n = 0, 1, · · · , N τ )
ψ (n) (q) = ψ(q, τ n ) ψ (n−1) , (1.42) τ = τ n
ψ (n) (q) = exp(−iH∆τ ) ψ (n−1) (q)
∆τ exp(−iH∆τ ) ≈ 1 − iH∆τ H
ϕ α (q)
ψ (0) (q) = ∑ α
c α (0) ϕ α (q) ,
Hϕ α (q) = E α ϕ α (q)
(1 − iH∆τ) n ϕ α (q) = (1 − iE α ∆τ) n ϕ α (q)
ψ (n) (q) ≈ (1 − iH∆τ) ψ (n−1) (q) ≈ · · · ≈ (1 − iH∆τ) n ψ (0) (q)
= ∑ α
c α (0) (1 − iE α ∆τ) n ϕ α (q) (3.35)
c α (t) = c α (0) e −iEατ
(3.35)
,
(
| c α (0) (1 − iE α ∆τ) n | = |c α (0)| 1 + (E α ∆τ) 2) n/2
, 1 + (Eα ∆τ) 2 > 1
, n ∆τ
, 1
exp(−iH∆τ ) ≈ 1 − iH∆τ/2
1 + iH∆τ/2 = 2
1 + iH∆τ/2 − 1 (3.36)
∆τ 2 exp(−iH∆τ) , 1 − iH∆τ
ψ (n) (q) ≈
( ) n 1 − iH∆τ/2
ψ (0) (q) = ∑ 1 + iH∆τ/2
α
c α (0)
( ) n 1 − iEα ∆τ/2
ϕ α (q)
1 + iE α ∆τ/2
∣ ∣∣∣ 1 − iE α ∆τ/2
1 + iE α ∆τ/2∣ = 1
,

3 1 53
(3.36)
(
)
ψ (n) 2
(q) ≈
1 + iH∆τ/2 − 1 ψ (n−1) (q) = φ(q) − ψ (n−1) (q)
φ(q) =
2
1 + iH∆τ/2 ψ(n−1) (q)
∆q = (q max − q min )/N q
(
1 + i∆τ
2 H )
φ(q) = 2ψ (n−1) (q) (3.37)
q = q k = q min + k∆q ,
k = 0, 1, · · · , N q
k q = q k
(
)
f k+1 + f k−1 − 2f k = (∆q) 2 f ′′ (q k ) + (∆q)2
12 f (4) (q k ) + · · ·
2
f ′′ (q k ) ≈ f k+1 + f k−1 − 2f k
(∆q) 2 , k = 1, 2, · · · , N q − 1
Hφ k = − φ ′′
k + U k φ k ≈ − φ k+1 + φ k−1 − 2φ k
(∆q) 2 + U k φ k
(3.37) k = 1, 2, · · · , N q − 1
φ k+1 − (a k − ib) φ k + φ k−1 = 2ib ψ (n−1)
k
, a k = 2 + (∆q) 2 U k , b = 2(∆q)2
∆τ
(3.38)
φ 0 , φ 1 φ k , ψ(q min , τ) = ψ(q max , τ) = 0 ,
φ 0 = φ Nx = 0 φ k
φ k+1 = C k φ k + D k (3.39)
(3.38)
1
φ k =
φ k−1 + D k − 2ib ψ (n−1)
k
a k − ib − C k a k − ib − C k
φ k = C k−1 φ k−1 + D k−1
C k−1 =
1
a k − ib − C k
,
D k−1 = D k − 2ib ψ (n−1) (
)
k
= C k−1 D k − 2ib ψ (n−1)
k
a k − ib − C k
(3.40)
, τ = τ n−1 ψ (n−1) τ = τ n ψ (n)
1. φ Nq = C Nq−1φ Nq−1 + D Nq−1 = 0 C Nq−1 = D Nq−1 = 0
(3.40) k = N q − 1 k = 1 k , ψ (n−1)
k
C k , D k
2. C k , D k , φ 0 = 0 (3.39) k = 0 k = N q − 1 ,
k φ k ψ (n) ψ (n)
k
= φ k − ψ (n−1)
k
C

3 1 54
•
qmin = q min , qmax = q max , dq = ∆q , dt = ∆τ , nq = N q , nt = N τ
a k k C
, ψ(q k , τ), C k , D k , φ k ... r , ... i
, GCC
• C k a k , a k
U(q) , U(q) ( pot(q) )
pot main
• ψ (n−1)
k
ψ (n)
k
, C k
D k D k ψ (n−1)
k
, φ 0 = 0
(3.39) φ k
int main()
{
/* , */
for( k=0; k=1; k-- ){
cr[k-1] = 1/( aa[k]-i*b-cr[k]-i*ci[k] ) ;
ci[k-1] = 1/( aa[k]-i*b-cr[k]-i*ci[k] ) ;
}
for( n=1; n=1; k-- ){
dr[k-1] = ..... ;
di[k-1] = ..... ;
}
phir[0] = phii[0] = 0; /* φ k */
for( k=0; k

3 1 55
}
return 0;
double pot( double q )
{
return 0; /* */
}
|ψ(q, τ)| 2 q
, , ,
q |ψ(q, τ)| 2 , , GNUPLOT
, PostScript TEX tpic
http://physics.s.chiba-u.ac.jp/~kurasawa/index.html
-->
51 , ψ(q, 0) ,
ψ(q, 0) =
(
1
(2πα 2 ) exp − (q − q 0) 2 )
{
1/4 4α 2 + ik 0 q , U(q) =
0 , q < 0
U 0 , q > 0
(3.41)
q min = − 110, q max = 100, τ max = 100, N τ = 1000, N q = 1500
q 0 = − 50, α = 8, k 0 = 0.6, U 0 = 0.38
Pentium4 ( 3.4GHz ) 0.1
3.3
1. , , U(q) = 0 (1.53)
(1.53)
(
|ψ(q, τ)| 2 1
= √
2πα2 (1 + τ 2 /α 4 ) exp − (q − q 0 − 2k 0 τ) 2 )
2α 2 (1 + τ 2 /α 4 )
2. 51
3. () , , () 121 ∼ 124
3.4
{
U0 q , q > 0
U(q) =
∞ , q < 0
ψ(q, τ) ψ(0, τ) = 0 q min = 0
q 0 τ = τ 0 q = 0 , τ 0 = √ q 0 /U 0
q 0 = τ 0 = 50 U 0 = 1/50 U 0
(3.41) k 0 = 0 q 0 = 50, U 0 = 0.025
, q = q 0 F = − U 0 q = 0

3 1 56
q cl (t) • × 〈 q 〉
〈 q 〉 =
∫ ∞
0
dq q|ψ(q, t)| 2
(1.11) F = 〈 q 〉 = q cl (t) ,
, ,
〈 q 〉 ≈ q cl (t) ,
〈 q 〉 ,
, |ψ(q, τ)| 2

4 57
4
x − kx
ω = √ k/m mω 2 x 2 /2 , H
Ĥ
H = p2
2m + 1 2 mω2 x 2
Ĥ = ˆp2
2m + 1 2 mω2 x 2 ,
ˆp = − i d
dx
,
, ,
, ˆ q
√ mω
q = α x , α =
H = − 2 d 2
2m dx 2 + 1 2 mω2 x 2 = ω 2
, H ϕ(x) = E ϕ(x)
(− d2
dq 2 + q2 )
d 2 ϕ
dq 2 + ( ε − q 2) ϕ(q) = 0 , ε = 2E
ω
(4.1)
4.1
(4.1)
q → ± ∞ ϕ(q)
) (
(− d2
dq 2 + q2 − ε q n e aq2 = 1 − 4a 2 + ε − 2a
)
n(n − 1)
q 2 −
q 4 q n+2 e aq2 ≈ (1 − 4a 2 )q n+2 e aq2
a = ±1/2 q n e ±q2 /2 q → ± ∞
(4.1)
ϕ(q) = h(q) e −q2 /2
d 2 h dh
− 2q + (ε − 1) h = 0 (4.2)
dq2 dq
x ,
h(q) = q l ( a 0 + a 1 q 2 + a 2 q 4 + · · · ) =
∞∑
a k q 2k+l , a 0 ≠ 0 (4.3)
k=0

4 58
l ,
q dh
∞ dq = ∑
(2k + l)a k q 2k+l
k=0
d 2 h
∞
dq 2 = ∑
(2k + l)(2k + l − 1)a k q 2k+l−2 =
k=0
∞∑
(2k + l + 2)(2k + l + 1)a k+1 q 2k+l
k=−1
∞∑
)
l(l − 1)a 0 q l−2 +
((2k + l + 2)(2k + l + 1)a k+1 − (2l + 4k + 1 − ε) a k q 2k+l = 0
k=0
l(l − 1)a 0 = 0 , a k+1 =
2(2k + l) + 1 − ε
(2k + l + 2)(2k + l + 1) a k (4.4)
a 0 ≠ 0 l = 0 l = 1 2 , k 2l+4k+1−ε ≠ 0
, k
a k 2l + 4k − 3 − ε
=
a k−1 (2k + l)(2k + l − 1) → 1 k
∞∑
e q2 = b k q 2k ,
k=0
b k (k − 1)!
= = 1 b k−1 k! k
q → ± ∞ h(q) → q l e q2
ϕ(q) = h(q) e −q2 /2 → q l e q2 /2
ϕ , (4.3) , k = k 0
2l + 4k 0 + 1 − ε = 0 (4.5)
a k0+1 = a k0+2 = · · · = 0 h(q) 2k 0 + l
q → ± ∞ ϕ(q) → 0 , (4.5)
E = 1 (
2 ωε = ω 2k 0 + l + 1 )
2
l = 0, 1 2k 0 + l n
(
E n = ω n + 1 )
, n = 0, 1, 2, · · · (4.6)
2
ε = 2n + 1 (4.2)
d 2 h dh
− 2q + 2nh = 0 (4.7)
dq2 dq
n = 0, 1, 2, · · · , (4.4)
h(q) =
∞∑
a k q 2k+l 2(2k + l − n)
, a k+1 =
(2k + 2 + l)(2k + l + 1) a k
k=0
h(q) k 0 = (n − l)/2 k 0
, n l = 0 , l = 1 k ≥ k 0 + 1 a k = 0

4 59
a k
k 0 − k + 1
a k = − 4
(2k + l)(2k + l − 1) a k−1 = (−4) k k 0 (k 0 − 1) · · · (k 0 − k + 1)
(2k + l)(2k + l − 1) · · · (l + 1) a 0
=
(−4) k k 0 ! l!
(k 0 − k)! (2k + l)! a 0
l = 0, 1 l! = 1 a 0 ≠ 0 , a k0 a k0 = 2 n = 2 2k0+l
a 0
a 0 = (−1) k 0 2l n!
k 0 !
, ∴ a k = (−1) k 0−k 2 2k+l n!
(k 0 − k)! (2k + l)! = (−1)k′ 2n−2k′ n!
k ′ ! (n − 2k ′ )!
k ′ = k 0 −k [n/2] = n/2 k 0 = [n/2]
h(q) = H n (q) ≡
[n/2]
∑
H n (q) n (Hermite)
k=0
(−1) k n!
k! (n − 2k)! (2q)n−2k (4.8)
d n
[n/2]
dq n F ∑ n!
(q2 ) =
k! (n − 2k)! (2q)n−2k F (n−k) (q 2 )
k=0
( I 9 ), F (q) = e −q
d n
[n/2]
∑ n!
dq n = e−q2 k! (n − 2k)! (2q)n−2k (−1) n−k e −q2
k=0
dn
H n (q) = (−1) n e q2
dq n (4.9)
e−q2
248 15.1
E n = ω(n + 1/2) ϕ n
ϕ n (x) = C n H n (q) e −q2 /2 = C n H n (αx) e −α2 x 2 /2 , α =
√ mω
(4.10)
C n
(4.7) 2 , H n (q) n
l = 0, n l = 1 k > [n/2] a k = 0
H n (q) , n l = 1 k a k ≠ 0 ,
h(q) H n (q) (4.7) n
, h(q) H n (q) 2
2 q → ± ∞ e q2 ,
4.1
(4.8) (4.9) n = 2, 3
4.2
(15.8) C n (4.22)

4 60
H a
√ ( mω
a = x +
i )
2 mω p = √ 1 (
q + d )
2 dq
p = − i d/dx
√ ( mω
a † = x −
i )
2 mω p = √ 1 (
q − d )
2 dq
(4.11)
(4.12)
a a † = 1 2
a † a = 1 2
2
H q
d
dq qf(q) = f(q) + q d dq f(q) , d dq q = 1 + q d dq
(
q + d ) (
q − d )
= 1 (q 2 + d dq dq 2 dq q − q d )
dq − d2
dq 2 = 1 2
(q 2 − d dq q + q d )
dq − d2
dq 2 = 1 )
(q 2 − 1 − d2
2
dq 2
H = − 2 mω
2m
d 2
(q 2 + 1 − d2
dq 2 )
a a † − a † a = [ a , a † ] = 1 (4.13)
d √ mω
dx =
d
dq
dq 2 + 1
2 mω2 mω q2 = ω 2
, N = a † a
) (
(− d2
dq 2 + q2 = ω a † a + 1 )
2
(4.14)
*** (4.13) , a a † ***
N ν , | ν 〉
| φ 〉 = a | ν 〉 (1.16)
N | ν 〉 = ν | ν 〉 , 〈 ν | ν 〉 = 1
N ν , ν = 0
ν = 〈 ν |N| ν 〉 = 〈 ν | a † a | ν 〉 = 〈 φ | φ 〉 ≥ 0 (4.15)
| φ 〉 = a | ν 〉 = 0
| ν 〉 | 0 〉 (4.13)
Na| ν 〉 = (aa † − 1)a| ν 〉 = aN| ν 〉 − a| ν 〉 = (ν − 1) a| ν 〉
Na † | ν 〉 = a † ( 1 + a † a ) | ν 〉 = (ν + 1) a † | ν 〉
a| ν 〉 N ν − 1 ν = 0 a | 0 〉 = 0
, a † | ν 〉 N ν + 1 | ν 〉 a , a † N
· · · a k | ν 〉 · · · a 2 | ν 〉 a| ν 〉 | ν 〉 a † | ν 〉 (a † ) 2 | ν 〉 · · ·
· · · ν − k · · · ν − 2 ν − 1 ν ν + 1 ν + 2 · · ·

4 61
ν , k ν − k ≠ 0 a k | ν 〉 ̸= 0
, k N ν − k N
, ν a ν | ν 〉 ∝ | 0 〉
a ν+1 | ν 〉 = a ν+2 | ν 〉 = · · · = 0
, N , N = a † a ν
ν n
a † a | n 〉 = n | n 〉 , n = 0, 1, 2, · · · (4.16)
H = ω (N + 1/2) E n = ω (n + 1/2)
(4.6)
| n 〉 c n
| n 〉 = c n
(
a
† ) n
| 0 〉 , 〈 0 | 0 〉 = 1
(1.16) Û = c n
(
a
† ) n
Û † = c ∗ na n
〈 n | n 〉 = |c n | 2 〈 0 | a n (a † ) n | 0 〉
I n = 〈 0 | a n (a † ) n | 0 〉 , (4.13)
I n = 〈 0 | a n−1 a a † (a † ) n−1 | 0 〉 = 〈 0 | a n−1 (N + 1) (a † ) n−1 | 0 〉
(a † ) n−1 | 0 〉 N n − 1
I n = n 〈 0 | a n−1 (a † ) n−1 | 0 〉 = n I n−1 = n(n − 1) I n−2 = · · · = n!I 0 = n!
〈 n | n 〉 = |c n | 2 n! = 1 ,
| n 〉 = 1 √
n!
(
a
† ) n
| 0 〉 , 〈 0 | 0 〉 = 1 (4.17)
| n 〉
a † | n 〉 = √ n + 1 | n + 1 〉 , a | n 〉 =
{ √ n | n − 1 〉 , n ≠ 0
0 , n = 0
(4.18)
a † N n 1 , a 1
(4.11), (4.12) a, a † ,
, (4.16) (4.17)
(4.13) , a, a †
, a † , a a| 0 〉 = 0
| 0 〉 , (4.17) | n 〉 n
4.3
(4.18)

4 62
(4.11), (4.12) (4.17) ϕ n ϕ 0
a ϕ 0 = √ 1 (
q + d )
ϕ 0 = 0
2 dq
C 0 ϕ 0 = C 0 e −q2 /2
∫ ∞
−∞
, ϕ 0
dx |ϕ 0 (q)| 2 = |C|2
α
∫ ∞
−∞
dq e −q2
= |C|2 √ π = 1
α
ϕ 0 (x) =
√ √ α
α
/2 x
=
2 /2
π 1/2 e−q2 π 1/2 e−α2
(4.19)
(4.17)
√ α (
ϕ n (x) =
a
† ) (
n
e
−q 2 /2 = C
π 1/2 n q − d ) n
e −q2 /2 , C n =
n!
dq
f(q)
e q2 /2 d dq e−q2 /2 f(q) =
( d
dq − q )
f(q) ≡ f 1 (q)
√
α
π 1/2 n! 2 n (4.20)
e q2 /2 d2
dq 2 e−q2 /2 f(q) = e q2 /2 d dq e−q2 /2 e q2 /2 d dq e−q2 /2 f(q)
= e q2 /2 d dq e−q2 /2 f 1 (q) =
( d
dq − q )
f 1 (q) =
( d
dq − q ) 2
f(q)
( d
dq − q ) n
f(q) = e q2 /2 dn
dq n e−q2 /2 f(q) , n = 0, 1, 2, · · · (4.21)
f(q) = e −q2 /2 (4.20) q = αx = √ mω/ x
√ α
ϕ n (x) =
/2 H
π 1/2 n! 2 n e−q2 n (q) , H n (q) = (−1) n e
q2 dn
dq n e−q2 (4.22)
(4.10) ,
4.2
E = 0 ,
, 0 ω/2
, x p ,
( 0 ) ,
0
H ϕ 〈 · · · 〉
∆x = √ 〈(x − 〈 x 〉) 2 〉 = √ 〈 x 2 〉 − 〈 x 〉 2 , ∆p = √ 〈(p − 〈 p 〉) 2 〉 = √ 〈 p 2 〉 − 〈 p 〉 2
ϕ(x) 〈 x 〉 = 〈 p 〉 = 0
〈 H 〉 = 1
2m 〈 p2 〉 + mω2
2 〈 x2 〉 = 1
2m (∆p)2 + mω2
2 (∆x)2

4 63
∆x∆p ≥ /2
〈 H 〉 ≥ 1 ( ) 2
+ mω2
2m 2∆x 2 (∆x)2 ≥ 2
(
1
2m
( ) ) 2 1/2
mω 2
2∆x 2 (∆x)2 = ω 2
2 ≥ 〈 H 〉 = ω/2
∆x∆p = ( ) 2
2 , 1
= mω2
2m 2∆x 2 (∆x)2 , ∆x =
√
2∆p = 2mω = √ 1
2 α
ω/2 ,
,
(4.17)
x = q α = a† + a
√
2 α
,
p = −iα d dq = i α √
2
(
a † − a )
x | n 〉 = 1 √
2 α
( √n
+ 1 | n + 1 〉 +
√ n | n − 1 〉
)
n ≠ n ′ 〈 n ′ | n 〉 = 0
〈 n | x | n 〉 = 0 , 〈 n | p | n 〉 = 0 , ϕ n (x)
, (4.13)
(
a † + a ) 2
= (a † ) 2 + a 2 + 2a † a + 1
x 2 | n 〉 = 1
2α 2 (√
(n + 1)(n + 2) | n + 2 〉 +
√
n(n − 1) | n − 2 〉 + (2n + 1)| n 〉
)
〈 n | x 2 | n 〉 = 1 (
α 2 n + 1 )
(
, 〈 n | p 2 | n 〉 = (α) 2 n + 1 )
2
2
(4.23)
(4.24)
(∆x) 2 = 〈 n | x 2 | n 〉 − 〈 n | x | n 〉 2 = 1 (
α 2 n + 1 )
2
(
(∆p) 2 = 〈 n | p 2 | n 〉 − 〈 n | p | n 〉 2 = (α) 2 n + 1 )
2
∆x∆p = (n + 1/2) , ( n = 0 ) α
∆x α 2 = 1/(2(∆x) 2 ) , (4.19)
( )
1
ϕ 0 (x) =
(2π(∆x) 2 ) exp −
x2
1/4 4(∆x) 2
, x 0 = k 0 = 0 (1.35)
α 2 = mω/ (4.24)
mω 2
2 〈 n | x2 | n 〉 = 1
2m 〈 n | p2 | n 〉 = ω 2
(
n + 1 )
= E n
2 2
(1.44) V (x) = mω 2 x 2 /2 x dV/dx = mω 2 x 2 = 2V (x)
(1.44)
1
2m 〈 n | p2 | n 〉 = 〈 n | V | n 〉 = 1 ( ) p
2
2 〈 n | 2m + V | n 〉 = E n
2

4 64
4.3
, E , v mv 2 /2 = E − mω 2 x 2 /2 ≥ 0
,
x 2 ≤ 2E
mω 2 , q2 = α 2 x 2 ≤ ε , ε = 2E
ω
, x x + dx P cl (x) dx ,
dt
P cl ∝ dt
dx = 1 v ∝ 1
√ , P C
cl = √
ε − q
2 ε − q
2
∫ √ ε/α
− √ ε/αdx P cl = 2C α
∫ √ ε
0
dq
√ = 2C [sin −1 q ] √ ε
√ = C
ε − q
2 α ε α π = 1
P cl =
0
α
π √ ε − q 2 (4.25)
E = ω(n + 1/2) ε = 2n + 1
0.6
n=0
0.2
n = 50
|ϕn(q)| 2 /α
0.4
0.2
n=3
0.1
0.0
0 1 2 3 q
0.0
0 1 2 3 4 5 6 7 8 9 10 q
n = 0, 3 (4.22) |ϕ n | 2 q ϕ n n
, q ≥ 0
P cl (q)/α q ,
| q | > √ 2n + 1 , , n |ϕ n | 2
P cl (q) ( n = 50 )ϕ n q > 0 n/2 , |ϕ n | 2
, P cl n → ∞ H n (q)
(15.105) WKB ( 154 )E n ∼ ω
, E n ≫ ω
t = 0 ψ(x, 0) , ψ(x, t) (1.39), (1.40)
∞∑
ψ(x, t) = c n e −iEnt/ ϕ n (x) , c n =
∫ ∞
n=0
−∞
dx ϕ n (x) ψ(x, 0) (4.26)

4 65
( ϕ ∗ n = ϕ n ) ψ(x, 0) , x 0 p 0
(1.35)
ψ(x, 0) =
(
1
(2π(∆x) 2 ) exp − (x − x 0) 2
1/4 4(∆x) 2 + i p )
0x
(4.27)
∆x ∆x (∆x) 2 = 1/(2α 2 ) x q = αx
ψ(x, 0) =
(15.7)
∞∑
n=0
√ α
π 1/2 e−(q−q0)2 /2+ik 0q , q 0 = αx 0 , k 0 = p 0
α = α p 0
mω
c n
√
n!
s n =
∫ ∞
−∞
dx
∞∑
n=0
ϕ n (x)
√
n!
s n ψ(x, 0)
= √ 1 ∫ ∞
dq exp
(− q2 + s 2
+ √ 2 sq − (q − q 0) 2
π 2
2
−∞
q
∞∑
n=0
(
c
√ n
s n = exp − q2 0 + k0 2 − 2iq 0 k 0
n! 4
(
= exp − q2 0 + k0 2 )
− 2iq 0 k 0
∑ ∞
4
n=0
+ q )
0 + ik
√ 0
s 2
1
n!
)
+ ik 0 q
( ) n q0 + ik
√ 0
s n
2
(4.28)
s n
c n = √ 1 ( ) n (
q0 + ik
√ 0
exp − q2 0 + k 2 )
0 − 2iq 0 k 0
n! 2
4
(4.29)
(4.22) (4.26) (15.6)
√ ( α
ψ(x, t) =
π exp − q2
1/2 2 − q2 0 + k0 2 − 2iq 0 k 0
− iωt )
∑ ∞
(λ/2) n
H n (q)
4
2 n!
n=0
√ ( α
=
π exp − q2
1/2 2 − q2 0 + k0 2 − 2iq 0 k 0
− iωt )
4
2 − λ2
4 + λq
(4.30)
λ =
(q 0 + ik 0
)
e −iωt
|e z | = exp(Re z)
|ψ(x, t)| 2 = √ α (
exp − q 2 − q2 0 + k0
2 − λ2 R − λ2 I
π 2 2
)
+ 2λ R q , λ R = Re λ , λ I = Im λ
λ |λ| 2 = λ 2 R + λ2 I = q2 0 + k 2 0
|ψ(x, t)| 2 =
√ α (
exp − (q − λ R ) 2) =
π
α √ π
exp
(
− α 2( ) ) 2
x − X(t)
(4.31)
X(t) = λ R
α = x 0 cos ωt + p 0
sin ωt
mω
X(t) t = 0 x 0 , p 0 , |ψ(x, t)| 2

4 66
ψ(x, t) 〈 · · · 〉 x x 2
〈 x 〉 = 1 ∫ ∞
α 2 dq q |ψ(x, t)| 2 = √ 1 ∫ ∞
dq (q + λ R ) e −q2
πα
〈 x 2 〉 = 1 √ πα
2
−∞
∫ ∞
−∞
−∞
dq (q + λ R ) 2 e −q2 = 1 ( 1
α 2 2 + λ2 R
(∆x) 2 = 1/(2α 2 ) ∆x (4.30)
∂
ψ(x, t) = α (λ − αx) ψ(x, t)
∂x
)
= λ R
α
= X(t) (4.32)
= 1
2α 2 + (X(t))2 (4.33)
( )
)
〈 p 〉 = − i λ − α〈 x 〉 = − iα
(λ − λ R = αλ I = p 0 cos ωt − mωx 0 sin ωt = m dX dt
(4.34)
∂ 2
∂x 2 ψ(x, t) = − α2 ( 1 − (λ − αx) 2) ψ(x, t) = − α 2( 1 − λ 2 + 2λαx − α 2 x 2) ψ(x, t)
(4.32), (4.33)
〈 p 2 〉 = 2 α 2 (1 − λ 2 + 2λλ R − 1 2 − λ2 R
) (
= 2 α 2 λ 2 I + 1 2
)
= 〈 p 〉 2 + 2 α 2
2
(∆p) 2 = 2 α 2 /2 ∆x∆p = /2 , (4.30) ψ(x, t)
, (∆x) 2 ≠ 1/(2α 2 ) , t = 0
, t ≠ 0 ψ(x, t) ( 4.4 )
K = p 2 /(2m) V (x) = mω 2 x 2 /2
〈 K 〉 = 1
2m 〈 p2 〉 = 1
2m 〈 p 〉2 + 2 α 2
4m = 1 (
) 2 ω
p 0 cos ωt − mωx 0 sin ωt +
2m
4
〈 V 〉 = mω2
2 〈 x2 〉 = mω2
2 〈 x 〉2 + mω2
4α 2
= mω2
2
〈 K 〉 〈 V 〉 , H
(
x 0 cos ωt + p ) 2
0
mω sin ωt ω +
4
〈 H 〉 = 〈 K 〉 + 〈 V 〉 = E cl + ω 2 , E cl = p2 0
2m + mω2
2 x2 0
〈 H 〉 E cl
(4.29)
|c n | 2 = ρn
n! e−ρ , ρ = q2 0 + k0
2
2
∞∑
|c n | 2 = 1 ,
n=0
n=0
n=1
= E cl
ω
∞∑
∑
∞
|c n | 2 n = ρ e −ρ ρ n−1
(n − 1)! = ρ
〈 H 〉 = ω
∞∑
n=0
(
|c n | 2 n + 1 ) (
= ω ρ + 1 )
= E cl + ω 2
2
2

4 67
4.4
i ∂ ψ(x, t) = Hψ(x, t) ,
∂t H
= p2
2m + mω2
2 x2
ψ(x, t) 〈 · · · 〉
1. (1.7)
D x (t) = mω 2( 〈 x 2 〉 − 〈 x 〉 2) , D p (t) = 1 (〈 p 2 〉 − 〈 p 〉 2)
m
D xp (t) = ω 2( )
〈 xp + px 〉 − 2〈 x 〉〈 p 〉
dD x
dt
= D xp ,
dD p
dt
= − D xp ,
dD xp
dt
= − 2ω 2( D x − D p
)
2. 1. D x (t) D p (t)
D x (0) = D y (0) , D xp (0) = 0
3. ψ(x, 0) (4.27) , 2. (∆x) 2 = 1/(2α 2 )
4.5
(4.26)
ψ(x, t) =
∞∑
e −iEnt/ ϕ n (x)
n=0
∫ ∞
−∞
G(x, x ′ , t) =
dx ′ ϕ n (x ′ ) ψ(x ′ , 0) =
∫ ∞
−∞
dx ′ G(x, x ′ , t) ψ(x ′ , 0)
∞∑
e −iEnt/ ϕ n (x)ϕ n (x ′ ) (4.35)
G(x, x ′ , t) 1 (15.10)
√
(
mω
G(x, x ′ , t) =
2πi sin ωt exp i (q2 + q ′2 ) cos ωt − 2qq ′ )
2 sin ωt
q = αx , q ′ = αx ′ , ω → 0
√ ( m
G(x, x ′ , t) →
2πit exp i m(x − x′ ) 2 )
2t
n=0
, (1.56)
(4.36)
4.6 3.3 pot(x) ,
(4.31) , (4.27) ∆x (∆x) 2 = 1/(2α 2 )
∆x , 4.4 D x (t)

4 68
4.4
( coherent state )
a | λ 〉
a| λ 〉 = λ| λ 〉 (4.37)
a , λ ,
| n=0 〉 λ = 0
, [ a , a † ] = 1 a
a † | λ 〉 , | λ 〉
〈 λ | a | λ 〉 = λ , 〈 λ | a 2 | λ 〉 = λ 2
,
a † a | λ 〉 = λ a † | λ 〉
〈 λ | a † | λ 〉 = λ ∗ , 〈 λ | (a † ) 2 | λ 〉 = (λ ∗ ) 2
〈 λ | a † a | λ 〉 = λ 〈 λ | a † | λ 〉 = λλ ∗ = |λ| 2 , 〈 λ | aa † | λ 〉 = 〈 λ | a † a + 1 | λ 〉 = |λ| 2 + 1
,
x = a + a†
√
2 α
, p = − i α a − a†
√
2
, α =
√ mω
〈 λ | x | λ 〉 = λ + λ∗
√
2 α
=
√
2
α Re λ ,
√
λ − λ∗ 2
〈 λ | p | λ 〉 = − i α √ = mω Im λ (4.38)
2 α
, λ x 2 p 2
,
〈 λ | x 2 | λ 〉 = 1 (
)
2α 2 〈 λ | a 2 + (a † ) 2 + aa † + a † a | λ 〉 = (λ + λ∗ ) 2 + 1
2α 2
〈 λ | p 2 | λ 〉 = 2 α 2 (1 − (λ − λ ∗ ) 2)
2
(∆x) 2 = 〈 λ | x 2 | λ 〉 − 〈 λ | x | λ 〉 2 = 1
2α 2 , (∆p)2 = 〈 λ | p 2 | λ 〉 − 〈 λ | p | λ 〉 2 = 2 α 2
2
(4.39)
λ , ∆x∆p = /2
H = ω(a † a + 1/2)
(
〈E 〉 = 〈 λ | H | λ 〉 = ω |λ| 2 + 1 )
2
(a † a) 2 = a † a a † a = a † (a † a + 1)a = (a † ) 2 a 2 + a † a
〈 λ | (a † a) 2 | λ 〉 = λ 2 〈 λ | (a † ) 2 | λ 〉 + |λ| 2 = |λ| 4 + |λ| 2

4 69
(
〈 λ | H 2 | λ 〉 = 2 ω 2 〈 λ | (a † a) 2 + a † a + 1 )
( (
| λ 〉 = 2 ω 2 |λ| 2 + 1 ) )
2
+ |λ| 2
4
2
(∆E) 2 = 〈 λ | H 2 | λ 〉 − 〈 λ | H | λ 〉 2 = 2 ω 2 |λ| 2
∆E
〈E 〉 = |λ|
|λ| 2 + 1/2
, |λ| ≫ 1, 〈E 〉/ω ≫ 1
| λ 〉 H | n 〉
(4.18)
n=0
| λ 〉 =
∞∑
c n (λ) | n 〉 (4.40)
n=0
∞∑
∞∑ √ ∑
∞ √ ∑
∞
a| λ 〉 = c n a| n 〉 = c n n | n − 1 〉 = c n+1 n + 1 | n 〉 = λ c n | n 〉
n=1
ϕ ∗ n ′ 〈 n′ | n 〉 = δ nn ′ √ n + 1 c n+1 = λ c n
c n =
n=0
√ λ
λ 2
c n−1 = √ c n−2 = · · · = √ λn
c 0
n n(n − 1) n!
n=0
c 0
〈 λ | λ 〉 =
∞∑
∑
∞
|c n | 2 = |c 0 | 2
n=0
n=0
|λ| 2
n!
= |c 0 | 2 e |λ|2 = 1
c 0 = e −|λ|2 /2
c n (λ) = λn
√
n!
e −|λ|2 /2
(4.41)
(4.18) (4.40)
| λ 〉 = e −|λ|2 /2
( ∞∑
) λa
† n (
)
| 0 〉 = exp λa † − |λ| 2 /2 | 0 〉 (4.42)
n!
n=0
, |c n (λ)| 2 = |λ| 2 e −|λ|2 /n! n, a † a |λ| 2
4.7
a †
( Campbell–Hausdorff )
A, B
[ A , [ A , B ] ] = [ B , [ A , B ] ] = 0 (4.43)
e A e B = e A+B+[A,B]/2 (4.44)

4 70
( Campbell–Hausdorff )
f(x) = e xA Be −xA
f ′ (x) = dexA
dx Be−xA + e xA B de−xA
= Ae xA Be −xA − e xA Be −xA A = [ A , f(x) ]
dx
f(0) = B
f ′′ (x) = [ A , f ′ (x) ] = [ A , [ A , f(x) ] ]
f (3) (x) = [ A , [ A , f ′ (x) ] ] = [ A , [ A , [ A , f(x) ] ] ]
· · · · · ·
f(x) = e xA Be −xA = f(0) + f ′ (0)x + f ′′ (0)
2
x 2 + f (3) (0)
x 3 + · · ·
3!
= B + x [ A , B ] + x2
2 [ A , [ A , B ] ] + x3
3! [ A , [ A , [ A , B ] ] ] + · · · (4.45)
, (4.43) (4.45)
g(x) = e xA e xB
e xA Be −xA = B + x [ A , B ] (4.46)
dg
(
dx = AexA e xB + e xA Be xB = A + e xA Be −xA) (
)
e xA e xB = A + B + x [ A , B ] g(x)
A+B [ A , B ] , g(0) =
1
x = 1 (4.44)
(
)
g(x) = exp (A + B)x + [ A , B ]x 2 /2
A = λ ( a † − a ) , B = λa [ A , B ] = − λ 2 (4.44)
e λa†
= e A+B = e λ2 /2 e λ(a† −a) e
λa
, (4.42)
a| 0 〉 = 0
( λ 2 − |λ| 2 )
| λ 〉 = exp
e
−a) λ(a† e λa | 0 〉
2
e λa | 0 〉 =
a † − a = − √ 2 d dq | λ 〉 ϕ λ
)
(1 + λa + λ2
2 a2 + · · · | 0 〉 = | 0 〉
( λ 2 − |λ| 2 ) (
ϕ λ (q) = exp
exp − √ 2 λ d )
ϕ 0 (q) = exp
2
dq
( λ 2 − |λ| 2
2
)
ϕ 0 (q − √ 2 λ)

4 71
, (1.25) (4.19)
√ ( α λ 2
ϕ λ (q) =
π exp − |λ| 2
− 1 (
q − √ ) ) 2
2 λ
1/2 2 2
(4.47)
(4.38)
q 0 = α 〈 λ | x | λ 〉 = √ 2 Reλ , k 0 = 1
α 〈 λ | p | λ 〉 = √ 2 Imλ
λ = (q 0 + ik 0 )/ √ 2 (4.47)
√ ( α
ϕ λ (q) =
π exp − (q − q 0) 2
)
+ ik 1/2 0 q − ik 0 q 0 /2
2
, e −ik0q0/2 (4.28)
t = 0 λ = λ 0 t |ψ(t)〉 (4.26)
c n (4.41)
|ψ(t)〉 =
∞∑
n=0
λ n 0
√
n!
e −|λ 0| 2 /2 e −iω(n+1/2)t | n 〉
= e −|λ0|2 /2−iωt/2
( ∞∑ λ0 e −iωt a †) n
| 0 〉 = e −iωt/2 | λ = λ 0 e −iωt 〉
n!
n=0
|ψ(t)〉 |ψ(t)〉 x p (4.38) λ λ 0 e −iωt
√
2
〈ψ(t)| x |ψ(t)〉 =
α Re ( λ 0 e −iωt) = 〈 λ 0 | x | λ 0 〉 cos ωt + 〈 λ 0 | p | λ 0 〉
sin ωt
mω
√
2
〈ψ(t)| p |ψ(t)〉 = mω
α Im ( λ 0 e −iωt) = 〈 λ 0 | p | λ 0 〉 cos ωt − mω 〈 λ 0 | p | λ 0 〉 sin ωt
(4.32), (4.34) ∆x, ∆p (4.39) ,
4.8
(4.37)
a ϕ λ = √ 1 (
q + d )
ϕ λ = λϕ λ
2 dq
(4.47) , a †
4.9 a H (t) = e iHt/ a e −iHt/ , H = ω ( a † a + 1/2 )
d
1.
dt a H = − iω a H a H (t) = e −iωt a
2. (4.45) a H (t) = e −iωt a
3. t = 0 | λ 0 〉 , t |ψ(t)〉 = e −iHt/ | λ 0 〉
a |ψ(t)〉 = λ 0 e −iωt |ψ(t)〉

4 72
(4.40), (4.41)
〈 λ | λ ′ 〉 = ∑ nn ′ c ∗ n(λ)c n ′(λ ′ )〈 n | n ′ 〉 =
∞∑
c ∗ n(λ)c n (λ ′ ) = exp
(− |λ|2 + |λ ′ | 2 )
∑ ∞
(λ ∗ λ ′ ) n
2
n!
n=0
= exp
(− |λ|2 + |λ ′ | 2 )
+ λ ∗ λ ′
2
n=0
|e z | = e Re z = e (z+z∗)/2
|〈 λ | λ ′ 〉| = exp
(− |λ|2 + |λ ′ | 2
2
+ λ∗ λ ′ + λλ ′∗ )
= exp
2
(− |λ − λ′ | 2 )
, , λ λ ′
S(x, x ′ ) = 1 ∫
d 2 λ ϕ λ (x) ϕ ∗
π
λ(x ′ )
λ R = Re λ , λ I = Im λ d 2 λ = dλ R dλ I (4.40), (4.41)
S(x, x ′ ) = 1 ∑
∫
√
π n! n′ ! ϕ n(x)ϕ n ′(x ′ ) d 2 λ e −|λ|2 (λ ∗ ) n λ n′
nn ′ 1
λ λ = re iθ
∫
d 2 λ e −|λ|2 (λ ∗ ) n λ n′ =
∫ ∞
0
dr
= 2π δ nn ′
(e −x ) ′ = − e −x
∫ ∞
0
∫ 2π
0
∫ ∞
dx x n e −x = [ − x n e −x] ∫ ∞
∞
+ n dx x n−1 e −x = n
0
0
S(x, x ′ ) =
0
dθ r n+n′ +1 e −r2 e i(n′ −n)θ
dr r 2n+1 e −r2 = π δ nn ′
∫ ∞
∞∑
ϕ n (x)ϕ n (x ′ ) = δ(x − x ′ )
n=0
0
∫ ∞
0
2
dx x n e −x
dx x n−1 e −x = · · · = n!
, ψ(x)
∫
ψ(x) = dx ′ ψ(x ′ ) δ(x − x ′ ) = 1 ∫ ∫
d 2 λ ϕ λ (x) dx ′ ϕ ∗
π
λ(x ′ )ψ(x ′ )
| ψ 〉 = 1 ∫
d 2 λ | λ 〉〈 λ | ψ 〉 , 〈 λ | ψ 〉 =
π
∫ ∞
−∞
dx ϕ ∗ λ(x)ψ(x)
∫ ∞
0
dx e −x = n!
| λ 〉 , | λ 〉
| λ 〉 = 1 ∫
d 2 λ ′ | λ ′ 〉〈 λ ′ | λ 〉 = 1 ∫
d 2 λ ′ | λ ′ 〉 exp
(− |λ|2 + |λ ′ | 2 )
+ λλ ′∗ (4.48)
π
π
2
, | n 〉
| n 〉 = 1 ∫
d 2 λ | λ 〉〈 λ | n 〉 = 1 ∫
d 2 λ | λ 〉 c ∗
π
π
n(λ) = 1 ∫
π
d 2 λ | λ 〉 (λ∗ ) n
√
n!
e −|λ|2 /2
(4.49)

4 73
4.10
(4.42)
| λ 〉 = D(λ)| 0 〉 , D(λ) = exp ( λa † − λ ∗ a )
, D
D † (λ) = D(−λ) ,
D † (λ)aD(λ) = a + λ
D(−λ) = D −1 (λ) D(λ)D † (λ) = D † (λ)D(λ) = 1 D(λ)
, 2 aD(λ) = D(λ)a + λD(λ)
a | λ 〉 = aD(λ)| 0 〉 =
(
)
D(λ)a + λD(λ) | 0 〉 = λD(λ)| 0 〉 = λ | λ 〉
4.11
(4.48), (4.49) (4.35) G(x, x ′ , t)
G(x, x ′ , t) = e−iωt/2
π
∫
d 2 λ ϕ λ(t) (x) ϕ ∗ λ(x ′ ) , λ(t) = λe −iωt
(4.36)

5 74
5
,
, ( Lie algebra )
(
)
,
5.1
, ˆ p p = −i∇ f(r)
( x 1 = x, x 2 = y, x 3 = z )
[ x i , p j ]f(r) ≡
(x i p j − p j x i
)
f(r) = −i
(
x i
∂f
− ∂ x ) (
if
= −i x i
∂x j ∂x j
i = j ∂x i /∂x j = 1 , i ≠ j ∂x i /∂x j = 0
[ x i , p j ]f(r) = i δ ij f(r)
f(r) f(r)
∂f
− ∂x )
i ∂f
f − x i
∂x j ∂x j ∂x j
[ x i , p j ] = i δ ij (5.1)
A, B, C, D
[ A , BC ] ≡ ABC − BCA = (AB − BA)C + B(AC − CA)
= [ A , B ] C + B [ A , C ] (5.2)
[ AB , C ] = A [ B , C ] + [ A , C ] B (5.3)
[ A + B , C + D ] = [ A , C ] + [ A , D ] + [ B , C ] + [ B , D ] (5.4)
L L = r×p
L
L = r×p ,
p = − i∇
x, y, z 1, 2, 3
[ L 1 , L 2 ] = [ x 2 p 3 − x 3 p 2 , x 3 p 1 − x 1 p 3 ]
= [ x 2 p 3 , x 3 p 1 ] − [ x 2 p 3 , x 1 p 3 ] − [x 3 p 2 , x 3 p 1 ] + [x 3 p 2 , x 1 p 3 ]
2 3 0
[ L 1 , L 2 ] = [ x 2 p 3 , x 3 p 1 ] + [ x 3 p 2 , x 1 p 3 ] = x 2 [ p 3 , x 3 ] p 1 + p 2 [ x 3 , p 3 ] x 1
= i (− x 2 p 1 + x 1 p 2 ) = i L 3

5 75
[ L 1 , L 2 ] = i L 3 , [ L 2 , L 3 ] = i L 1 , [ L 3 , L 1 ] = i L 2
[ L i , L j ] = i
3∑
ε ijk L k
ε ijk ( Levi–Civita ) ( 237 )
k=1
⎧
⎪⎨ 0 2
ε ijk = +1 ijk 123
⎪⎩
−1 ijk 123
ε 123 = ε 231 = ε 312 = 1 , ε 132 = ε 213 = ε 321 = −1 , ε ijk = 0
L = r×p
[ L 1 , L 2 ] = iL 3 , [ L 2 , L 3 ] = iL 1 , [ L 3 , L 1 ] = iL 2
, J
[J x , J y ] = iJ z , [J y , J z ] = iJ x , [J z , J x ] = iJ y (5.5)
[ J i , J j ] = i ∑ k
ε ijk J k
J ( angular momentum ) J r p
,
, , J
,
5.2
6 | 〉 2 A, B
| n 〉 , | n 〉 A a n
B b n
A | n 〉 = a n | n 〉 , B | n 〉 = b n | n 〉
|ψ〉
|ψ〉 = ∑ n
α n | n 〉 ,
α n = 〈 n |ψ〉
a n , b n , α n , A, B
AB |ψ〉 = A ∑ n
α n B | n 〉 = A ∑ n
α n b n | n 〉 = ∑ n
α n b n A | n 〉 = ∑ n
α n b n a n | n 〉

5 76
BA |ψ〉 = ∑ n
α n a n b n | n 〉
[ A , B ]|ψ〉 = 0
|ψ〉 [ A , B ] = 0
, [ A , B ] = 0 A, B | n 〉 a n
A
A | n 〉 = a n | n 〉
AB = BA
AB| n 〉 = BA| n 〉 = a n B| n 〉
B| n 〉 a n A a n ,
, a n 1 , B| n 〉 | n 〉 , b
B| n 〉 = b| n 〉
, | n 〉 B , A B
A a k , a | a, i 〉 , ( i =
1, 2, · · · , k )
A|a, i 〉 = a |a, i 〉 ,
〈a, i |a, j 〉 = δ ij
AB = BA
AB|a, j 〉 = BA|a, j 〉 = aB|a, j 〉
B|a, j 〉 a A
k∑
B|a, j 〉 = |a, i 〉 b ij , b ij = 〈a, i |B|a, j 〉
i=1
( A, B B|a, j 〉 a A
) (b ij ) {c i }, b :
k∑
b ij c j = b c i , det(b ij − b δ ij ) = 0 (5.6)
j=1
k∑
|ψ〉 = c j |a, j 〉
, |ψ〉 a A , , (5.6)
k∑
B|ψ〉 = c j B|a, j 〉 =
k∑
c j
j=1
j=1 i=1
j=1
k∑
|a, i 〉 b ij =
k∑ k∑
|a, i 〉 b ij c j = b
i=1 i=1
i=1
k∑
c i |a, i 〉 = b |ψ〉
B , A ,
A, B A B

5 77
, , B ,
A, B
[ A , B ] = 0 , [ A , C ] = 0 , A A |a〉
B, C b, c , |a〉
[ B , C ]|ψ〉 = ∑ a
(BC − CB) |a〉〈a|ψ〉 ∑ a
(bc − cb) |a〉〈a|ψ〉 = 0
[ B , C ] = 0 , A , A
, A, B |ψ ab 〉 A, C |ψ ac 〉 , |ψ ab 〉
|ψ ac 〉 , |ψ ab 〉 C , |ψ ac 〉 B
[ B , C ] = 0 , [ J 2 , J x ] = 0, [ J 2 , J z ] = 0 J 2
[ J x , J z ] ≠ 0
5.3
(5.2) ,
[ J x , J 2 ] = [ J x , Jy 2 + Jz 2 ]
= J y [ J x , J y ] + [ J x , J y ]J y + J z [ J x , J z ] + [ J x , J z ]J z
)
= i
(J y J z + J z J y − J z J y − J y J z = 0
[J i , ∑ j
J j J j ] = ∑ j
(
)
[J i , J j ] J j + J j [J i , J j ] = i ∑ jk
ε ijk
(
J k J j + J j J k
)
2 j k, k j ε ikj = − ε ijk
[J i , ∑ j
J j J j ] = i ∑ jk
(ε ijk + ε ikj ) J k J j = 0
, J J 2 = J 2 x + J 2 y + J 2 z
2
J ± = J x ± iJ y ,
J † + = J −
(5.5)
[ J z , J ± ] = [ J z , J x ] ± i [ J z , J y ] = iJ y ± J x = ± J ± (5.7)
[ J + , J − ] = [iJ y , J x ] + [J x , −iJ y ] = −2i [J x , J y ] = 2J z (5.8)
(5.8) J 2 = (J + J − + J − J + )/2 + Jz 2
J − J + = J 2 − J z (J z + 1) , J + J − = J 2 − J z (J z − 1) (5.9)
| a 〉 | b 〉 = J k | a 〉 ( k = x, y, z )J k
(1.16) 〈 a |Jk 2| a 〉 = 〈 b | b 〉 ≥ 0

5 78
2 , J 2 J 1
1 J z J 2 λ
, J z m λ, m |λ m〉
J 2 |λ m〉 = λ|λ m〉 , J z |λ m〉 = m|λ m〉 , 〈λ m|λ m〉 = 1
J 2 , J z λ, m
〈λ m|J 2 |λ m〉 = λ ≥ 0
| a 〉 = J + |λ m〉 J † + = J − 〈 a | a 〉 = 〈λ m|J − J + |λ m〉 , (5.9)
〈 a | a 〉 = 〈λ m| ( J 2 − Jz 2 ) )
− J z |λ m〉 = 〈λ m|
(λ − m 2 − m |λ m〉
λ − m 2 − m
〈 a | a 〉 =
(
)
λ − m 2 − m 〈λ m|λ m〉 = λ − m 2 − m ≥ 0 (5.10)
| b 〉 = J − |λ m〉
〈 b | b 〉 = 〈jm|J + J − |jm〉 = 〈λ m| ( J 2 − J 2 z + J z
)
|λ m〉 = λ − m 2 + m ≥ 0 (5.11)
(5.10), (5.11)
−1 − √ 1 + 4λ
2
≤ m ≤ −1 + √ 1 + 4λ
2
,
1 − √ 1 + 4λ
2
≤ m ≤ 1 + √ 1 + 4λ
2
− j ≤ m ≤ j , j = −1 + √ 1 + 4λ
2
, λ m m max , m min
− j ≤ m min ≤ m max ≤ j
, m max m min , j λ
λ = (2j + 1)2 − 1
4
= j(j + 1) , j ≥ 0
λ j :
J 2 |jm〉 = j(j + 1)|jm〉 ,
J z |jm〉 = m|jm〉
J ± J 2
J 2 J ± |jm〉 = J ± J 2 |jm〉 = j(j + 1)J ± |jm〉
(5.7)
J z J ± |jm〉 = (J ± J z ± J ± ) |jm〉 = (m ± 1)J ± |jm〉
J ± |jm〉 J 2 , J z
, j(j + 1), m ± 1
J + |jm max 〉 ̸= 0 J + |jm max 〉 J z m max + 1 m max

5 79
J + |jm max 〉 = 0 J − |jm min 〉 = 0
, (5.10), (5.11)
j(j + 1) − m 2 max − m max = 0 , j(j + 1) − m 2 min + m min = 0
m max = j, m min = −j |jm max 〉 J − , J− 2 , · · · , J− n , J z
m max − 1 , m max − 2 , · · · , m max − n , m max − n = m min n
m max = j, m min = −j 2j = n, , 2j
,
J 2 |jm〉 = j(j + 1)|jm〉 , j = 0, 1/2, 1, 3/2, 2, · · · (5.12)
J z |jm〉 = m|jm〉 , m = −j, −j + 1, · · · , j − 1, j (5.13)
(1.18)
〈jm|j ′ m ′ 〉 = δ jj ′ δ mm ′ (5.14)
, √ j(j + 1) , j
j J 2 j(j + 1)
| a 〉 = J + |jm〉 J 2 , J z , j(j + 1), m + 1 ,
|j m+1〉 c jm | a 〉 = c jm |j m+1〉
〈 a | a 〉 = |c jm | 2 〈j m+1|j m+1〉 = |c jm | 2
(5.10)
|c jm | 2 = (j − m)(j + m + 1)
c jm c jm
J − (5.9)
J + |jm〉 = √ (j − m)(j + m + 1) |j m+1〉 (5.15)
J − |jm〉 = √ (j + m)(j − m + 1) |j m−1〉 (5.16)
J + J z 1 , J − ,
(5.15) (5.16) , 1 |j m 0 〉 2j + 1
|jm〉
|j j−1〉 = 1 √ 2j
J − |j j〉
|j j−2〉 =
|j j−3〉 =
1
1
√ J − |j j−1〉 = √ (J − ) 2 |j j〉
(2j − 1)·2 2j(2j − 1)·2
1
1
√ J − |j j−2〉 = √ (J − ) 3 |j j〉
(2j − 2)·3 2j(2j − 1)(2j − 2)·3·2
,
|j j−k〉 =
√
1
√ (J − ) k (2j − k)!
|j j〉 =
(J − ) k |j j〉
2j(2j − 1) · · · (2j − k + 1) k! (2j)! k!

5 80
m = j − k
√
(j + m)!
|jm〉 =
(2j)! (j − m)! (J −) j−m | j j 〉 (5.17)
√
(j − m)!
|jm〉 =
(2j)! (j + m)! (J +) j+m | j −j 〉 (5.18)
−i ∇ 3 , ,
, 3 exp(ik·r) , J , J x ,
J y , J z ( j = 0 )
|jm〉 J z ,
J x , J y
∆J k =
√
〈 jm |J 2 k | jm 〉 − 〈 jm |J k| jm 〉 2
J x = (J + + J − )/2 (5.15), (5.16)
J x |jm〉 = 1 (
)
c + |j m+1〉 + c − |j m−1〉 , c ± = √ (j ∓ m)(j ± m + 1)
2
〈jm|jm ′ 〉 = δ mm ′
〈jm|J x |jm〉 = 1 (
)
c + 〈jm|j m+1〉 + c − 〈jm|j m−1〉 = 0 , 〈jm|J y |jm〉 = 0
2
|ψ〉 = J x |jm〉 〈ψ|ψ〉 = 〈jm|Jx|jm〉 2
〈jm|Jx|jm〉 2 = 1 (
)
c 2
4
+〈j m+1|j m+1〉 + c 2 −〈j m−1|j m−1〉
+ c (
)
+c −
〈j m−1|j m+1〉 + 〈j m+1|j m−1〉
4
= c2 + + c 2 −
4
=
j(j + 1) − m2
2
, 〈jm|J 2 x|jm〉 = 〈jm|J 2 y |jm〉
〈jm|J 2 x|jm〉 = 〈jm|J 2 y |jm〉 = 1 2 〈jm|(J 2 − J 2 z )|jm〉 =
j(j + 1) − m2
2
√
j(j + 1) − m
2
∆J x = ∆J y =
2
|m| ≤ j ∆J x = ∆J y = 0 j = m = 0 , , J z J x ,
J y
∆J x ∆J y =
j(j + 1) − m2
2
≥
|m|(|m| + 1) − m2
2
(1.27)
= |m|
2 = 1 ∣
∣〈 jm |[ J x , J y ]| jm 〉 ∣
2
, z J 2 z = J 2
m 2 = j(j + 1) , , m 2 = j(j + 1)
( j = 0 ), j → ∞ j(j + 1) ≈ j 2 , |j ±j 〉
z

5 81
5.4
z
P(x, y, z)
3 e r , e θ , e φ
e φ xy P r
,
θ , φ
e r = e x sin θ cos φ + e y sin θ sin φ + e z cos θ
e θ = e x cos θ cos φ + e y cos θ sin φ − e z sin θ (5.19)
e φ = − e x sin φ + e y cos φ
x
φ
θ
e θ
e r
e φ
θ
φ
y
x ′
∂ x = ∂
∂x ∇ ≡ e x ∂ x + e y ∂ y + e z ∂ z
f(r)
df = dx ∂ x f + dy ∂ y f + dz ∂ z f =
) (
)
(dx e x + dy e y + dz e z · e x ∂ x f + e y ∂ y f + e z ∂ z f
= dr·∇f (5.20)
, r = re r , e r θ, φ
)
dr = dr e r + r de r = dr e r + r
(dθ ∂ θ e r + dφ ∂ φ e r = dr e r + r dθ e θ + r sin θ dφ e φ
f(r) r, θ, φ
(5.20)
df = dr ∂ r f + dθ ∂ θ f + dφ ∂ φ f
) (
1
=
(dr e r + r dθ e θ + r sin θ dφ e φ · e r ∂ r f + e θ
(
)
1
= dr· e r ∂ r + e θ
r ∂ 1
θ + e φ
r sin θ ∂ φ f
r ∂ θf + e φ
1
)
r sin θ ∂ φf
∇ = e r ∂ r + e θ
1
r ∂ θ + e φ
1
r sin θ ∂ φ (5.21)
e r ×e θ = e φ , e φ ×e r = e θ
(
) (
)
1
L = − i re r × e r ∂ r + e θ
r ∂ 1
θ + e φ
r sin θ ∂ 1
φ = − i e φ ∂ θ − e θ
sin θ ∂ θ
(5.19)
L = ie x
(
sin φ ∂ θ + cot θ cos φ ∂ φ
)
− ie y
(
cos φ ∂ θ − cot θ sin φ ∂ φ
)
− ie z ∂ φ (5.22)
(
L ± ≡ L x ± iL y = e ±iφ ± ∂ ∂θ + i cot θ ∂ )
, L z = − i ∂
∂φ
∂φ
L 2 = L +L − + L − L +
2
(5.23)
+ L 2 z = − ∂2
∂θ 2 − cot θ ∂ ∂θ − 1 ∂ 2
sin 2 θ ∂φ 2 (5.24)
, r

5 82
, j l L = − i r×∇ θ, φ
, θ, φ Y lm (θ, φ)
L 2 Y lm (θ, φ) = l(l + 1) Y lm (θ, φ) , L z Y lm (θ, φ) = m Y lm (θ, φ)
, f(r) ψ(r) = f(r)Y lm (θ, φ) L 2 , L z f(r)
, , ψ(r)
L z = −i∂/∂φ
−i ∂Y lm(θ, φ)
∂φ
= m Y lm (θ, φ) , ∴ Y lm (θ, φ) = F lm (θ) e imφ
r 1 Y lm (θ, φ) = Y lm (θ, φ + 2π), e 2πim = 1
, m l
, l , 1 ,
l
L 2 F lm (θ) e imφ = l(l + 1) F lm (θ) e imφ
F lm (θ) (5.24)
(− d2
dθ 2 − cot θ d )
dθ +
m2
sin 2 F lm (θ) = l(l + 1) F lm (θ)
θ
(15.25) ,
L − Y l −l = L − F l −l (θ) e −ilφ = 0
F l −l (θ) , L + Y l −l (θ, φ) Y lm (θ, φ) ( L + Y l l = 0 )
, L ± Y l ±l = 0, L z Y l ±l = ± l Y l ±l
L 2 Y l ±l = ( L ∓ L ± + L 2 z ± L z
)
Yl ±l = l(l + 1)Y l ±l
, L 2 Y l ±l = l(l + 1)Y l ±l , L − Y l −l = 0
(5.23) L − F (θ) e −ilφ = 0 ( F l −l (θ) F (θ) )
∫ ∫
∫
dF
F
= l dθ cot θ = l dθ
∫ π
|C| 2 dθ sin θ
0
∫ 2π
0
(sin θ)′
sin θ
∫
d 3 r · · · =
∫ π
0
dF
dθ − l cot θ F = 0
∫ ∞
0
dθ sin θ
= l log | sin θ | , ∴ F (θ) = C sin l θ , C =
∫ π
dr r 2 dθ sin θ
∫ 2π
dφ sin 2l θ = 2π|C| 2 I l = 1 ,
0
0
∫ 2π
dφ |Y lm (θ, φ)| 2 = 1
I l ≡
0
∫ π
0
dφ · · ·
dθ sin 2l+1 θ =
∫ 1
−1
dt (1 − t 2 ) l

5 83
t = cos θ
I l = [ t(1 − t 2 ) l ] ∫ 1
1
−1 − dt t d dt (1 − t2 ) l = 2l
−1
∫ 1
−1
dt t 2 (1 − t 2 ) l−1 = 2l (− I l + I l−1 )
I l =
2l
2l + 1 I 2l · 2(l − 1)
l−1 =
(2l + 1)(2l − 1) I l−2
=
2l · 2(l − 1) · 2(l − 2) · · · 2
(2l + 1)(2l − 1)(2l − 3) · · · 3 I 0 = 2
(
2 l l! ) 2
(2l + 1)!
(5.25)
Y l −l (θ, φ) = F (θ)e −ilφ = 1
√
(2l + 1)!
2 l l! 4π
C , C
(5.18), (5.26)
sin l θ e −ilφ (5.26)
√
Y lm (θ, φ) = 1 2l + 1 (l − m)!
2 l l! 4π (l + m)! (L +) l+m sin l θ e −ilφ (5.27)
L + , (5.23)
( ∂
L + f(θ)e inφ = e iφ ∂θ + i cot θ ∂ )
( )
df(θ)
f(θ)e inφ = e i(n+1)φ − n cot θ f(θ)
∂φ
dθ
( t = cos θ )
d
dt f(θ) sin−n θ =
dθ
d cos θ
( )
d
df(θ)
dθ f(θ) sin−n θ = − sin −n−1 θ − n cot θ f(θ)
dθ
L + f(θ)e inφ = g(θ)e i(n+1)φ , g(θ) = − sin n+1 θ d dt f(θ) sin−n θ (5.28)
L + , f → g, n → n + 1
(L + ) 2 f(θ)e inφ = L + g(θ)e i(n+1)φ = − e i(n+2)φ sin n+2 θ d dt sin−n−1 θ g(θ)
= (−1) 2 e i(n+2)φ sin n+2 θ d2
dt 2 f(θ) sin−n θ
(L + ) k f(θ)e inφ = (−1) k e i(n+k)φ sin n+k θ dk
dt k f(θ) sin−n θ (5.29)
n = − l, f(θ) = sin l θ, k = l + m , (5.27)
√
Y lm (θ, φ) = (−1)l+m
2 l l!
√
= (−1) m
P m l (x) = 1
2 l l!
( ) l+m
2l + 1 (l − m)!
d
4π (l + m)! eimφ sin m θ
sin 2l θ
d cos θ
2l + 1 (l − m)!
4π (l + m)! eimφ Pl m (cos θ) (5.30)
(
1 − x
2 ) m/2 d l+m
dx l+m (x2 − 1) l , − l ≤ m ≤ l , |x| ≤ 1 (5.31)

5 84
Y lm (θ, φ) , Pl
m(x) ( Legendre ) , P l 0(x)
P l (x) P l (x) l
250 15.2 255 15.3
Y 00 = √ 1
√ √
3
3
, Y 1 ±1 = ∓
4π 8π sin θ e±iφ , Y 10 =
4π cos θ
Y 2 ±2 =
√
15
32π sin2 θ e ±2iφ ,
√
15
Y 2 ±1 = ∓
8π sin θ cos θ e± iφ , Y 20 =
√
5 (
3 cos 2 θ − 1 )
16π
|Y lm (θ, φ)| 2 φ z ,
l = 0 |Y 00 | 2 = 1/(4π)
() θ |Y lm (θ, φ)| 2 , l = 2, 3
( 1 0.5 )
θ
|Y lm | 2
r(t), p(t) , L cl
L cl = r(t)×p(t) L cl , r(t) L cl ,
L cl m = ± l , L cl z
xy m = ± l xy (5.26)
|Y l ±l | 2 ∝ sin 2l θ l → ∞ θ ≠ π/2 |Y l ±l | 2 = 0 , , xy
m 0 , L cl z
, |Y lm | 2 xy m = 0 L cl z
, |Y l 0 | 2 z
32π
15 |Y 2 ±2| 2 32π
15 |Y 2 ±1| 2 4π
5 |Y 2 0| 2
64π
35 |Y 3 ±3| 2 72π
35 |Y 3 ±2| 2 45π
28 |Y 3 ±1| 2 4π
7 |Y 3 0| 2
5.1
5.2
x , y , z rY 1m
ψ x (r) = xf(r) , ψ y (r) = yf(r) , ψ z (r) = zf(r)

5 85
(
1. L x ψ x (r) = − i y ∂ ∂z − z ∂ )
ψ x (r) ψ x (r) L x
∂y
ψ y (r) , ψ z (r) L y , L z
2. L y ψ x (r) = − i ψ z (r) , L z ψ x (r) = i ψ y (r) ψ y , ψ z ,
ψ x , ψ y , ψ z L 2
3. ψ x , ψ y , ψ z Y 1m
5.5
,
Hψ(r) = E ψ(r) ,
H = − 2
2m ∇2 + V (r)
{ ϕ n (r) } ψ(r) ϕ n (r)
|ψ〉 = ∑ n
c n | n 〉
, ϕ ∗ m(r)
∑
〈 m |H| n 〉c n = E ∑ 〈 m | n 〉 c n = E c m
n
n
, 〈 m | n 〉 = δ mn h mn = 〈 m |H| n 〉 ,
⎛
⎞ ⎛ ⎞ ⎛ ⎞
h 11 h 12 h 13 · · · c 1
c 1
⎜ h
⎝ 21 h 22 h 23 · · · ⎟ ⎜ c
⎠ ⎝ 2 ⎟
⎠ .
. . . ..
. = E ⎜ c
⎝ 2 ⎟
⎠
. (h mn ) , H
〈 m |H| n 〉 ∗ = 〈 n |H † | m 〉 = 〈 n |H| m 〉 h ∗ mn = h nm
, (h mn )
2 ψ α , ψ β A 〈ψ α |A|ψ β 〉
|ψ α 〉 = ∑ n
α n | n 〉 ,
|ψ β 〉 = ∑ n
β n | n 〉
∫
〈ψ α |A|ψ β 〉 =
d 3 r ψ ∗ α(r)Aψ β (r) = ∑ mn
α ∗ mβ n
∫
d 3 r ϕ ∗ m(r)Aϕ n (r)
= ∑ mn
α ∗ m〈 m |A| n 〉β n (5.32)
a mn = 〈 m |A| n 〉
⎛
〈ψ α |A|ψ β 〉 = ∑ αma ∗ mn β n = ( α1 ∗ α2 ∗ · · · ) ⎜
⎝
mn
⎞ ⎛
a 11 a 12 a 13 · · ·
a 21 a 22 a 23 · · · ⎟ ⎜
⎠ ⎝
.
. . . ..
β 1
β 2
.
⎞
⎟
⎠ (5.33)

5 86
,
⎛ ⎞
α =
⎜
⎝
α 1
α 2
.
⎟
⎠ ,
⎛
β = ⎜
⎝
β 1
β 2
.
⎞
⎟
⎠ ,
⎛
A = ⎜
⎝
⎞
a 11 a 12 a 13 · · ·
a 21 a 22 a 23 · · · ⎟
⎠
.
.
.
. ..
〈ψ α |A|ψ β 〉 = α † A β , 〈ψ α |ψ β 〉 = α † β (5.34)
→ , → , → (5.35)
, ,
{ϕ n } , ,
ϕ n :
⎛ ⎞
0
.
ϕ n →
1
⎜
⎝ .
⎟
⎠
0
n
{ϕ n }
, (5.32)
α ∗ m = 〈 m |ψ α 〉 ∗ = 〈ψ α | m 〉 , β n = 〈 n |ψ β 〉
〈ψ α |A|ψ β 〉 = ∑ mn〈ψ α | m 〉〈 m |A| n 〉〈 n |ψ β 〉
,
(∑ )
〈ψ α |A|ψ β 〉 = 〈ψ α | | m 〉〈 m | A
m
(∑
n
)
| n 〉〈 n | |ψ β 〉
∑
| n 〉〈 n | = 1 (5.36)
, (5.36) , (5.32)
∑
mn
α ∗ m〈 m |A| n 〉β n = ∑ mn
n
〈ψ α | m 〉〈 m |A| n 〉〈 n |ψ β 〉 = 〈ψ α |A|ψ β 〉 (5.37)
, (5.36) | a 〉
(∑ )
| a 〉 = | n 〉〈 n | | a 〉 = ∑ n
n
| n 〉〈 n | a 〉
(1.21) (5.36) (1.22)
∑
ϕ n (r) ϕ ∗ n(r ′ ) = δ(r − r ′ ) (5.38)
n

5 87
, { ϕ n (r) } ,
, (5.37)
∑
αm〈 ∗ m |A| n 〉β n
mn
= ∑ ∫
∫
∫
d 3 r 1 ψa(r ∗ 1 )ϕ m (r 1 ) d 3 r 2 ϕ ∗ m(r 2 )A(r 2 , p 2 )ϕ n (r 2 ) d 3 r 3 ϕ ∗ n(r 3 )ψ b (r 3 )
mn } {{ } } {{ } } {{ }
αm
∗ 〈 m |A| n 〉
β n
∫
(∑
) (∑
)
= d 3 r 1 d 3 r 2 d 3 r 3 ψa(r ∗ 1 ) ϕ m (r 1 )ϕ ∗ m(r 2 ) A(r 2 , p 2 ) ϕ n (r 2 )ϕ ∗ n(r 3 ) ψ b (r 3 )
m
(5.38)
∑
∫
αm〈 ∗ m |A| n 〉β n = d 3 r 1 d 3 r 2 d 3 r 3 ψa(r ∗ 1 )δ(r 1 − r 2 ) A(r 2 , p 2 ) δ(r 2 − r 3 ) ψ b (r 3 )
mn
∫ ∫
= d 3 r 2
∫
d 3 r 1 ψa(r ∗ 1 )δ(r 1 − r 2 ) A(r 2 , p 2 ) d 3 r 3 δ(r 2 − r 3 ) ψ b (r 3 )
∫
= d 3 r 2 ψa(r ∗ 2 )A(r 2 , p 2 )ψ b (r 2 ) = 〈ψ α |A|ψ β 〉
J 2 , J z |jm〉
| n 〉 = | j , m=j+1−n 〉 , n = 1, 2, · · · , 2j + 1
j J x
J x = (J + + J − )/2 (5.15), (5.16) J x
〈 n | J x | n ′ 〉 = 1 2 〈 m=j+1−n | J + | m=j+1−n ′ 〉 + 1 2 〈 m=j+1−n | J − | m=j+1−n ′ 〉
n
= 1 2√
(n′ − 1)(2j + 2 − n ′ ) 〈 m=j+1−n | m=j+2−n ′ 〉
+ 1 2√
n′ (2j + 1 − n ′ ) 〈 m=j+1−n | m=j−n ′ 〉
= 1 2
j = 1
〈 n | J x | n ′ 〉 = 1 2
〈 n | J x | 1 〉 = 1 2
√
(n′ − 1)(2j + 2 − n ′ ) δ n n′ −1 + 1 2
√
(n′ − 1)(4 − n ′ ) δ n n′ −1 + 1 2
√
n′ (2j + 1 − n ′ ) δ n n′ +1
√
n′ (3 − n ′ ) δ n n′ +1 , n , n ′ = 1, 2, 3
√
2 δn 2 , 〈 n | J x | 2 〉 = 1 2√
2
(δ n 1 + δ n 3
)
, 〈 n | J x | 3 〉 = 1 2√
2 δn 2
J x
⎛
1 ⎜
√ ⎝ 2
0 1 0
1 0 1
0 1 0
⎞
⎟
⎠

5 88
〈 n | J z | n ′ 〉 = 〈 m=j+1−n | J z | m=j+1−n ′ 〉 =
( )
j + 1 − n δ nn ′
J z
J x m x , | m x 〉
J x | m x 〉 = m x | m x 〉 , |m x 〉 =
2j+1
∑
n=1
c n |n〉 (5.39)
j = 1 ,
⎛
1 ⎜
√ ⎝ 2
0 1 0
1 0 1
0 1 0
⎞ ⎛
⎟ ⎜
⎠ ⎝
c 1
c 2
c 3
⎞ ⎛
⎟ ⎜
⎠ = m x ⎝
c 1
c 2
c 3
⎞
⎟
⎠
⎛
⎜
⎝
− m x 1/ √ ⎞ ⎛
2 0
1/ √ 2 − m x 1/ √ ⎟ ⎜
2
0 1/ √ ⎠ ⎝
2 − m x
c 1 = c 2 = c 3 = 0
− m x 1/ √ ∣
2 0 ∣∣∣∣∣∣ 1/ √ 2 − m x 1/ √ 2
∣ 0 1/ √ = −m 3 x + m x = 0
2 − m x
m x = 0 , ±1 J z
5.3 m x = 0, ±1
|m x =0〉 = √ 1
)
(|m=1〉 − |m=−1〉 , |m x =±1〉 = 1 2 2
c 1
c 2
c 3
⎞
⎟
⎠ = 0 (5.40)
(
|m=1〉 ± √ )
2 |m=0〉 + |m=−1〉
c 1 ,
5.6
l , (5.5) ,
j 1/2, 3/2, · · ·
,
j = 1/2 , ,
, , 1/2 , 1
, 3
, 3
l ,
J S S (5.5)
[S x , S y ] = iS z , [S y , S z ] = iS x , [S z , S x ] = iS y

5 89
S r ˆp
S s
S 2 |s m s 〉 = s(s + 1)|s m s 〉 , S z |s m s 〉 = m s |s m s 〉 , m s = −s, −s + 1, · · · , s (5.41)
S 2 , S z |s m s 〉 |s m s 〉 ,
|s m s 〉 r (5.41)
, ,
∑
m s
ψ ms (r, t) |s m s 〉
ψ ms (r, t) ψ ms (r, t) r ,
, ( s = 0 ) , ψ(r, t)
s = 1/2 m s ±1/2 2 ,
|+〉 = |s= 1 2 , m s = 1 2 〉 , |−〉 = |s= 1 2 , m s =− 1 2 〉
|+〉, |−〉 2 , 2
ψ + (r, t)|+〉 + ψ − (r, t)|−〉 (5.42)
S j = 1/2 S
S = 1 2 σ
σ S 2 |±〉 = 3 4 |±〉 σ2 |±〉 = 3 |±〉
, |ψ〉 = ψ + |+〉 + ψ − |−〉
σ 2 |ψ〉 = ψ + σ 2 |+〉 + ψ − σ 2 |−〉 = 3 |ψ〉
σ 2 = 3 σ 2 z|±〉 = |±〉 σ 2 z = 1 (5.15), (5.16) j = 1/2
( σ ± = σ x ± i σ y )
σ + |+〉 = 0 , σ + |−〉 = 2|+〉 , σ − |+〉 = 2|−〉 , σ − |−〉 = 0 (5.43)
σ 2 +|±〉 = 0 , σ 2 −|±〉 = 0
)
σ± 2 = σx 2 − σy 2 ± i
(σ x σ y + σ y σ x = 0 , ∴ σx 2 = σy 2 , σ x σ y + σ y σ x = 0
σ 2 x + σ 2 y = σ 2 − σ 2 z = 2 σ 2 x = σ 2 y = 1 σ x σ y + σ y σ x = 0
σ x σ y − σ y σ x = 2iσ z
σ x σ y = − σ y σ x = iσ z
σ x σ y = iσ z σ y σ z σ y = − iσ x
σ y σ z = − σ z σ y = iσ x ,
σ z σ x = − σ x σ z = iσ y

5 90
σ i σ j = δ ij + i ∑ k
ε ijk σ k (5.44)
( )
( )
1
0
|+〉 =⇒ , |−〉 =⇒
0
1
(5.45)
(5.42)
ψ(r, t) =
(
ψ + (r, t)
ψ − (r, t)
)
2 ψ(r, t) ,
(5.34)
∫
∫
d 3 r ψ † (r, t)ψ(r, t) =
∫
=
d 3 r ( ψ ∗ +(r, t) ψ ∗ −(r, t) ) ( ψ + (r, t)
ψ − (r, t)
(
d 3 r |ψ + (r, t)| 2 + |ψ − (r, t)| 2) = 1
|ψ + (r, t)| 2 d 3 r ( m s = 1/2 ) r d 3 r
|ψ − (r, t)| 2 d 3 r
, , ψ + (r, t) ψ − (r, t) r ,
( )
c + (t)
ψ(r, t)
(5.46)
c − (t)
c ± (t) r
(
|c+ (t)| 2 + |c − (t)| 2) ∫ d 3 r |ψ(r, t)| 2 = 1
)
|c + (t)| 2 + |c − (t)| 2 = 1 ,
∫
d 3 r |ψ(r, t)| 2 = 1
σ σ (5.45) σ
(
)
〈+|σ|+〉 〈+|σ|−〉
σ =
〈−|σ|+〉 〈−|σ|−〉
(5.43)
σ + =
(
0 2
0 0
)
, σ − =
(
0 0
2 0
)
, σ z =
(
1 0
0 −1
)
σ x = (σ + + σ − )/2, σ y = −i(σ + − σ − )/2
( ) ( ) (
0 1
0 − i
σ x =
, σ y =
, σ z =
1 0
i 0
1 0
0 −1
)
(5.47)
( Pauli matrix ) (5.44), (5.48)
2 × 2 1 , 2 × 2 I , , σ 2 x = I
, I 1 , σ 2 x = 1

5 91
5.4
(5.44)
σ·A σ·B = A·B + i σ·(A×B) (5.48)
A, B σ
5.5
n σ n ≡ n·σ σ 2 n = 1
exp(iθ σ n ) = cos θ + i σ n sin θ (5.49)
, A e A
e A =
∞∑
k=0
A k
k!
(5.50)
5.6
5.7
2 × 2 σ·A σ·B (5.48)
σ x σ y σ z = i , Tr σ k = 0 , Tr(σ i σ j ) = 2δ ij , det σ i = − 1
n σ n = n·σ n
n = e x sin θ cos φ + e y sin θ sin φ + e z cos θ
σ n = σ x sin θ cos φ + σ y sin θ sin φ + σ z cos θ =
σ n m , χ m =
(
cos θ − m
e iφ sin θ
(
(
cos θ
e iφ sin θ
e −iφ sin θ
− cos θ
)
c +
σ n χ m = m χ m
c −
e −iφ sin θ
− cos θ − m
) (
)
(5.51)
)
c +
= 0 (5.52)
c −
, c + = c − = 0
cos θ − m e −iφ sin θ
∣ e iφ sin θ − cos θ − m
∣ = m2 − 1 = 0
m = ±1 σ n σ 2 n = 1 , m m 2 = 1
, (5.52)
iφ m − cos θ
c − = c + e
sin θ
1 − cos θ = 2 sin 2 (θ/2), 1 + cos θ = 2 cos 2 (θ/2)
⎧
⎨ c + e iφ tan(θ/2) , m = 1
c − =
⎩ − c + e iφ cot(θ/2) , m = − 1

5 92
|c + | 2 + |c − | 2 = 1
⎧
⎨ e iα cos(θ/2) , m = 1
c + =
⎩ e iβ sin(θ/2) , m = − 1
α, β α = β = 0 , n·σ ± 1 χ ±
(
)
(
)
cos(θ/2)
sin(θ/2)
χ + (θ, φ) =
, χ
e iφ − (θ, φ) =
(5.53)
sin(θ/2)
− e iφ cos(θ/2)
,
χ † +χ − = ( cos(θ/2) e −iφ sin(θ/2) ) (
)
sin(θ/2)
= 0 ,
− e iφ cos(θ/2)
∴ χ † mχ m ′ = δ mm ′
χ + χ − , θ , φ r
, n
θ → π − θ , φ → π + φ n → − n
(−n)·σ χ + (π − θ, π + φ) = + χ + (π − θ, π + φ) , ∴ σ n χ + (π − θ, π + φ) = − χ + (π − θ, π + φ)
χ + (π − θ, π + φ) = e iα χ − (θ, φ) ,
α =
, (5.53) χ + (π − θ, π + φ) = χ − (θ, φ)
θ = 0 σ n = σ z , ± 1 σ z χ ± (z)
( )
( )
1
χ + (z) = , χ − (z) = − e iφ 0
0
1
− e iφ β = π − φ , θ = 0 , φ
n z φ = π
θ = π/2 , φ = 0 σ n = σ x , ± 1 σ x χ ± (x)
( )
χ ± (x) = √ 1 1
|+〉 ± |−〉
, √ (5.54)
2 ± 1
2
(5.47) σ x θ = π/2 , φ = π/2 ± 1
σ y χ ± (y)
(
χ ± (y) = √ 1
2
1
± i
)
,
|+〉 ± i |−〉
√
2
χ ± (θ, φ) σ 〈σ〉 ± , (5.33) 〈σ x 〉 + = χ † +σ x χ + (5.53)
〈σ x 〉 + = ( cos(θ/2) e −iφ sin(θ/2) ) ( 0 1
1 0
〈σ y 〉 + = sin θ sin φ , 〈σ z 〉 + = cos θ
) (
cos(θ/2)
e iφ sin(θ/2)
)
= sin θ cos φ
〈σ〉 ± = ± n (5.55)

5 93
χ − − n·σ + 1 , n − n χ ± σ
± n , , σ x , σ y , σ z , 3
n ′ σ n ′
χ ± (θ, φ) ∆σ ± (σ n ′) 2 = 1 , 〈σ n ′〉 ± = ± n ′·n
√
∆σ ± = 〈(σ n ′) 2 〉 ± − 〈σ n ′〉 2 ± = √ 1 − (n ′·n) 2
∆σ ± = 0 n ′·n = ± 1, , n ′ = ± n
= n ′ ·σ ,
, χ + (θ, φ) σ x 9 3 ′ ,
σ x +1 −1 9 3 ′ c i (t) ϕ i → χ ± (x) ,
ψ → χ + (θ, φ) (5.34) , ± 1 P ±
∣
P ± = ∣χ † ±(x) χ + (θ, φ) ∣ 2 (5.56)
(5.53), (5.54)
(
P ± = 1 2 ∣ ( 1 ± 1 ) cos(θ/2)
e iφ sin(θ/2)
)∣ ∣∣∣∣
2
=
1 ± sin θ cos φ
2
P + + P − = 1 χ + (θ, φ) σ x 〈σ x 〉 +
〈σ x 〉 + = (+1)×P + + (−1)×P − = sin θ cos φ
, 〈σ x 〉 + σ x 1
P + = 1 , P − = 0 ( sin θ cos φ = 1 ) P + = 0 , P − = 1 ( sin θ cos φ = − 1 )
n x , x ,
χ + (θ, φ) σ x
5.8
χ ±
χ + (θ, φ)χ † +(θ, φ) + χ − (θ, φ)χ † −(θ, φ) = 1 =
, χ
χ = χ + (θ, φ) χ † +(θ, φ)χ + χ − (θ, φ) χ † −(θ, φ)χ = c + χ + (θ, φ) + c − χ − (θ, φ)
c ± = χ † ±(θ, φ)χ
5.7
A(r) ( = − e ) H ,
H 0 = p 2 /2m p P ≡ p + (e/c)A
H = 1 (
p + e ) 2
2m c A , p = − i∇ (5.57)
, H 0 (5.48) ∇×∇ = 0
(σ·p) 2 = p 2 , H 0 = (σ·p) 2 /2m p → P
H = 1
2m (σ·P )2 = 1
2m P 2 +
i
2m σ·(P ×P ) , P = − i ∇ + e c A (5.58)

5 94
P ∇ A(r) ∇×∇ = 0 , A×A = 0
P ×P =
(− i ∇ + e )
c A ×
(− i ∇ + e )
c A
= − ie (
)
A×∇ + ∇×A
c
ψ(r)
(
)
A×∇ + ∇×A ψ = A×(∇ψ) + ∇×(Aψ) = A×(∇ψ) + (∇ψ)×A + (∇×A) ψ = B ψ
B(r) = (∇×A) , (5.58)
H = 1
2m P 2 + µ B σ·B(r) , µ B = e = (5.59)
2mc
(σ·p) 2 /2m p → P ,
,
B P 2 /2m , µ B σ·B r
, (5.59) ,
ϕ(r) r 2 χ
ψ(r) = ϕ(r) χ Hψ = Eψ
χ H r ϕ(r) + ϕ(r)H s χ = E ϕ(r) χ , H r = 1
2m P 2 ,
χ † χ † χ χ † H s χ
H s = µ B σ·B
H r ϕ(r) = E r ϕ(r) , H s χ = E s χ , E = E r + E s
5.10 , , B
r , H s r , ,
, B ,
( B = ∇×A = 0 ) , A ≠ 0
229
b B , B = |B| , σ b = b·σ H s = µ B B σ b σ b
± 1
H s χ ± (θ b , φ b ) = ± µ B B χ ± (θ b , φ b )
χ ± (5.53) , θ b , φ b B χ + ,
χ − P 2 /2m ϕ i (r),
E i , (5.59) H
E = E i ± µ B B , ψ(r) = ϕ i (r) χ ± (θ b , φ b )
µ B B
( B = 0 ), 2
, θ
E i b , φ b ,
2 ,
µ B B
, 2 E i ± µ B B
( ) B = 0 B ≠ 0
E i B
, B = 0
, 2

5 95
, H s = µ B σ·B
( )
i d dt χ(t) = H c + (t)
sχ(t) , χ(t) =
c − (t)
(5.60)
3
1. , ω = µ B B/ (5.51) (5.60)
( ) (
) (
i d c + (t)
cos θ b e −iφ b
sin θ b c + (t)
= ω
dt c − (t) e iφ b
sin θ b − cos θ b c − (t)
)
)
)
i ċ + = ω
(cos θ b c + + e −iφ b
sin θ b c − , i ċ − = ω
(e iφ b
sin θ b c + − cos θ b c −
1
c − (t) = eiφ b
sin θ b
( i ċ+
ω − c + cos θ b
)
2 ¨c + = − ω 2 c + H 2 s = (µ B B) 2
(5.61)
(5.62)
(i) 2 d2
dt 2 χ(t) = i d dt H sχ(t) = H 2 s χ(t) = (µ B B) 2 χ(t) , ∴ ¨c ± = − ω 2 c ±
, c ± a, b
(5.62)
c + (t) = a cos ωt + b sin ωt
c − (t) = eiφ b (
)
(ib − a cos θ b ) cos ωt + (−ia + b cos θ b ) sin ωt
sin θ b
t = 0 a, b c ± (0)
a = c + (0) ,
ib = c + (0) cos θ b + c − (0) sin θ b e −iφ b
(
c + (t)
c − (t)
)
=
(
) (
cos ωt − i sin ωt cos θ b − i sin ωt sin θ b e −iφ b
c + (0)
− i sin ωt sin θ b e iφ b
cos ωt + i sin ωt cos θ b c − (0)
)
(5.63)
2. (1.39) χ ± = χ ± (θ b , φ b ) , e ∓iωt χ ± (5.60)
, (5.60)
χ(t) = u e −iωt χ + + v e iωt χ − ,
u, v =
χ ± χ † +χ(0) = u , χ † −χ(0) = v
χ(t) =
(
)
e −iωt χ + χ † + + e iωt χ − χ † − χ(0) (5.64)
2×2 e −iωt χ + χ † + + e iωt χ − χ † − (5.53) (5.63)

5 96
3. , i d | t 〉 = H| t 〉 , H
dt
, (1.42) | t 〉 = e −iHt/ | t = 0 〉 , ()
(5.50) H i d dt e−iHt/ = He −iHt/ , | t 〉 = e −iHt/ | t = 0 〉
,
, H = ω σ b (5.49)
| t 〉 = exp (−iωt σ b ) | t = 0 〉 =
(
)
cos ωt − iσ b sin ωt | t = 0 〉
σ b (5.51) (5.63) , e −iH st/ χ ± = e ∓iωt χ ± (5.64)
χ(t) = e −iHst/ ( χ + χ † + + χ − χ † −
, 5.8
, z ( θ b = 0 ) (5.63)
)
χ(0) = e −iHst/ χ(0)
c + (t) = e −iωt c + (0) , c − (t) = e iωt c − (0)
( (5.61) θ b = 0 , ) (5.53) φ = 0
c + (0) = cos(θ/2) , c − (0) = sin(θ/2) , θ
χ(t) =
(
c + (t)
c − (t)
)
=
(
e −iωt cos(θ/2)
e iωt sin(θ/2)
)
(5.65)
(5.53) χ + χ(t) = e −iωt χ + (θ, 2ωt)
, t σ 〈σ〉 (5.55) φ = 2ωt
〈σ〉 = χ † (t)σχ(t) = χ † +(θ, φ)σχ + (θ, φ) = n
cos θ
B
〈σ〉
〈σ x 〉 = sin θ cos 2ωt , 〈σ y 〉 = sin θ sin 2ωt , 〈σ z 〉 = cos θ
2ωt
〈σ〉 σ ( z )
2ω ( Larmor precession )
5.9
χ + , (5.65) t 〈σ〉
5.10
B B = (0, 0, B) B z
∇×A = B A
A(r) = 1 2 B×r = 1 ( −By, Bx, 0 ) , A(r) = ( −By, 0, 0 )
2
A(r) = ( −By, 0, 0 )
H = 1 (p + e ) 2
2m c A 2
= −
2m
[ ( ) ]
2 ∂
∂x − ieB c y + ∂2
∂y 2 + ∂2
∂z 2

5 97
H p y p x , p z , H ψ(r)
p x , p z ψ(r) = e i(k xx+k z z) ϕ(y) H E
(
E = ω L n + 1 )
+ 2 kz
2
2 2m , ω L = eB , n = 0, 1, 2, · · ·
mc
xy
5.11
A(r) = 1 ( −By, Bx, 0 ) , xy 2
2
130 6.9 ,
5.12 B = ∇×A 2 A 1 (r) , A 2 (r)
[ 1
(
p + e ) ]
[ 2 1
(
2m c A 1 + V (r) ψ 1 (r) = E ψ 1 (r) , p + e ) ]
2
2m c A 2 + V (r) ψ 2 (r) = E ψ 2 (r)
∇×(A 2 − A 1 ) = 0 A 2 (r) = A 1 (r) + ∇f(r)
(
ψ 2 (r) = exp − ie )
c f(r) ϕ 2 (r)
ϕ 2 (r) ψ 1 (r) , A
2 , ,
( )
z xy
B(t) = ( B 1 cos ωt , B 1 sin ωt , B 0 )
ω 0 = µ B B 0 / , ω 1 = µ B B 1 /
(
)
)
ω 0 ω 1 e
H(t) = µ B B·σ = ω 1
(σ −iωt
x cos ωt + σ y sin ωt + ω 0 σ z =
ω 1 e iωt − ω 0
(5.66)
H 2., 3. H(t)
(
)
cos θ b sin θ b e −iωt
H(t) = Ω
, cos θ
sin θ b e iωt b = ω 0
− cos θ b Ω , sin θ b = ω 1
Ω ,
Ω = √
ω 2 0 + ω2 1
(5.51) θ = θ b , φ = ωt , H(t)
± Ω , χ ± (θ b , ωt) , 2. e ∓iΩt χ ± (θ b , ωt)
(i d dt − H )
e ∓iΩt χ ± (θ b , ωt) = i e ∓iΩt d dt χ ±(θ b , ωt) ≠ 0
, ( 170 )
i d dt
(
c + (t)
c − (t)
H (5.66)
)
= H
(
c + (t)
c − (t)
i ċ + = ω 0 c + + ω 1 e −iωt c − , i ċ − = ω 1 e iωt c + − ω 0 c − (5.67)
)

5 98
2
1
c + = e−iωt
ω 1
(
i ċ − + ω 0 c −
)
)
¨c − − iωċ − +
(ω0 2 + ω1 2 − ωω 0 c − = 0
(5.68)
√
2 x 2 − iωx + ω0 2 + ω1 2 − ωω 0 = 0 i ω/2 ± i (ω 0 − ω/2) 2 + ω1 2 ,
a , b
(5.68)
c − (t) = e iωt/2( a e iγt + b e −iγt) , γ =
c + (t) = e −iωt/2 (
ω0 − ω/2 − γ
ω 1
√
(ω 0 − ω/2) 2 + ω 2 1
a e iγt + ω 0 − ω/2 + γ
ω 1
b e −iγt )
t = 0 z c + (0) = 1 , c − (0) = 0
a = − b = − ω 1 /(2γ)
c − (t) = − i ω (
1
γ eiωt/2 sin γt , c + (t) = e −iωt/2 cos γt − i ω )
0 − ω/2
sin γt
γ
z P − (t) , (5.56)
( )∣ P − (t) =
∣ ( 0 1 ) c + (t) ∣∣∣∣
2
= |c − (t)| 2
c − (t)
(5.69)
P − (t) = A sin 2 (√
(ω 0 − ω/2) 2 + ω 2 1 t )
, A =
ω 2 1
(ω 0 − ω/2) 2 + ω 2 1
(5.70)
( ω 1 = 0 ), P − = 0
1.0
, ω 1 ≠ 0 ω = 2ω 0
P − (t) = sin 2 ω 1 t
A
, 0.5
( ω 1 ≪ ω 0 ) |ω − 2ω 0 | ≤ 2ω 1
0.01 0.2
A ≥ 1/2 , ω 1 ≪ ω 0 ,
ω ω ≈ 2ω 0 0.0
1 2 ω/ω 0
ω 1 /ω 0 = 0.01, 0.2 A(ω) , ω 0 = µ B B 0 / ,
µ B
5.13 H (5.66) ,
1. D(θ) = exp(−iθσ z /2) (5.49)
D(θ) σ x D † (θ) = σ x cos θ + σ y sin θ
( z θ ) (5.66)
)
H = D(ωt)
(ω 1 σ x + ω 0 σ z D † (ωt)

[ J 1i , J 1j ] = i ∑ k
[ J 2i , J 2j ] = i ∑ k
5 99
2. t | t 〉 i d dt | t 〉 = H| t 〉 | ˜t 〉 = D † (ωt)| t 〉
d
dt | ˜t 〉 = − iS| ˜t 〉 ,
S = ω 1 σ x + (ω 0 − ω/2) σ z
S | ˜t 〉 = e −itS | t = 0 〉
3. n
n = ω 1
γ e x + ω √
0 − ω/2
e z , γ = (ω 0 − ω/2) 2 + ω1
2 γ
S = γ n·σ
(
)
| t 〉 = D(ωt) cos γt − i n·σ sin γt | t = 0 〉
c ± (t) (5.69)
5.8
2 1, 2 J 1 , J 2
(5.5)
ε ijk J 1k ,
ε ijk J 2k
2 [ J 1i , J 2j ] = 0 , 2
L 1 L 2 , S , L
S , [ L i , S j ] = 0
,
J 1 + J 2 ,
, ,
|j 1 m 1 , j 2 m 2 〉 ≡ |j 1 m 1 〉|j 2 m 2 〉
, |j 1 m 1 〉, |j 2 m 2 〉 J 1 , J 2
J 2 1 |j 1 m 1 〉 = j 1 (j 1 + 1)|j 1 m 1 〉 , J 1z |j 1 m 1 〉 = m 1 |j 1 m 1 〉
J 2 2 |j 2 m 2 〉 = j 2 (j 2 + 1)|j 2 m 2 〉 , J 2z |j 2 m 2 〉 = m 2 |j 2 m 2 〉
2 , J 1 J 2 |j 2 m 2 〉 , J 2
J 1 |j 1 m 1 〉
( )( )
J 1x J 2x |j 1 m 1 , j 2 m 2 〉 = J 1x |j 1 m 1 〉 J 2x |j 2 m 2 〉
(J 1x + J 2x ) |j 1 m 1 , j 2 m 2 〉 =
( )
( )
J 1x |j 1 m 1 〉 |j 2 m 2 〉 + |j 1 m 1 〉 J 2x |j 2 m 2 〉
J J = J 1 + J 2 J (5.5)
[ J x , J y ] = [ J 1x + J 2x , J 1y + J 2y ] = [ J 1x , J 1y ] + [ J 2x , J 2y ] = i (J 1z + J 2z ) = i J z

5 100
, J = J 1 + J 2 , J 2 J z
|jm〉 :
J 2 |jm〉 = j(j + 1) |jm〉 ,
J z |jm〉 = m |jm〉
J z J z = J 1z + J 2z
J z |j 1 m 1 , j 2 m 2 〉 =
( )
( )
J 1z |j 1 m 1 〉 |j 2 m 2 〉 + |j 1 m 1 〉 J 2z |j 2 m 2 〉 = (m 1 + m 2 ) |j 1 m 1 , j 2 m 2 〉
, J z , J 2 = J 2 1 + J 2 2 + 2J 1·J 2 J 2 1 + J 2 2
, |j 1 m 1 , j 2 m 2 〉 J 2
|j 1 m 1 , j 2 m 2 〉 = |m 1 , m 2 〉 , |j 1 m 1 〉 = |m 1 〉 , |j 2 m 2 〉 = |m 2 〉
J 2 |m 1 , m 2 〉 = ( J1 2 + 2J 1·J 2 + J2 2 )
|m1 , m 2 〉
(
)
= j 1 (j 1 + 1) + j 2 (j 2 + 1) |m 1 , m 2 〉 + 2J 1·J 2 |m 1 , m 2 〉
J 1x J 2x + J 1y J 2y J 1± = J 1x ± iJ 1y , J 2± = J 2x ± iJ 2y
)
2J 1·J 2 = 2
(J 1x J 2x + J 1y J 2y + J 1z J 2z = J 1+ J 2− + J 1− J 2+ + 2J 1z J 2z
(5.15), (5.16)
2J 1·J 2 |m 1 , m 2 〉 = (J 1+ |m 1 〉) (J 2− |m 2 〉) + (J 1− |m 1 〉)(J 2+ |m 2 〉) + 2 (J 1z |m 1 〉)(J 2z |m 2 〉)
= c + (j 1 , m 1 )c − (j 2 , m 2 ) |m 1 +1〉|m 2 −1〉 + c − (j 1 , m 1 )c + (j 2 , m 2 ) |m 1 −1〉|m 2 +1〉
+ 2m 1 m 2 |m 1 〉|m 2 〉 , c ± (j, m) = √ (j ∓ m)(j ± m + 1) (5.71)
J 2 |m 1 , m 2 〉 =
(j 1 (j 1 + 1) + j 2 (j 2 + 1) + 2m 1 m 2
)
|m 1 , m 2 〉
+ c + (j 1 , m 1 )c − (j 2 , m 2 ) |m 1 +1〉|m 2 −1〉
+ c − (j 1 , m 1 )c + (j 2 , m 2 ) |m 1 −1〉|m 2 +1〉 (5.72)
1 |m 1 , m 2 〉 J 2 ,
|jm〉 |m 1 , m 2 〉 c(m 1 , m 2 )
|jm〉 =
j 1
∑
m 1 =−j 1
, J z
J z |jm〉 =
∑
j 2
∑
m 1,m 2
c(m 1 , m 2 ) (J 1z + J 2z ) |m 1 , m 2 〉 =
m 2 =−j 2
c(m 1 , m 2 ) |m 1 , m 2 〉 (5.73)
∑
m 1,m 2
c(m 1 , m 2 ) (m 1 + m 2 ) |m 1 , m 2 〉

5 101
, m 1 + m 2 = = m |m 1 , m 2 〉
∑
m 1 ,m 2
m 1+m 2=m
J z
J 2 |jm〉 =
|jm〉 =
∑
m 1 ,m 2
m 1+m 2=m
c(m 1 , m 2 ) (m 1 + m 2 ) |m 1 , m 2 〉 = m
∑
m 1 ,m 2
m 1+m 2=m
c(m 1 , m 2 )|m 1 , m 2 〉 (5.74)
∑
m 1 ,m 2
m 1+m 2=m
c(m 1 , m 2 ) J 2 |m 1 , m 2 〉 = j(j + 1)
c(m 1 , m 2 )|m 1 , m 2 〉 = m |jm〉
∑
m 1 ,m 2
m 1+m 2=m
c(m 1 , m 2 ) |m 1 , m 2 〉
c(m 1 , m 2 ) (5.72) c(m 1 , m 2 )
, c(m 1 , m 2 )
j 12 = j 1 + j 2
|m 1 | ≤ j 1 , |m 2 | ≤ j 2 , J z m = m 1 + m 2 |m| ≤ j 12 m
j 12 j j 12 m = j 12 m 1 , m 2 m 1 = j 1 ,
m 2 = j 2 , (5.74) 1
|j =j 12 , m=j 12 〉 = |m 1 =j 1 , m 2 =j 2 〉 (5.75)
m 1 = j 1 , m 2 = j 2 ( m 1 = −j 1 , m 2 = −j 2 ) , 1 J 2
, (5.72) m 1 = j 1 , m 2 = j 2 , (5.72) 2 3 0
, |m 1 =j 1 , m 2 =j 2 〉 J 2
j = j 1 + j 2
J 2 | m 1 =j 1 , m 2 =j 2 〉 = (j 1 + j 2 ) (j 1 + j 2 + 1) | m 1 =j 1 , m 2 =j 2 〉
j = j 12 |jm〉 |j 1 m 1 , j 2 m 2 〉 , (5.75) (5.75)
J − = J 1− + J 2−
J − |j =j 12 , m=j 12 〉 = (J 1− + J 2− ) | m 1 =j 1 , m 2 =j 2 〉
(5.16) j = m = j 12
,
(J 1− + J 2− ) | m 1 =j 1 , m 2 =j 2 〉 =
J − |j =j 12 , m=j 12 〉 = √ 2j 12 |j =j 12 , m=j 12 −1〉
(
)
(
)
J 1− | m 1 =j 1 〉 | m 2 =j 2 〉 + | m 1 =j 1 〉 J 2− | m 2 =j 2 〉
= √ 2j 1 | m 1 =j 1 −1 , m 2 =j 2 〉 + √ 2j 2 | m 1 =j 1 , m 2 =j 2 −1〉
√
√
j 1
j 2
|j =j 12 , m=j 12 −1〉 = |m 1 =j 1 −1 , m 2 =j 2 〉 + |m 1 =j 1 , m 2 =j 2 −1〉 (5.76)
j 12 j 12

5 102
J − = J 1− + J 2− |j =j 12 , m=j 12 −2 〉
m = j 12 , · · · , − j 12 |j =j 12 , m〉 |m 1 , m 2 〉
j 1 |j =j 12 −1 , m 〉 m m = j 12 − 1 J z
j 12 − 1 |m 1 , m 2 〉 |m 1 =j 1 , m 2 =j 2 −1〉 |m 1 =j 1 −1 , m 2 =j 2 〉 2
|j =j 12 −1 , m=j 12 −1〉 = c 1 |m 1 =j 1 −1 , m 2 =j 2 〉 + c 2 |m 1 =j 1 , m 2 =j 2 −1〉
|j =j 12 , m=j 12 −1〉 J 2 (5.76)
c 1
√
j1 /j 12 + c 2
√
j2 /j 12 = 0 |c 1 | 2 + |c 2 | 2 = 1
√
√
j 2
j 1
|j =j 12 −1 , m=j 12 −1〉 = − |m 1 =j 1 −1 , m 2 =j 2 〉 + |m 1 =j 1 , m 2 =j 2 −1〉 (5.77)
j 12 j 12
|j = j 12 −1 , m = j 12 −1〉 , J −
|j =j 12 −1 , m〉 |m 1 , m 2 〉
j = j 12 −n , n = 1, 2, · · ·
• |jm〉
, J − ,
m
j 12
j 12 −1
j 12 −2
j 12
j 12 −1
j 12 −2
2 1 j = m = j 12
, (5.75) J −
, j = j 12 1 ,
j = m = j 12 − 1 j = j 12 , m = j 12 − 1
, J − , j = j 12 − 1
2 , j = m = j 12 − 2 ,
j = j 12 , j = j 12 − 1 m = j 12 −2 2
−j 12 +2
−j 12 +1
, j = m = j 12 −n j > j 12 − n m = j 12 −n n
j j j min j |jm〉 2j + 1
j = j min , j min + 1, · · · , j 1 + j 2 ,
N =
j∑
1+j 2
j=j min
(2j + 1)
j 1 + j 2 ( )
N =
∑
(2j + 1) −
j 1 +j 2
j=0
∑
j min −1
j=0
(2j + 1)
−j 12
)
= (j 1 + j 2 )(j 1 + j 2 + 1) + j 1 + j 2 + 1 −
(j min (j min − 1) + j min
= (j 1 + j 2 + 1) 2 − jmin
2
, |jm〉 |j 1 m 1 , j 2 m 2 〉 ,
|j 1 m 1 , j 2 m 2 〉 (2j 1 + 1)(2j 2 + 1)
(j 1 + j 2 + 1) 2 − jmin 2 = (2j 1 + 1)(2j 2 + 1) , j min = |j 1 − j 2 |

5 103
j
j = |j 1 − j 2 |, |j 1 − j 2 | + 1, · · · , j 1 + j 2 (5.78)
5.14
(5.77) J + = J 1+ + J 2+ 0
c(m 1 , m 2 ) (5.74) , J 2 , J z , J 2 1 ,
J2 2
J1 2 |jm〉 =
∑
c(m 1 , m 2 ) J1 2 |m 1 , m 2 〉 =
m 1 ,m 2
m 1 +m 2 =m
∑
m 1 ,m 2
m 1 +m 2 =m
c(m 1 , m 2 ) j 1 (j 1 + 1) |m 1 , m 2 〉 = j 1 (j 1 + 1) |jm〉
J 2 2 |jm〉 = j 2 (j 2 + 1) |jm〉 J 2 , J z , J 2 1 , J 2 2
, J 2 1 J 1
[ J 2 , J 2 1 ] = [ J 2 1 + 2J 1·J 2 + J 2 2 , J 2 1 ] = 2[J 1·J 2 , J 2 1 ] = 2 ∑ k
[J 1k , J 2 1 ]J 2k = 0
[ J 2 , J 2 2 ] = 0
J 2 J 1 , J 2
[ J 2 , J 1z ] = 2 [ J 1·J 2 , J 1z ] = 2 [ J 1x , J 1z ]J 2x + 2 [ J 1y , J 1z ]J 2y = i (J 1x J 2y − J 1y J 2x ) ≠ 0
, J 2 J 1z , J 2z |m 1 , m 2 〉 J 1z , J 2z
J 2 , , |jm〉 J 2 , J 1z , J 2z
( J z = J 1z + J 2z )
|m 1 , m 2 〉 J 2 |jm〉 , (5.76) (5.77)
|m 1 =j 1 −1 , m 2 =j 2 〉
√
√
j 1
j 2
= |j =j 1 +j 2 , m=j 1 +j 2 −1〉 − |j =j 1 +j 2 −1 , m=j 1 +j 2 −1〉
j 1 + j 2 j 1 + j 2
|m 1 , m 2 〉 |jm〉 ,
, ( , )
(5.73) c(m 1 , m 2 ) (Clebsch–Gordan)
〈 j 1 m 1 j 2 m 2 | jm 〉
|jm〉 =
∑
〈 j 1 m 1 j 2 m 2 | jm 〉 |j 1 m 1 , j 2 m 2 〉
m 1,m 2
CG
CG
m = m 1 + m 2 |j 1 − j 2 | ≤ j ≤ j 1 + j 2 〈 j 1 m 1 j 2 m 2 | jm 〉 = 0 CG
|jm〉 |j 1 m 1 , j 2 m 2 〉
〈jm|j ′ m ′ 〉 =
∑
〈 j 1 m 1 j 2 m 2 | jm 〉〈 j 1 m ′ 1 j 2 m ′ 2 | j ′ m ′ 〉〈j 1 m 1 , j 2 m 2 |j 1 m ′ 1 , j 2 m ′ 2〉
m 1 ,m 2
m ′ 1 ,m′ 2
= ∑
m 1 ,m 2
〈 j 1 m 1 j 2 m 2 | jm 〉〈 j 1 m 1 j 2 m 2 | j ′ m ′ 〉 = δ jj ′ δ mm ′ (5.79)

5 104
| jm 〉
〈 j 1 m 1 j 2 m 2 | j 1 m ′ 1 j 2 m ′ 2 〉 = δ m1 m ′ 1 δ m 2 m ′ 2
∑
〈 j 1 m 1 j 2 m 2 | jm 〉〈 j 1 m ′ 1 j 2 m ′ 2 | jm 〉 = δ m1 m ′ δ 1 m 2 m ′ (5.80)
2
j m
J ± |jm〉 = (J 1± + J 2± ) ∑ 〈 j 1 m 1 j 2 m 2 | jm 〉 |j 1 m 1 , j 2 m 2 〉
= ∑ (( )
( ))
〈 j 1 m 1 j 2 m 2 | jm 〉 J 1± |j 1 m 1 〉 |j 2 m 2 〉 + |j 1 m 1 〉 J 2± |j 2 m 2 〉
(5.15), (5.16)
√
(j ∓ m)(j ± m + 1) 〈 j1 m 1 j 2 m 2 | j m±1 〉
= √ (j 1 ∓ m 1 + 1)(j 1 ± m 1 ) 〈 j 1 m 1 ∓1 j 2 m 2 | jm 〉
+ √ (j 2 ∓ m 2 + 1)(j 2 ± m 2 ) 〈 j 1 m 1 j 2 m 2 ∓1 | jm 〉 (5.81)
, 〈 j 1 j 1 j 2 j 2 | j 1 +j 2 j 1 +j 2 〉 = 1 CG
, CG m = m 1 + m 2
〈 j 1 m 1 j 2 m 2 | jm 〉
√
(2j 1 + 1)(j 1 + j 2 − j)! (j 1 − j 2 + j)! (−j 1 + j 2 + j)!
=
(j 1 + j 2 + j + 1)!
× √ (j 1 + m 1 )! (j 1 − m 1 )! (j 2 + m 2 )! (j 2 − m 2 )! (j + m)! (j − m)!
× ∑ k
(−1) k
k! (j 1 + j 2 − j − k)! (j 1 − m 1 − k)! (j 2 + m 2 − k)! (j − j 2 + m 1 + k)! (j − j 1 − m 2 + k)!
, k
j 1 = j 2 = 1/2
|j 1 = 1 2 m 1 =± 1 2 〉 = |±〉 1
|+〉 1 |+〉 2 = | + + 〉 , |+〉 1 |−〉 2 = | + − 〉
j = j 1 + j 2 , · · · , |j 1 − j 2 | j = 1, j = 0 j = 1
(5.76) j 1 = j 2 = 1/2
|j =1 m=1 〉 = | + + 〉
|j =1 m=0 〉 = | − + 〉 + | + − 〉 √
2
(5.82)
J −
J − |j =1 m=0 〉 = √ (1 + 0)(1 − 0 + 1) |j =1 m=−1 〉 = √ 2 |j =1 m=−1 〉

5 105
, J 1− |−〉 1 = 0, J 2− |−〉 2 = 0, J 1− |+〉 1 = |−〉 1 , J 2− |+〉 2 = |−〉 2 (5.82)
) )
)
(J 1− + J 2− )
(|−〉 1 |+〉 2 + |+〉 1 |−〉 2 = |−〉 1
(J 2− |+〉 2 +
(J 1− |+〉 1 |−〉 2 = 2 | − − 〉
|j =1 m=−1 〉 = | − − 〉
m = 0 j = 0 (5.82)
|j =0 m=0 〉 = | + − 〉 − | − + 〉 √
2
|j 1 m 1 〉|j 2 m 2 〉 2 × 2 = 4 |jm〉 4 , 3 j = 1,
j = 0
|j =0 m=0 〉 = | + − 〉 − | − + 〉 √
2
, |j =1 m 〉 =
⎧
⎪⎨
| + + 〉 , m = 1
| − + 〉 + | + − 〉
√
2
, m = 0
⎪⎩
| − − 〉 , m = −1
j = 0 1 1 , j = 1 3 3
, 2
H A, B
H = AJ 1·J 2 + B (J 1z + J 2z )
, H , |m 1 , m 2 〉
| jm 〉 (5.72) , J 1·J 2 |m 1 , m 2 〉
J 2 |jm〉 = j(j + 1) |jm〉 , J 2 1 |jm〉 = j 1 (j 1 + 1) |jm〉 , J 2 2 |jm〉 = j 2 (j 2 + 1) |jm〉
2J 1·J 2 |jm〉 = ( J 2 − J1 2 − J2
2 ) (
)
|jm〉 = j(j + 1) − j 1 (j 1 + 1) − j 2 (j 2 + 1) |jm〉
H |jm〉 = E jm |jm〉 , E jm = A 2
, |jm〉 H j 1 = j 2 = 1/2
E jm = A 2
⎧
(
j(j + 1) − 3 ) ⎪⎨
+ Bm =
2
⎪⎩
(
)
j(j + 1) − j 1 (j 1 + 1) − j 2 (j 2 + 1) + Bm
A
4
+ Bm , j = 1
− 3 4 A , j = 0 (5.83)
B = 0 1 3 , 3 3 B ≠ 0
3

5 106
j 1 = j 2 = 1/2 , 4 ,
H (5.71)
J 1·J 2 | ± ± 〉 = 1 4 | ± ± 〉 , J 1·J 2 | ± ∓ 〉 = 1 2 | ∓ ± 〉 − 1 4 | ± ∓ 〉
H| ± ± 〉 =
( A
4 ± B )
| ± ± 〉 , H| ± ∓ 〉 = A 2 | ∓ ± 〉 − A 4 | ± ∓ 〉
| ± ± 〉 H A/4 ± B (5.83) j = 1 ,
m = ± 1 , | ± ∓ 〉 H 2
c 1 , c 2
H|ψ〉 = E|ψ〉 , |ψ〉 = c 1 | + − 〉 + c 2 | − + 〉
H|ψ〉 = c 1
( A
2 | − + 〉 − A 4 | + − 〉 )
+ c 2
( A
2 | + − 〉 − A 4 | − + 〉 )
= A 4
(
) (2c 2 − c 1 | + − 〉 + A ) (2c 1 − c 2 | − + 〉
4
A
) (2c 2 − c 1 = Ec 1 ,
4
−A/4 − E A/2
A/2 −A/4 − E
A
) (2c 1 − c 2 = Ec 2 (5.84)
4
) (
)
c 1
= 0
c 2
−A/4 − E A/2
∣ A/2 −A/4 − E ∣ = (A/4 + E)2 − (A/2) 2 = 0 , ∴ E = − A 4 ± A 2
E = − 3A/4 (5.84) c 2 = − c 1 |c 1 | 2 + |c 2 | 2 = 2|c 1 | 2 = 1 c 1 = 1/ √ 2
|ψ〉 = | + − 〉 − | − + 〉 √
2
, E = − 3 4 A
(5.83) j = 0 E = A/4 c 2 = c 1
(5.83) j = 1 , m = 0
|ψ〉 = | + − 〉 + | − + 〉 √
2
, E = A 4
5.15 j 1 = 1, j 2 = 1 |jm〉 |j 1 m 1 〉|j 2 m 2 〉 |j =0 , m=
0 〉 |j =2 , m=0 〉 , |j =1 , m=0 〉
5.16 5.15 m 1 , m 2 = 0, ±1 3×3 = 9 |j 1 m 1 〉|j 2 m 2 〉
m = 0 (m 1 , m 2 ) = (1, −1), (−1, 1), (0, 0) 3
| 1 〉 = | m 1 =1 〉| m 2 =−1 〉 , | 2 〉 = | m 1 =−1 〉| m 2 =1 〉 , | 3 〉 = | m 1 =0 〉| m 2 =0 〉

5 107
1. (5.72) J 2 | k 〉 , k = 1, 2, 3
2. m = 0 c k
3∑
c k | k 〉
k=1
J 2 5.15 m = 0
,
5.17 L 1/2
J = L + S J 2 J z |jm〉 L 2 , L z | l m l 〉
| ± 〉 | l m l 〉 Y l ml (θ, φ) 5.8 J −
, | ± 〉 2
l ≠ 0
1. J z = L z + S z m | l m− 1 2 〉|+〉 | l m+ 1 2
〉|−〉 2
(5.72)
J 2 |l m− 1 2 〉|+〉 = (¯l2 + m ) |l m− 1 2
√¯l2 〉|+〉 + − m 2 |l m+ 1 2 〉|−〉
J 2 |l m+ 1 2 〉|−〉 = (¯l2 − m ) |l m+ 1 2
√¯l2 〉|−〉 + − m 2 |l m− 1 2 〉|+〉
¯l = l + 1/2
2. 1. 2
|jm〉 = c 1 |l m− 1 2 〉|+〉 + c 2 |l m+ 1 2 〉|−〉
c 1 , c 2 J z , c 1 , c 2
J 2 |jm〉 = j(j + 1) |jm〉
j = l ± 1/2 , , |jm〉
| j =l± 1 2 m 〉 = ± √
l ± m + 1/2
2l + 1
|l m− 1 2 〉|+〉 + √
l ∓ m + 1/2
2l + 1
|l m+ 1 2 〉|−〉
c 2 > 0 2
| j =l± 1 2 m 〉 ⎛
⎜
⎝
√
l ± m + 1/2
±
Y l m−1/2 (θ, φ)
2l + 1
√
l ∓ m + 1/2
Y l m+1/2 (θ, φ)
2l + 1
⎞
⎟
⎠
5.9
1. ,
2. ,

5 108
2 ,
1.
n θ θ > 0 n
a a ′ , θ = ε
a ′ = a + ε n×a (5.85)
, z
a ′ x = a x cos ε − a y sin ε = a x − a y ε , a ′ y = a x sin ε + a y cos ε = a x ε + a y , a ′ z = a z
a ′ = a + ε (−a y , a x , 0) = a + ε n×a , n = (0, 0, 1)
,
ψ(r), ˜ψ(r)
ψ(r) ˜ψ(r) R
˜ψ(r) = R(n, θ) ψ(r)
r ,
r r 1 :
˜ψ(r) = ψ(r 1 )
r
θ
r 1
˜ψ
ψ
r 1 r (5.85) r 1 = r − ε n×r
˜ψ(r) = ψ(r − ε n×r) = ψ(r) − ε (n×r)·∇ψ(r)
(n×r)·∇ = n·(r×∇)
(
)
˜ψ(r) = 1 − εn·(r×∇) ψ(r) =
( )
1 − iεn·L ψ(r)
L ,
R(n, ε) = 1 − iεn·L
,
J , n
R(n, ε) = 1 − iεn·J (5.86)
θ θ N ( N → ∞ ),
θ θ/N N
(
R(n, θ) = lim 1 − iθn·J ) N
= exp (−iθ n·J) (5.87)
N→∞ N
, θ + dθ θ dθ
R(n, θ + dθ) = R(n, dθ)R(n, θ) = (1 − i dθ n·J) R(n, θ)

5 109
dR(n, θ)
dθ
= −in·JR(n, θ)
, R(n, 0) = 1 (5.87) J
R † (n, θ) = exp (+ iθ n·J)
RR † = R † R = 1 (5.88)
R ( unitary operator ) | α 〉 | α ′ 〉
| α ′ 〉 = R| α 〉
〈 α ′ | α ′ 〉 = 〈 α |R † R| α 〉 = 〈 α | α 〉
, S ,
〈 α |S| α 〉 = 〈 α ′ |S| α ′ 〉 = 〈 α | (1 + iεn·J) S (1 − iεn·J) | α 〉
(
)
= 〈 α | S + iε [n·J , S ] | α 〉
[ S , J ] = 0 (5.89)
J
V ,
(5.85) (5.86)
〈 α ′ |V | α ′ 〉 = 〈 α |V | α 〉 + ε n×〈 α |V | α 〉 , | α ′ 〉 = (1 − iεn·J) | α 〉
(1 + iεn·J) V (1 − iεn·J) = V + ε n×V
[ V , n·J ] = i n×V
n j ( n m = δ jm ),
[ V i , J j ] = i (n×V ) i = i ∑ mk
ε imk n m V k = i ∑ k
ε ijk V k (5.90)
V = J , (5.86)
H J H, J 2 , J z |n j m〉 ,
E njm :
H|n j m〉 = E njm |n j m〉 , J 2 |n j m〉 = j(j + 1)|n j m〉 , J z |n j m〉 = m |n j m〉

5 110
n j, m [ H , J ± ] = 0
HJ ± |n j m〉 = J ± H|n j m〉 = E njm J ± |n j m〉
J ± |n j m〉 ∝ J ± |n j m±1 〉
H|n j m±1〉 = E njm |n j m±1〉
E njm±1 = E njm m j
H 2j + 1
, ψ(r) a ˜ψ(r)
˜ψ(r) = ψ(r − a)
ψ(r − a) = ψ(x − a x , y − a y , z − a z ) =
p p = − i∇
=
∞∑
(
) n
1 ∂
− a x
n! ∂x − a ∂
y
∂y − a ∂
z ψ(r)
∂z
n=0
∞∑
n=0
˜ψ(r) = ψ(r − a) = exp (− ia·p/) ψ(r)
1
n! (− a·∇)n ψ(r) = exp (− a·∇) ψ(r)
(5.87)
5.18
(5.90) , A, B A·B
5.19 n = ( sin θ cos φ , sin θ sin φ , cos θ ) z y
θ , z φ , (5.53)
n·σ , σ z |±〉 2
, |±〉 y θ , (5.87) J = S = σ/2
exp (−iθσ y /2) |±〉 , z φ
U(θ, φ) |±〉 , U(θ, φ) = exp (−iφσ z /2) exp (−iθσ y /2) (5.91)
(5.49) (5.91) (5.53) ,
U(θ, φ)σ z U † (θ, φ) = σ n
σ z |±〉 = ± |±〉 Uσ z |±〉 = ± U|±〉
Uσ z U † U|±〉 = ± U|±〉 , σ n U|±〉 = ± U|±〉
exp (−iθσ y /2) exp (−iφσ z /2) |±〉

5 111
5.20 (S k ) mn = − i ε kmn 3 3 × 3 S 1 , S 2 , S 3
S , ε kmn (14.1)
1. A(r) n ε , A ′ (r)
r r 1 , r A ′ , r 1 A
A ′ (r) = A(r 1 ) + ε n×A(r 1 ) , r 1 = r − ε n×r
A ′ (r) =
, A =
(
)
1 − i ε n·(L + S) A(r) ,
⎛
⎜
⎝
A 1
A 2
⎞
⎟
⎠
L = − ir×∇
A 3
2. S 1 , S 2 , S 3 , S 2 = 2 ,
s = 1
3. S 1 , S 2 , S 3 3 × 3 S 3
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎠
0
S 1 , S 2 , S 3 0
0
, 0, ± 1 S 3 ,
S 1 , S 2 , S 3 0
1
5.10
2k + 1 T kq ( q = −k, −k + 1, · · · , k ) J
[ J ± , T kq ] = √ (k ∓ q)(k ± q + 1) T k q±1 , [ J z , T kq ] = q T k q (5.92)
, T kq k ( spherical tensor operator of rank k )
2 , n
[ n·J , T kq ] = ∑ T kq ′〈 kq ′ |n·J| kq 〉
q ′
| kq 〉 J 2 , J z
V
J 2 | kq 〉 = k(k + 1)| kq 〉 , J z | kq 〉 = q| kq 〉
V 1 1 = − V 1 + iV 2
√
2
, V 1 0 = V 3 , V 1 −1 = V 1 − iV 2
√
2
(5.93)
, (5.90) V 1 q , (5.92) k = 1, T 1 q = V 1 q
, 1 J
0

5 112
F = F (θ, φ)
L + Y kq F =
(L + Y kq
)
F + Y kq L + F = √ (k − q)(k + q + 1) Y k q+1 F + Y kq L + F
[ L + , Y kq ] = √ (k − q)(k + q + 1) Y k q+1
[ L ± , Y kq ] = √ (k ∓ q)(k ± q + 1) Y k q±1 , [ L z , Y kq ] = q Y k q
, Y kq k
, J 2 , J z
|α j m〉
〈α ′ j ′ m ′ |T kq |α j m〉 = 〈 j m k q | j ′ m ′ 〉 〈α′ j ′ ‖ T k ‖α j 〉
√ 2j + 1
(5.94)
α j, m
, m, m ′ , q CG 〈α ′ j ′ ‖ T k ‖ α j 〉
( reduced matrix element )
(5.92)
J + T kq |jm〉 = [ J + , T kq ] |jm〉 + T kq J + |jm〉
= √ (k − q)(k + q + 1) T k q+1 |jm〉 + √ (j − m)(j + m + 1) T kq |j m+1〉
|(jk)j ′ m ′ 〉 ≡ ∑ m q
T kq |j m〉〈 j m k q | j ′ m ′ 〉
J + |(jk)j ′ m ′ 〉
= ∑ (√
〈 j m k q | j ′ m ′ 〉 (k − q)(k + q + 1) Tk q+1 |jm〉 + √ )
(j − m)(j + m + 1) T kq |j m+1〉
m q
= ∑ m q
(
T k q |jm〉 〈 j m k q−1 | j ′ m ′ 〉 √ (k − q + 1)(k + q)
CG (5.81)
+ 〈 j m−1 k q | j ′ m ′ 〉 √ )
(j − m + 1)(j + m)
J + |(jk)j ′ m ′ 〉 = √ (j ′ − m ′ )(j ′ + m ′ + 1) ∑ m q
T k q |jm〉〈 j m k q | j ′ m ′ +1 〉
= √ (j ′ − m ′ )(j ′ + m ′ + 1) |(jk)j ′ m ′ +1〉
J ± |(jk)j ′ m ′ 〉 = √ (j ′ ∓ m ′ )(j ′ ± m ′ + 1) |(jk)j ′ m ′ ± 1〉 (5.95)
J z |(jk)j ′ m ′ 〉 = m ′ |(jk)j ′ m ′ 〉 (5.96)

5 113
(5.15), (5.16)
CG (5.80)
∑
〈j ′ m ′ |(jk)j ′′ m ′′ 〉 = δ j′ j ′′δ m ′ m ′′〈j′ m ′ |(jk)j ′ m ′ 〉
∑
|(jk)j ′ m ′ 〉〈j m k q|j ′ m ′ 〉 = ∑ T kq ′′|j m ′′ 〉〈j m ′′ k q ′′ |j ′ m ′ 〉〈j m k q|j ′ m ′ 〉
j ′ m ′ j ′ m ′ m ′′ q ′′
= ∑
T kq ′′|j m ′′ 〉 δ mm ′′δ qq ′′
m ′′ q ′′
〈j ′ m ′ |T kq |j m〉 = ∑
= T kq |j m〉
j ′′ m ′′ 〈j ′ m ′ |(jk)j ′′ m ′′ 〉〈j m k q|j ′′ m ′′ 〉 = 〈j m k q|j ′ m ′ 〉C m ′
C m ′ = 〈j ′ m ′ |(jk)j ′ m ′ 〉 C m ′ m ′
(
〈j ′ m ′ |J − J + |(jk)j ′ m ′ 〉 = 〈j ′ m ′ |
)
J 2 − J z (J z + 1) |(jk)j ′ m ′ 〉
(
)
= j ′ (j ′ + 1) − m ′ (m ′ + 1) C m ′
, (5.15), (5.95)
〈j ′ m ′ |J − J + |(jk)j ′ m ′ 〉 = (j ′ − m ′ )(j ′ + m ′ + 1)〈j ′ m ′ +1 |(jk)j ′ m ′ +1 〉
= (j ′ − m ′ )(j ′ + m ′ + 1)C m′ +1
, C m ′ +1 = C m ′ , C m ′ m ′
〈j ′ m ′ |(jk)j ′ m ′ 〉 = 〈j′ ‖ T k ‖j 〉
√ 2j + 1
〈j ′ m ′ |T kq |j m〉 = 〈 j m k q | j ′ m ′ 〉 〈j′ ‖ T k ‖j 〉
√ 2j + 1
(5.94)
CG 〈 j m k q | j ′ m ′ 〉 CG
, 〈α ′ j ′ m ′ |T kq |α j m〉 0
m + q = m ′ ,
|j − k| ≤ j ′ ≤ j + k
, 0 , , T 00 = S
〈α ′ j ′ m ′ | S |α j m〉 = 〈 j m 0 0 | j ′ m ′ 〉 〈α′ j ′ ‖ S ‖α j 〉
〈α
√ = δ jj ′δ ′ j ‖ S ‖α j 〉
mm ′ √ 2j + 1 2j + 1
j m , m
V j ′ = j
〈α ′ j m ′ |V |α j m〉 = 〈α′ j m| J ·V |α j m〉
j(j + 1)
〈j m ′ | J |j m〉 (5.97)

5 114
( projection theorem ) , V J V J
, J e J e J = J/|J|
V J = V ·e J e J = V ·J
J 2 J
, (5.97) V J
V ·J (5.93)
J ·V = ∑ q
(−1) q J 1q V 1 −q
,
〈α ′ j m| J ·V |α j m〉 = m 〈α ′ j m| V 1 0 |α j m〉
+ √ (j + m)(j − m + 1)/2 〈α ′ j m−1| V 1 −1 |α j m〉
− √ (j − m)(j + m + 1)/2 〈α ′ j m+1| V 1 1 |α j m〉
= C j
〈α ′ j ‖ V ‖α j 〉
√ 2j + 1
(5.98)
C j = m 〈 j m 1 0 | j m 〉 + √ (j + m)(j − m + 1)/2 〈 j m 1 −1 | j m−1 〉
− √ (j − m)(j + m + 1)/2 〈 j m 1 1 | j m+1 〉
J ·V m , C j m
(5.98)
V 1q = J 1q
〈α ′ j m ′ |V 1 q |α j m〉 = 〈 j m 1 q | j m ′ 〉 〈α′ j ‖ V ‖α j 〉
√ 2j + 1
= 〈 j m 1 q | j m′ 〉
C j
〈α ′ j m| J ·V |α j m〉 (5.99)
〈j m ′ |J 1q |j m〉 = 〈 j m 1 q | j′ m ′ 〉
C j
〈j m| J 2 |j m〉 = 〈 j m 1 q | j′ m ′ 〉
C j
j(j + 1) (5.100)
, (5.99) (5.100)
〈α ′ j m ′ |V 1q |α j m〉 = 〈α′ j m| J ·V |α j m〉
j(j + 1)
〈j m ′ |J 1q |j m〉
5.21 2 T k1 q 1
, T k2 q 2
(T k1 T k2 ) kq
≡ ∑ q 1 q 2
〈k 1 q 1 k 2 q 2 |k q〉 T k1 q 1
T k2 q 2
(T k1 T k2 ) kq
k

5 115
5.22 1/2 µ µ = g l L + g s S
〈 jm | µ z | jm 〉 | jm 〉 5.17
J 2 | jm 〉 = j(j + 1)| jm 〉 , L 2 | jm 〉 = l(l + 1)| jm 〉 , j = l ± 1/2

6 116
6
6.1
V (r) r V (r)
t r(t) ,
m d2 r
dt 2
= − ∇V (r) = − r r
v = dr
dt E = mv2 /2 + V (r)
dE
dt
dv
= mv·
dt + dr
dt
dV
dr
·∇V = v·(
m dv
dt + ∇V )
= 0
L = m r×v
dL
dt
dv
= m v×v + m r×
dt = − r×r 1 dV
r dr = 0
, , L = m r × v =
xy z = 0 x, y 2
x = r cos θ , y = r sin θ
ẋ = ṙ cos θ − r ˙θ sin θ ,
ẏ = ṙ sin θ + r ˙θ cos θ
E = m 2
(
ẋ 2 + ẏ 2) + V (r) = m 2
|L| = |mr 2 ˙θ| =
( ) ( )
ṙ 2 + r 2 ˙θ2 + V (r) , L = m ( 0, 0, xẏ − ẋy ) = 0, 0, mr 2 ˙θ
E = m 2 ṙ2 + V eff (r) , V eff (r) = V (r) + L2
2mr 2
√
dr 2
( )
dt = ± E − V eff (r) , ∴
m
∫
√ ∫
dr
2
√
E − Veff (r) = ± dt
m
V eff (r) 1 L 2 /(2mr 2 )
6.2
V (r)
H ψ(r) = E ψ(r) ,
H = p2
2m + V (r) ,
p = − i∇
r , x, y, z
r, θ, φ (5.21) ∇ 2
,
(A×B)·(C×D) = A·C B·D − A·D B·C =
3∑
)
(A i C i B j D j − A i D i B j C j
i, j=1

6 117
, ,
(A×B)·(C ×D) =
3∑
i=1 j=1
3∑
)
(A i B j C i D j − A i B j C j D i
A = C = r , B = D = ∇ L = − i r×∇
L 2 = − ∑ )
(x i ∂ j x i ∂ j − x i ∂ j x j ∂ i , ∂ i = ∂
∂x
ij
i
1 ∂ j x i = x i ∂ j + δ ij , 2 x i ∂ j = ∂ j x i − δ ij
L 2 = − ∑ ij
x 2 i ∂ 2 j − 2 ∑ i
x i ∂ i + ∑ ij
∂ j x j x i ∂ i = − r 2 ∇ 2 − 2r·∇ + ∑ ij
(x j ∂ j + 1) x i ∂ i
∑
x i ∂ i = 3 ∑
ij
i
L 2 = − r 2 ∇ 2 + r·∇ + (r·∇) 2
∂
∂r = ∂x ∂
∂r ∂x + ∂y
∂r
∂
∂y + ∂z
∂r
x i ∂ i = 3r·∇
∂
∂z = x r
∂
∂x + y r
∂
∂y + z r
∂
∂z = 1 r r·∇
L 2 = − r 2 ∇ 2 + r ∂ ∂r + r ∂ ∂r r ∂ ∂r = − r2 ∇ 2 + r ∂2
∂r 2 r , ∴ ∇2 = 1 ∂ 2
r ∂r 2 r − 1 r 2 L2 (6.1)
r ≠ 0 L 2 (5.24) , L 2
L 2 r ,
V (r) H r ≠ 0
H = − 2
2m ∇2 + V (r) = − 2
2m
1 ∂ 2
r ∂r 2 r + 2 L 2
+ V (r) (6.2)
2mr2 L , (6.2) r
, [ L 2 , L ] = 0 , H, L 2 , L z ,
ψ(r) L 2 , L z Y l ml (θ, φ) , H, L 2 , L z
ψ(r)
ψ(r) = R(r) Y l ml (θ, φ) ,
m l = − l, − l + 1, · · · , l − 1, l
( m L z m l ) R(r) R(r) Y lml
L 2 , L z , H R(r)
L 2 Y l ml (θ, φ) = l(l + 1)Y l ml (θ, φ)
H ψ(r) = E ψ(r)
(− 2 1 d 2
2m r dr 2 r + 2 l(l + 1)
2mr 2
)
+ V (r) R l (r) = E R l (r) (6.3)
(− 2
d 2
2m dr 2 + 2 l(l + 1)
2mr 2
)
+ V (r) χ l (r) = E χ l (r) , χ l (r) = rR l (r) (6.4)

6 118
,
r ≥ 0 r = 0 ψ(r)
r d 3 r
lim rR l(r) = χ l (0) = 0 (6.5)
r→0
|ψ(r)| 2 d 3 r = |R l (r)| 2 r 2 dr |Y l ml (θ, φ)| 2 dΩ = |χ l (r)| 2 dr |Y l ml (θ, φ)| 2 dΩ
|Y l ml (θ, φ)| 2 1 , r r + dr |χ l (r)| 2 dr
, χ l (r)
∫ ∞
0
dr |χ l (r)| 2 = 1
3 , χ l (0) = 0
( r ≥ 0 )
V eff (r) = V (r) + 2 l(l + 1)
2mr 2 (6.6)
1 V eff
, l = 0 V eff (r) = V (r) V (r)
1 , χ(0) = 0 , V (r) = V (− r)
3
V (r) lim
r→0
r 2 V (r) = 0 χ l V (r)
χ l (0) = 0 r → 0 χ l (r) = c r α , α > 0
(6.4)
(
− α(α − 1) + l(l + 1) + 2m
2 (
r 2 V (r) − r 2 E) ) r α−2 = 0
lim
r→0 r2 V (r) = 0 − α(α − 1) + l(l + 1) = 0 α = − l , l + 1 , α > 0
α = l + 1
r→0
χ l (r) −−−−→ c r l+1 (6.7)
r 2(l+1) , l l
2 l(l + 1)/(2mr 2 ) ,
(6.4) 2 , χ l (r) 2 , 2
r l+1 r −l , (6.4) 2
, (6.5) r −l
1 , E (6.4) χ l (r) 1
, (6.3) (6.4) l m l , E
l , m l = l, l − 1, · · · , − l 2l + 1
R l (r) Y l ml (θ, φ) , 2l + 1 H
a m
l∑
m=−l
a m R l (r)Y lm (θ, φ)
1 χ l (r)

6 119
, (6.1) r = 0 ∇ 2 1 = − 4π δ(r) , L r
r
, (6.1)
(
− 1 r
∂ 2
∂r 2 r + L2
r 2 ) 1
r = − 1 r
∂ 2
∂r 2 1 = 0
, 1/r (6.1) r = 0 , (6.5)
, (6.1) r = 0
∇ 2 ψ(r) = 0 , V (r) = 0 , E = 0
χ ∝ r l+1 χ ∝ r −l (6.4) ∇ 2 r l Y lml = 0 , ∇ 2 r −l−1 Y lml = 0
, r l Y lml r −l−1 Y lml
χ l (r) r r ν
∫
∫ ∞
〈r ν 〉 = d 3 r r ν |ψ(r)| 2 = dr r ν χ l (r) 2
lim
r→0
r 2 V (r) = 0 r → 0 χ l → r l+1 , , r → ∞
χ l → 0 ν + 2l + 1 > 0 (6.4)
χ ′′
l(l + 1)
l (r) − F (r)χ l (r) = 0 , F (r) =
r 2 + 2m ( )
2 V (r) − E
r ν+1 χ ′ l χ′ l χ′′ l = (χ′ l 2)′
/2
∫ ∞
0
dr r ν+1 χ ′ l χ ′′
l = 1 2
∫ ∞
0
dr r ν+1 (χ ′ 2
l ) ′ = 1 2
ν + 2l + 1 > 0 r → 0 χ l ∝ r l+1 r ν+1 χ ′ 2
l
∫ ∞
χ ′ l χ l = (χ 2 l )′ /2
2
∫ ∞
0
dr r ν+1 χ ′ l
∫ ∞
0
0
dr r ν+1 χ ′ l χ ′′
l = − ν + 1
2
dr r ν+1 F (r)χ ′ l χ l = − 1 2
(
χ ′′
l − F χ l
)
=
∫ ∞
[
0
] ∞
r ν+1 χ ′ l
2 − ν + 1
0 2
∫ ∞
, (6.8) r ν χ l
∫ ∞
0
∫ ∞
0
dr r ν χ l χ ′′
l = −
dr r ν χ l
(
χ ′′
∫ ∞
0
)
l − F χ l =
0
dr χ ′ l (r ν χ l ) ′ = − ν
ν(ν − 1)
2
∫ ∞
0
0
∫ ∞
0
dr r ν χ ′ 2
l
∝ r ν+2l+1 → 0
∫ ∞
0
dr r ν χ ′ 2
l
dr χ 2 (
l r ν+1 F ) ′
(6.8)
dr χ 2 (
l r ν+1 F ) ∫ ∞
′
− (ν + 1) dr r ν χ ′ l 2 = 0 (6.9)
=
∫ ∞
0
ν(ν − 1)
2
dr r ν−2 χ 2 l −
dr r ν−1 χ ′ l χ l −
∫ ∞
0
∫ ∞
0
0
∫ ∞
dr r ν−2 χ 2 l −
dr r ν χ ′ 2
l
−
dr r ν χ ′ 2
l
0
∫ ∞
0
∫ ∞
0
dr r ν χ ′ 2
l
dr r ν F χ 2 l = 0

6 120
r ν χ ′ 2
l
2 (6.9)
ν(ν 2 − 1)
〈r ν−2 〉 − 2(ν + 1)〈r ν F 〉 − 〈r ν+1 F ′ 〉 = 0
2
V (r) = V 0 r κ ( κ > − 2 ), F (r) , ν + 2l + 1 > 0
ν
(
)
ν 2 − 1 − 4l(l + 1) 〈r ν−2 〉 − 2mV ( )
0
2
2 2ν + 2 + κ 〈r ν+κ 〉 + 4mE
2 (ν + 1)〈rν 〉 = 0 (6.10)
( κ = −1 )
, ν = 0 〈r 0 〉 = 1
E = 2 + κ 〈V (r)〉 (6.11)
2
(1.44) V (r) = V 0 r κ , r·∇V = r dV/dr = κV , (1.44)
〈p 2 〉/2m = κ 〈V 〉/2 E = 〈p 2 〉/2m + 〈V 〉 = (1 + κ/2)〈V 〉
6.3 3
V (r) =
{
− V 0 , r < a
0 , r > a , V 0 > 0
r < a , (6.3)
( 1
ρ
d 2
)
l(l + 1)
ρ + 1 −
dρ2 ρ 2 R l (r) = 0 , ρ = kr , k =
√
2m(E + V0 )
2 (6.12)
−V 0 < E < 0 k (15.36)
, j l (ρ) n l (ρ) R l (ρ) = A l j l (ρ) + B l n l (ρ)
(15.47) (6.5) B l
r < a R l (r) = A l j l (kr) r > a (6.3)
( 1
z
d 2
)
l(l + 1)
z + 1 −
dz2 z 2 R l (r) = 0 , z = iKr , K =
= 0
√
− 2mE
z (15.36) ,
R l (r) = B l h (1)
l
(iKr) + C l h (2)
l
(iKr) r → ∞ (15.48)
h (1)
l
(iKr) → − i−l
Kr e−Kr h (2)
l
(iKr) → il
Kr eKr
, r → ∞ R l (r) C l = 0
⎧
⎨
√ √
A l j l (kr) , r < a
2m(E +
R l (r) =
⎩ B l h (1)
l
(iKr) , r > a , k = V0 )
−2mE
2 , K =
2
r = a R l (r)
A l j l (ka) = B l h (1)
dj l (kr)
l
(iKa) , A l dh (1)
l
(iKr)
dr ∣ = B l (6.13)
r=a
dr ∣
r=a
2

6 121
2 (15.43)
(6.13)
) (
)
A l
(l j l (ka) − ka j l+1 (ka) = B l l h (1)
l
(iKa) − iβa h (1)
l+1 (iKa)
ka j l+1(ka)
j l (ka)
= iKa h(1) l+1 (iKa)
h (1)
l
(iKa) , k2 + K 2 = 2mV 0
2 (6.14)
V 0 K , E = − 2 K 2 /(2m)
(15.42) l = 0 (6.14)
Ka = − ka cot(ka) (6.15)
(2.8) α , β α = − β cot β , 1
(2.16) , l = 0 χ(0) = 0 1
(6.14) ,
√
2mV0
v 0 = a
2 , ε = 2ma2 E
2
v 0 ≤ π/2
v 0 > π/2 l = 0
, v 0 > π
l = 1
2 l(l + 1)/(2mr 2 ) l
ε
0
−10
−20
l = 0
l = 1
l = 2
l = 3
0 π/2 π 3π/2 v 0
, l v 0 v 0 > 3π/2
2 l = 0
6.1
l E → 0 (6.14)
, l = 0 cos v 0 = 0 , l > 0 j l−1 (v 0 ) = 0 (15.43),
(15.46), (15.47)
6.2
V 0 → + ∞ , a K → ∞ ( E →
− ∞ ) , − V 0 E + V 0 = 2 k 2 /2m
(6.14) k j l (ka) = 0 , V 0 → + ∞
,
6.4
,
144 Ze , −e
V (r)
V (r) = − 1 Ze 2
= − Zαc , α = e2
4πε 0 r r
4πε 0 c = 1
137.08

6 122
α (6.4)
(− 2
d 2
2m dr 2 + 2 l(l + 1)
2mr 2
E < 0
− Zαc
r
)
− E χ l (r) = 0
ρ = ar , a = 2 √ −2mE/ 2 (6.16)
( d
2
l(l + 1)
−
dρ2 ρ 2 + λ ρ − 1 )
√
mc
2
χ l = 0 , λ = Zα
4
−2E
(6.17)
ρ → ∞ χ ′′
l − χ l/4 = 0 χ l → e ±ρ/2 ,
χ l → e −ρ/2 , χ l → ρ l+1 2
χ l (ρ) = ρ l+1 e −ρ/2 v l (ρ)
(
dχ l
d
dρ = ρl+1 e −ρ/2 dρ + l + 1 − 1 )
v l
ρ 2
(6.17)
d 2 (
χ l
d
dρ 2 = ρl+1 e −ρ/2 dρ + l + 1 − 1 ρ 2
[ d
= ρ l+1 e −ρ/2 2 (
v l 2(l + 1)
dρ 2 + − 1
ρ
) ( d
dρ + l + 1 − 1 ρ 2
) dvl
)
v l
( 1
dρ + 4 − l + 1
ρ
+
) ]
l(l + 1)
ρ 2 v l
ρ d2 v
( )
l
dρ 2 + dvl
( )
2l + 2 − ρ
dρ − l + 1 − λ v l = 0 (6.18)
(15.59) a = l + 1 − λ, b = 2l + 2 ,
v l (ρ) = CM(l + 1 − λ, 2l + 2, ρ) (6.19)
l + 1 − λ ≠ − n r , ( n r = 0, 1, 2, · · · ) , ρ → ∞
M(l + 1 − λ, 2l + 2, ρ) → e ρ , χ l → ρ l+1 e ρ/2
χ l , l + 1 − λ = − n r M(− n r , 2l + 2, ρ) n r χ l
0 λ = n r + l + 1 n (6.17)
E n = − (Zα)2 mc 2
2n 2 , n = n r + l + 1 = 1, 2, · · · (6.20)
0
l = 0 l = 1 l = 2 l = 3
n l n
, 3
, l
Z = 1 E n ( n ≤ 5 ,
l ≤ 3 ) (n, l) = (1, 0)
(n, l) = (2, 0), (2, 1) l 2l + 1
, 4 n = 3 2
(n, l) = (3, 0), (3, 1), (3, 2) 9 E n n 2
, l = 0, 1, 2, 3, · · · 1 s, p, d, f, g, · · ·
En ( eV )
−5
−10
−15

6 123
, n 1s, 2s, 2p
χ nl (r) = Cρ l+1 e −ρ/2 M(− n r , 2l + 2, ρ) , n r = n − l − 1 ≥ 0 (6.21)
(6.20) (6.16)
ρ = 2 r
, a z = a B
n a z Z , a B =
mcα =
ρ n , a B = 0.529 × 10 −10 m χ nl
0
∫ ∞
0
dr χ 2 nl = C 2 na z
2
∫ ∞
0
dρ ρ 2l+2 e −ρ( M(− n r , 2l + 2, ρ)
(15.71) b = 2l + 2 , ν = 1
∫ ∞
dr χ 2 nl = C 2 na (
z n r ! Γ (2l + 3)
1 + n )
r
= C 2 n 2 [(2l + 1)!] 2 (n − l − 1)!
a z
2 (2l + 2) nr l + 1
(n + l)!
,
√
1 1
χ nl (ρ) =
n(2l + 1)! a z
√
= 1 1
n
a z
) 2
= 1 (6.22)
(n + l)!
(n − l − 1)! ρl+1 e −ρ/2 M(l + 1 − n, 2l + 2, ρ) (6.23)
(n + l)!
(n − l − 1)! ρl+1 e −ρ/2 L 2l+1
n−l−1
(ρ) (6.24)
L k n(ρ) (15.89) ( Laguerre )
n = 1 l = 0 , n = 2 l = 0, 1 (15.66)
χ 10 (r) = 1 √
az
ρ e −ρ/2 = 2 √
az
χ 21 (r) =
χ 20 (r) = 1 √ 2az
ρ
1
2 √ 6a z
ρ 2 e −ρ/2 =
r
a z
e −r/az (6.25)
1
2 √ r 2
e −r/(2az) (6.26)
6a z
a 2 z
(
1 − ρ )
e −ρ/2 = √ 1
2
2az
(
r
1 − r )
e −r/(2az) (6.27)
a z 2a z
χ 2 nl
(r)
χ 2 21 n
ρ n
e −ρ/2 = e −r/(na z)
, M(− n r , 2l + 2, ρ) n r
n r ,
n r = n − l − 1 χ nl (r) = 0 r ≠ 0
0.5
azχ 2 nl
0
(10)
(21) (20)
5 10 r/a z
, L , E
E = m 2 ṙ2 +
L2
2mr 2 − Zαc , m r
2 ṙ2 = E − L2
2mr 2 + Zαc ≥ 0
r

6 124
r L 2 = 2 l(l + 1) E (6.20)
E −
L2
2mr 2 + Zαc = − (Zα)2 mc 2 (
)
r 2n 2 r 2 r 2 − 2n 2 a z r + n 2 l(l + 1)a 2 z
r − ≤ r ≤ r + , r ± = a z
(n 2 ± n √ )
n 2 − l(l + 1)
r − , r + (2.37), (4.25)
, P cl (r)
∫ r+
r −
dr
P cl (r) ∝ 1
dr/dt ∝ r
√
(r+ − r)(r − r − )
[
r
√
(r+ − r)(r − r − ) = − √ (r + − r)(r − r − ) − r + + r −
sin −1 r ] r+
+ + r − − 2r
2
r + − r − r −
= π ) (r + + r − = πa z n 2
2
P cl (r) = 1
r
√
πa z n 2 (r+ − r)(r − r − )
(6.28)
P cl (r) 1 , P cl χ 2 nl ,
r , n r χ 2 nl , P cl ( )
0.005
(n, l) = (20, 0)
(n, l) = (20, 12)
azχ 2 nl
0
200 400 600 800 r/a z
(6.10)
2m
2 V 0 = − 2m
2 Zαc = − 2 ,
a z
4mE
2
= − 4m
2 (Zα) 2 mc 2
2n 2 = − 2
n 2 a 2 z
κ = − 1 (6.10) ν + 2l + 1 > 0
ν
(
)
ν 2 − 1 − 4l(l + 1) a 2
4
z〈r ν−2 〉 + (2ν + 1) a z 〈r ν−1 〉 − ν + 1
n 2 〈rν 〉 = 0 , 〈r ν 〉 ≡
〈r 0 〉 = 1 ν = 0, 1, 2, · · ·
∫ ∞
0
dr r ν χ 2 nl
〈r −1 〉 = 1
n 2 , 〈r〉 = 3n2 − l(l + 1)
a z , 〈r 2 〉 = n 2 5n2 + 1 − 3l(l + 1)
a 2 z , · · · (6.29)
a z 2
2
r −1 , r, r 2 , · · · χ nl , 〈r −2 〉

6 125
( MeV = 10 6 eV , fm = 10 −15 m , m = )
α =
e2
4πε 0 c = 1
137.08 ≈ 1
137 , α2 = 0.5322 × 10 −4 ≈ 1
20000
(6.30)
c = 197.327 MeV fm ≈ 200 MeV fm , mc 2 = 0.511 MeV ≈ 0.5 MeV (6.31)
, e 2 , m ,
, ( Z = 1 ) , (6.20)
α 2 mc 2
2
≈ 1 0.5 50
MeV = eV = 12.5 eV , ( 13.60 eV )
20000 2 4
6.3 a B =
mcα
6.4
(6.20) n 2 ,
6.5 (15.71) 〈r −1 〉 (6.29) , 〈r −2 2
〉 =
(2l + 1)n 3 a 2 z
6.5 3
V (r) = mω 2 r 2 /2 , a = √ mω/ (6.4)
( d
2
l(l + 1)
−
dq2 q 2 − q 2 + 2E )
√ mω
χ l (q) = 0 , q = ar =
ω
r
q → ∞ ( d
2
dq 2 − q2 )
χ l (q) = 0
d 2
dq 2 e−q2 /2 = ( q 2 − 1 ) e −q2 /2 → q 2 e −q2 /2
( q → ∞ )
q → ∞ χ l → e −q2 /2 χ l (q) = q l+1 e −q2 /2 v l (q)
(
dχ l
dq = ql+1 e −q2 /2 d
dq + l + 1 )
− q v l
q
d 2 (
χ l
dq 2 = ql+1 e −q2 /2 d
dq + l + 1 ) ( d
− q
q dq + l + 1 )
− q v l
q
( ( ) )
= q l+1 e −q2 /2 d
2 l + 1 d
dq 2 + 2 l(l + 1)
− q +
q dq q 2 − 2l − 3 + q 2 v l
ρ = q 2
d 2 ( ) (
v l l + 1
dq 2 + 2 dvl
− q
q dq − 2l + 3 − 2E )
v l = 0
ω
d
dq = dρ d
dq dρ = 2√ ρ d dρ , d 2
dq 2 = 4√ ρ d (
√ d ρ
dρ dρ = 4 ρ d2
dρ 2 + 1 2
)
d
dρ

6 126
(
ρ d2 v l
dρ 2 + l + 3 )
2 − ρ dvl
dρ − a′ v l = 0 , a ′ ≡ 1 (
l + 3 2 2 − E )
ω
(15.59) a = a ′ , b = l + 3/2 (15.65)
(6.32)
χ l (q) =
v l (ρ) = C M(a ′ , l + 3/2, ρ) + Dρ −l−1/2 M(a ′ − l − 1/2, 1/2 − l, ρ)
(
)
Cq l+1 M(a ′ , l + 3/2, q 2 ) + Dq −l M(a ′ − l − 1/2, 1/2 − l, q 2 ) e −q2 /2
χ l (0) = 0 D = 0 a ′ ≠ − n , ( n = 0, 1, 2, · · · ) q → ∞
χ l (q) = C q l+1 e −q2 /2 M(a ′ , l + 3/2, q 2 ) → C q l+1 e −q2 /2 e q2 = C q l+1 e q2 /2
a ′ = − n
(
E nl = ω 2n + l + 3 )
2
(6.33)
ψ nlml (r) = χ nl(r)
r
ρ = q 2
∫ ∞
0
Y l ml (θ, φ) , χ nl (r) = A nl q l+1 e −q2 /2 M ( −n , l + 3/2 , q 2 ) (6.34)
dr χ nl (r) 2 = A2 nl
2a
∫ ∞
0
dρ ρ l+1/2 e −ρ( M(−n, l + 3/2, ρ)) 2
= 1
(15.71) b = l + 3/2 , ν = 0
A 2 nl
2a
√
n! Γ (l + 3/2)
Γ (l + n + 3/2)
= 1 , ∴ A nl = 2a
(l + 3/2) n n! Γ 2 (l + 3/2)
(6.35)
n χ nl (r) = 0 r = 0
(6.33)
N = 2n + l (n, l )
,
, l
N , l
l = N − 2n , n = 0, 1, 2, · · · , [N/2]
[N/2] n, l
(6.34) 2l + 1 ,
ω(N + 3/2) D N
Enl/ω
11/2
9/2
7/2
5/2
3/2
l = 0 l = 1 l = 2 l = 3 l = 4
(2, 0) (1, 2) (0, 4)
(1, 1) (0, 3)
(1, 0) (0, 2)
(0, 1)
(0, 0)
[N/2]
[N/2]
∑
∑ (
D N = (2l + 1) = 2 ( N − 2n ) )
+ 1 =
n=0
n=0
(N + 1)(N + 2)
2
(6.36)

6 127
χ 2 02
N = 2 , χ 2 nl (r) () χ2 10,
(6.34), (6.35)
√ ( 6a
χ 10 =
π q 1 − 2 ) √
16a
1/2 3 q2 e −q2 /2 , χ 02 =
15π 1/2 q3 e −q2 /2
(6.28) , P cl (r)
P cl (r) = 2 r
√
, (ar ± ) 2 = N + 3 √(N
π
(r+ 2 − r 2 )(r 2 − r−)
2 2 ± + 3/2) 2 − l(l + 1) (6.37)
, n P cl χ 2 nl
, r , n χ 2 nl , P cl(r)
( )
1
0.5
(n, l) = (10, 6)
χ 2 nl /a
0.5
χ 2 nl /a
0
1 2 3 ar
0
2 4 6 8 ar
6.6
(6.37)
6.7
l = 0 , , 1 χ(0) = 0
, 1 , l = 0
, (6.33) 1 ω(n + 1/2) n 2n + 1
(6.34) χ n0 1 (4.22)
√
ϕ n (r) = C n e −a2 r 2 a
/2 H n (ar) , C n =
π 1/2 n! 2 n
χ n0 (r) = (−1) n√ 2 ϕ 2n+1 (r) 268 15.12
6.8 ψ nlml r 2 〈r 2 〉
〈r 2 〉 =
∫ ∞
0
dr r 2 χ 2 nl = A2 nl
a 3
∫ ∞
0
dq q 2l+4 e −q2( M(−n, l + 3/2, q 2 )
) 2
(15.71) 〈r 2 〉
〈V (r)〉 = ω 2
(
2n + l + 3 )
= E nl
2 2
1 , (1.44) , (6.11) κ = 2

6 128
3 , 3 1 ( x 1 = x, x 2 = y, x 3 = z )
H = H 1 + H 2 + H 3 ,
H k = p2 k
2m + 1 2 mω2 x 2 k ( k = 1, 2, 3 )
, ϕ ϕ(r) = X(x) Y (y) Z(z) , Hϕ = Eϕ ,
, ϕ −1 Hϕ = E
1
X H 1X + 1 Y H 2Y + 1 Z H 3Z = E
1 , 2 , 3 x, y, z ,
, , E 1 , E 2 , E 3
H 1 X = E 1 X , H 2 Y = E 2 Y , H 3 Z = E 3 Z , E = E 1 + E 2 + E 3
1 , H
(
H ϕ n1 n 2 n 3
(r) = ω n 1 + n 2 + n 3 + 3 )
ϕ n1 n
2
2 n 3
(r) , ϕ n1 n 2 n 3
(r) = ϕ n1 (x) ϕ n2 (y) ϕ n3 (z)
n 1 , n 2 , n 3 , ϕ n (x) 1 (4.22) H
N = n 1 + n 2 + n 3 , N ( N = 0, 1, 2, 3, · · · ) n 1 + n 2 + n 3 = N
ϕ n1 n 2 n 3
, ( N = 0 ) , D N
N ◦ 2 •
1
◦} ◦ ·{{ · · ◦ ◦}
• ◦} ◦ ·{{ · · ◦ ◦}
• ◦} ◦ ·{{ · · ◦ ◦}
n 1 n 2 n 3
D N = N+2 C 2 =
(6.36)
(N + 1)(N + 2)
2
(6.38)
(6.34) ψ nlml (r)
ϕ n1 n 2 n 3
(r) = C n1 C n2 C n3 e −a2 r 2 /2 H n1 (x)H n2 (y)H n3 (z)
H ψ nlml (r) L 2 , L z , ϕ n1 n 2 n 3
(r)
H 1 , H 2 , H 3 , ψ nlml
H 1 , H 2 , H 3
, ϕ n1 n 2 n 3
(r) L 2 , L z ω(N + 3/2)
ψ nlml (r) ϕ n1 n 2 n 3
(r) D N
∑
∑
C n1n 2n 3
ϕ n1n 2n 3
(r) , D nlml ψ nlml (r)
n 1 n 2 n 3
n 1 +n 2 +n 3 =N
n l m l
2n+l=N
H ω(N + 3/2) , , ψ nlml (r) ϕ n1n 2n 3
(r)
N = 0, 1
N = 0 , n 1 = n 2 = n 3 = 0 , n = l = m l = 0 ψ 000 (r) = ϕ 000 (r)
, M( 0, b, x ) = 1 , Y 00 (θ, φ) = 1/ √ 4π , H 0 (x) = 1
ψ 000 (r) = A e −a2 r 2 /2 , ϕ 000 (r) = C0 3 e −α2 r 2 /2 , A = aA √
00 a
3
√ = 4π π = 3/2 C3 0

6 129
N = 1 3
( n 1 , n 2 , n 3 ) = ( 1, 0, 0 ) , ( 0, 1, 0 ) , ( 0, 0, 1 ) n = 0, l = 1, m l = 0, ±1
√
4π
3 rY 10 = r cos θ = z ,
√
4π
3 rY 1 ±1 = ∓
r sin θ e±iφ
√
2
= ∓ x − iy √
2
M( 0, b, x ) = 1 (6.34)
⎧
⎪⎨ z , m l = 0
ψ 01ml = A e −a2 r 2 /2 rY 1 ml = A e −a2 r 2 /2 × ∓ x − iy
⎪⎩ √ , 2
m l = ±1 , √
A = 2a 5
π √ π
, , H 0 (x) = 1 , H 1 (x) = 2x
ϕ 100 = C x e −a2 r 2 /2 , ϕ 010 = C y e −a2 r 2 /2 , ϕ 001 = C z e −a2 r 2 /2 , C = 2aC 2 0C 1 = A
ψ 010 = ϕ 001 , ψ 01 ±1 = ∓ ϕ 100 − iϕ 010
√
2
6.6 2
H
H = − 2
2m
( ∂
2
∂x 2 + ∂2
∂y 2 )
+ V (r) , r = √ x 2 + y 2
2 x = r cos θ , y = r sin θ
H
∂ 2
∂x 2 + ∂2
∂y 2 = ∂2
∂r 2 + 1 ∂
r ∂r + 1 ∂ 2
r 2 ∂θ 2 = √ 1 ∂ 2 √ 1 r + r ∂r 2 4r 2 + 1 ∂ 2
r 2 ∂θ 2
(
H = − 2 1√r ∂ 2 √ 1 r +
2m ∂r 2 4r 2 + 1 ∂ 2 )
r 2 ∂θ 2 + V (r) (6.39)
(
L z = − i x ∂ ∂y − y ∂ )
= − i ∂ ∂x ∂θ
, H L z L z ϕ(θ), m z
− i dϕ
dθ = m zϕ , ∴ ϕ = Ne im zθ
r ϕ(θ) = ϕ(θ + 2π) , , e 2πm zi = 1
m z , H L z ψ(r)
ψ(r) = χ(r) √ r
e imzθ
√
2π
, m z = 0, ±1, ±2, · · ·
(6.39)
H χ(r)
(
√ e imzθ = eim zθ
√ − 2
r r 2m
d 2
dr 2 + 2
2m
m 2 z − 1/4
r 2
)
+ V (r) χ(r)

6 130
Hψ(r) = Eψ(r)
(− 2
2m
∫ ∞
0
d 2
dr 2 + 2
2m
dr r
∫ 2π
0
m 2 z − 1/4
r 2
dθ |ψ(r)| 2 =
)
+ V (r) χ(r) = E χ(r) (6.40)
∫ ∞
0
dr |χ(r)| 2 = 1
r → 0 χ(r) → r α
2 (
− α(α − 1) + m 2 z − 1 ) ( )
r α−2 + V (r) − E r α = 0
2m
4
− α(α − 1) + m 2 z − 1 4 + 2m ( )
2 V (r) − E r 2 = 0
r → 0 r 2 V (r) → 0
− α(α − 1) + m 2 z − 1 4 = 0 , ∴ α = ± m z + 1 2
α ≥ 0 α = |m z | + 1/2
2 , , 1 (6.40)
l(l + 1) = (l + 1/2) 2 − 1/4 l = |m z | − 1/2 , (6.40) (6.4) ,
r → 0 r |mz|+1/2 = r l+1 , 3 , 3
l = |m z | − 1/2 , 2
, (6.19) l = |m z | − 1/2
v = CM(|m z | + 1/2 − λ, 2|m z | + 1, ρ)
|m z | + 1/2 − λ = − n r , n r = 0, 1, 2, · · ·
E = − (Zα)2 mc 2
2λ 2 = − (Zα)2 mc 2
2(n − 1/2) 2 , n = |m z| + n r + 1 = 1, 2, 3, · · ·
, 3 (6.21)
χ(r) = C 1 ρ |mz|+1/2 e −ρ/2 M(− n r , 2|m z | + 1, ρ)
6.9 2
V (r) =
(x mω2 2 + y 2)
2
(
)
E = ω 2n + |m z | + 1 , ψ(r) = C r |mz| e −a2 r 2 /2 M(−n, |m z | + 1, a 2 r 2 ) e im zθ
( )
E = ω N + 1 ,
(
)
E = ω n 1 + n 2 + 1 , ψ(r) = ϕ n1 (x)ϕ n2 (y)

6 131
6.7
λ
(6.4)
q = r λ ,
ε = 2mλ2 E
2 , U l (q) = 2mλ2 l(l + 1)
2 V (r) +
q 2
d 2 χ l
dq 2
( )
+ ε − Ul (q) χ l = 0 (6.41)
(2.38) , 2.4 1
, q ≥ 0 l(l + 1)/q 2
q min ≤ q ≤ q max , q max q = q max 1
, q min = 0 l ≠ 0 U(0) , q min
, q min = 10 −6 r → 0 r 2 V (r) → 0 (6.7) χ l → r l+1
,
χ l (q min ) = q l+1
min , χ l(q min + ∆q) = (q min + ∆q) l+1 (6.42)
U l (q) l , 37
, l main
.....
int nc, nq, l; // l()
int main()
{
.....
qmax=qmin+nq*dq; qc=qmin+nc*dq;
for( l=0; l

6 132
d = 0.1, 0.5, 1 U(q)/u 0 d → + 0
e (q−1)/d → ∞ ( q − 1 > 0 ) , 0 ( q − 1 < 0 )
U(q)
0 1 2 3 q
{
0 , q > 1
U(q) →
−0.5
− u
1
0 , q < 1
(6.14)
0.5
l = 0, 1 , d −1.0 0.1
ε (6.14)
√
2ma2 (E + V 0 )
ka =
2 = √ √
− 2ma2 E
ε + u 0 , Ka =
2
l = 0 (6.14) (6.15) , l = 1 (15.42)
= √ − ε
√
− ε (ε + u0 ) + ε √ ε + u 0 cot √ ε + u 0 + u 0 = 0
d 1 , U(q) q = 1
U(q) ≈ U(1) + U ′ (1)(q − 1) = − u (
0
1 + 1 )
+ u 0
2 2d 4d q (6.43)
l = 0 (6.41) q = (4d/u 0 ) 1/3 z
d 2 χ 0
dz 2 + (ε′ − z) χ 0 = 0 , ε ′ =
( ) 2/3 [ 4d
ε + u 0
u 0 2
(
1 + 1 ) ]
2d
(2.33) a n Ai(x) ε ′ = − a n
, a n (2.35)
ε ≈ − u (
0
1 + 1 ) [ 3πu0
+
2 2d 8d
l = 0 ε (6.44)
(
n − 1 4) ] 2/3
, n = 1, 2, 3, · · · (6.44)
l = 0 ε d , ε ,
(6.44) n = 1, 2, · · · (6.15)
d = 1 χ 2 0 d = 1 , (6.43)
q 2.5 n = 1, 2 ,
, (6.43) , n = 3
0
ε
−10
χ 2 0
1.0
n = 1
n = 2
n = 3
u 0 = 30.0
l = 0
d = 1.0
−20
u 0 = 30.0
l = 0
0.5
0 1 2
d
0.0
0 1 2 3 4 q

7 133
7
, A F (A) a A | a 〉
A| a 〉 = a | a 〉
| a 〉 F (A)
F (A)| a 〉 = F (a) | a 〉
F (a) F (A) A ,
A n | a 〉 = a n | a 〉 |ψ〉
|ψ〉
|ψ〉 = ∑ c a | a 〉 , c a =
a
F (A)|ψ〉 = ∑ a
c a F (A) | a 〉 = ∑ a
c a F (a) | a 〉
F (A)
, 1
z + ∇ 2 , ∇2 e ik·r = − k 2 e ik·r
1
eik·r
z + ∇ 2 = 1
eik·r
z − k 2
ψ(r)
∫
ψ(r) = d 3 k a(k) e ik·r
∫
1
z + ∇ 2 ψ(r) = d 3 k a(k)
∫
1
eik·r
z + ∇ 2 = d 3 k
a(k)
z − k 2 eik·r
a(k) , ψ(r)
7.1
H ,
( 34 , 131 ), ,
,
H 2
H = H 0 + H ′
, H 0 H 0 E n ,
| n 〉 | n 〉 :
H 0 | n 〉 = E n | n 〉 ,
〈 m | n 〉 = δ mn

7 134
H ′ H 0 , H 0 , H ′
, H |ψ〉 W
H|ψ〉 = W |ψ〉
H ′ H
H λ |ψ〉 = W |ψ〉 , H λ = H 0 + λ H ′ (7.1)
, |ψ〉 W λ λ = 0 , |ψ〉 W
, λ = 1 |ψ〉, W
∞∑
∞∑
|ψ〉 = λ k |ψ k 〉 , W = λ k W k
k=0
k=0
(7.1)
∞∑
∞∑
∞∑
λ k H 0 |ψ k 〉 + λ k H ′ |ψ k−1 〉 =
λ k
k=0
k=1
k=0 k ′ =0
k∑
W k−k ′|ψ k ′〉
λ
k∑
H 0 |ψ 0 〉 = W 0 |ψ 0 〉 , H 0 |ψ k 〉 + H ′ |ψ k−1 〉 = W k−k ′|ψ k ′〉 , ( k ≥ 1 )
(H 0 − W 0 ) |ψ 0 〉 = 0 (7.2)
(H 0 − W 0 ) |ψ 1 〉 = (W 1 − H ′ ) |ψ 0 〉 (7.3)
(H 0 − W 0 ) |ψ 2 〉 = (W 1 − H ′ ) |ψ 1 〉 + W 2 |ψ 0 〉 (7.4)
(H 0 − W 0 ) |ψ k 〉 = (W 1 − H ′ ) |ψ k−1 〉 + W 2 |ψ k−2 〉 + · · · + W k |ψ 0 〉 (7.5)
(7.3) ∼ (7.5) |ψ k 〉
|ψ k〉 ′ = |ψ k 〉 + α|ψ 0 〉 , α =
k ′ =0
(H 0 − W 0 ) |ψ ′ k〉 = (H 0 − W 0 ) |ψ k 〉 + α (H 0 − W 0 ) |ψ 0 〉 = (H 0 − W 0 ) |ψ k 〉
, (7.3) ∼ (7.5) , |ψ k 〉 α|ψ 0 〉
,
〈ψ 0 |ψ k〉 ′ = 〈ψ 0 |ψ k 〉 + α〈ψ 0 |ψ 0 〉 = 0
|ψ
k ′ 〉 |ψ k〉
〈ψ 0 |ψ k 〉 = 0 , k ≥ 1 (7.6)

7 135
H 0 (7.2)
〈ψ 0 | (H 0 − W 0 ) |ψ k 〉 ∗ = 〈ψ k | (H 0 − W 0 ) |ψ 0 〉 = 0
(7.5) |ψ 0 〉 (7.6)
0 = − 〈ψ 0 |H ′ |ψ k−1 〉+W 1 〈ψ 0 |ψ k−1 〉+· · ·+W k−1 〈ψ 0 |ψ 1 〉+W k 〈ψ 0 |ψ 0 〉 = − 〈ψ 0 |H ′ |ψ k−1 〉+W k 〈ψ 0 |ψ 0 〉
W k = 〈ψ 0|H ′ |ψ k−1 〉
〈ψ 0 |ψ 0 〉
(7.7)
7.2
1
(7.2) |ψ 0 〉 H 0 |ψ 0 〉
|ψ 0 〉 = | i 〉 , W 0 = E i
E i , , E i | i 〉
, |ψ 0 〉 , |ψ 0 〉 = | i 〉
E i
(7.7) k = 1
W 1 = 〈ψ 0|H ′ |ψ 0 〉
〈ψ 0 |ψ 0 〉
= 〈 i |H ′ | i 〉
1 , H ′
, |ψ 1 〉 |ψ 1 〉 H 0
|ψ 1 〉 = ∑ n
c n | n 〉
(7.3)
∑
c n (H 0 − E i ) | n 〉 = (W 1 − H ′ ) | i 〉
n
H 0 | n 〉 = E n | n 〉
∑
c n (E n − E i ) | n 〉 = (W 1 − H ′ ) | i 〉
n
| m 〉
= ∑ n
c n (E n − E i ) 〈 m | n 〉 = ∑ n
c n (E n − E i ) δ mn = c m (E m − E i )
= W 1 〈 m | i 〉 − 〈 m |H ′ | i 〉 = W 1 δ mi − 〈 m |H ′ | i 〉
c m (E m − E i ) = 〈 i |H ′ | i 〉 δ mi − 〈 m |H ′ | i 〉 (7.8)

7 136
| i 〉 m ≠ i E m ≠ E i
c m = 〈 m |H′ | i 〉
E i − E m
, m ≠ i (7.9)
m = i (7.8) 0 = 0 c i
(7.6)
〈 i |ψ 1 〉 = ∑ n
c n 〈 i | n 〉 = ∑ n
c n δ in = c i = 0
|ψ 1 〉 = ∑ m≠i
| m 〉 〈 m |H′ | i 〉
E i − E m
(7.10)
, 1
|ψ〉 = | i 〉 + |ψ 1 〉 = | i 〉 + ∑ m≠i
|ψ〉 = | i 〉 + |ψ 1 〉 〈 i |ψ 1 〉 = 〈ψ 1 | i 〉 ∗ = 0
| m 〉 〈 m |H′ | i 〉
E i − E m
(7.11)
〈ψ|ψ〉 = 〈 i | i 〉 + 〈 i |ψ 1 〉 + 〈ψ 1 | i 〉 + 〈ψ 1 |ψ 1 〉 = 1 + 〈ψ 1 |ψ 1 〉
〈ψ 1 |ψ 1 〉 H ′ 2 , 1 (7.11) 1
|ψ〉 m ≠ i | m 〉 , 1
∣ ∣∣∣ 〈 m |H ′ | i 〉
E i − E m
∣ ∣∣∣
≪ 1
(7.10) H 0 | m 〉 = E m | m 〉
1
1
| m 〉 = | m 〉
E i − E m E i − H 0
1/(E i − E m ) , | m 〉
1
| m 〉 = | m 〉
1
= | m 〉
E i − E m E i − E m E i − E m
, 1/(E i − H 0 ) , 1/(E i − H 0 ) | m 〉
(7.10)
|ψ 1 〉 = ∑ m≠i
| m 〉
1
, | m 〉
E i − H 0 E i − H 0
1
E i − H 0
| m 〉 〈 m |H ′ | i 〉 =
1 ∑
| m 〉 〈 m |H ′ | i 〉
E i − H 0
1/(E i − E m ) m , 1/(E i − H 0 ) m
m≠i
, Q i
Q i | m 〉 =
{
0 , m = i
| m 〉 , m ≠ i
Q i
1 ∑
|ψ 1 〉 = Q i | m 〉 〈 m |H ′ | i 〉
E i − H 0
m

7 137
Q i | i 〉 = 0 m m = i , m ≠ i
| m 〉 (5.36)
,
∑
| m 〉〈 m | = 1
n
|ψ 1 〉 =
1 H (7.11)
(
|ψ〉 = 1 +
1
E i − H 0
Q i H ′ | i 〉 (7.12)
)
1
(
Q i H ′ | i 〉 = 1 + G i H ′) | i 〉 , G i =
E i − H 0
(7.11)
2
(7.7) k = 2
(7.12)
W 2 = 〈 i |H ′ |ψ 1 〉 = ∑ m≠i
〈 i |H ′ | m 〉 〈 m |H ′ | i 〉
E i − E m
= ∑ m≠i
1
E i − H 0
Q i
|〈 m |H ′ | i 〉| 2
E i − E m
(7.13)
W 2 = 〈 i |H ′ 1
E i − H 0
Q i H ′ | i 〉 (7.14)
H ′ 1/(E i − H 0 ) ,
|ψ 2 〉 = ∑ n
d n | n 〉 , d i = 0
(7.4)
(7.10)
(E m − E i ) d m = W 1 c m − 〈 m |H ′ |ψ 1 〉 + W 2 〈 m | i 〉
〈 m |H ′ |ψ 1 〉 = ∑ n≠i
〈 m |H ′ | n 〉〈 n |H ′ | i 〉
E i − E n
, m ≠ i
|ψ 2 〉 = ∑ m≠i
d m = −
G i
⎛
1
⎝ 〈 m |H′ | i 〉〈 i |H ′ | i 〉
− ∑ E i − E m E i − E m
n≠i
d m | m 〉 = ∑ m≠i
∑
n≠i
⎞
〈 m |H ′ | n 〉〈 n |H ′ | i 〉
⎠
E i − E n
| m 〉 〈 m |H′ | n 〉〈 n |H ′ | i 〉
(E i − E m )(E i − E n ) − ∑ | m 〉 〈 m |H′ | i 〉〈 i |H ′ | i 〉
(E i − E m ) 2
m≠i
1
|ψ 2 〉 = Q i H ′ 1
Q i H ′ 1
| i 〉 −
E i − H 0 E i − H 0 (E i − H 0 ) 2 Q iH ′ | i 〉〈 i |H ′ | i 〉
(
)
= G i H ′ G i H ′ − G 2 i H ′ 〈 i |H ′ | i 〉 | i 〉

7 138
|ψ〉
|ψ〉 = | i 〉 + |ψ 1 〉 + |ψ 2 〉
〈 i |ψ 1 〉 = 〈 i |ψ 2 〉 = 0
〈ψ|ψ〉 = 〈 i | i 〉 + 〈ψ 1 |ψ 1 〉 + 〈ψ 1 |ψ 2 〉 + 〈ψ 2 |ψ 1 〉 + 〈ψ 2 |ψ 2 〉
H ′ 2 , H ′ 2
|ψ 1 〉, |ψ 2 〉 H ′ 1 , 2 , 3, 4 H ′ 3 , 5 4
〈ψ|ψ〉 = 1 + ∑ n≠i
⎛
N |ψ〉 , N = ⎝1 + ∑ n≠i
|c n | 2 = 1 + ∑ n≠i
|〈 n |H ′ | i 〉| 2
(E i − E n ) 2
⎞−1/2
|〈 n |H ′ | i 〉| 2
⎠
(E i − E n ) 2 ≈ 1 − 1 ∑ |〈 n |H ′ | i 〉| 2
2 (E i − E n ) 2
n≠i
, N H ′ 2 ,
⎛
⎞
⎝1 − 1 ∑ |〈 n |H ′ | i 〉| 2
⎠
2 (E i − E n ) 2 | i 〉 + |ψ 1 〉 + |ψ 2 〉
n≠i
1
d 2
H 0 = − 2
2m dx 2 + 1 2 kx2
n = 0, 1, 2, · · ·
H 0 | n 〉 = E n | n 〉 ,
(
E n = ω n + 1 ) √
k
, ω =
2
m
a † = √ 1 (
q − d )
, a = 1 (
√ q + d ) √ mω
, q =
2 dq
2 dq
x
a † | n 〉 = √ n + 1 | n + 1 〉 , a | n 〉 = √ n | n − 1 〉 (7.15)
H ′ = b 2 x2 =
b (
a † + a ) 2 b (
= (a † ) 2 + a 2 + 2a † a + 1 )
4mω
4mω
H ′ | n 〉 =
b (√ √ )
(n + 1)(n + 2) | n + 2 〉 + n(n − 1) | n − 2 〉 + (2n + 1) | n 〉
4mω

7 139
W 1 = 〈 n |H ′ | n 〉 =
b (
n + 1 )
2mω 2
W 2 = ∑ n ′ ≠n
|〈 n ′ |H ′ | n 〉| 2
E n − E n ′
= |〈 n + 2 |H′ | n 〉| 2
E n − E n+2
+ |〈 n − 2 |H′ | n 〉| 2
E n − E n−2
( ) 2 ( b (n + 1)(n + 2)
=
− +
4mω
2ω
= − 1 ( ) 2 ( b
8 mω 2 ω n + 1 )
2
)
n(n − 1)
2ω
2
(
W = ω n + 1 )
(
+ W 1 + W 2 = ω n + 1 ) ( 1 + b
2
2 2mω 2 − 1 ( ) ) 2 b
8 mω 2
(
= ω n + 1 ) ( 1 + b
2 2k − 1 ( ) ) 2 b
8 k
(7.16)
2 1
(7.15)
|ψ 1 〉 = ∑ n ′ ≠n
| n ′ 〉 〈 n′ |H ′ | n 〉
E n − E n ′
= | n + 2 〉 〈 n + 2 |H′ | n 〉
E n − E n+2
+ | n − 2 〉 〈 n − 2 |H′ | n 〉
E n − E n−2
= b (
8mω 2 − √ (n + 1)(n + 2) | n + 2 〉 + √ )
n(n − 1) | n − 2 〉
√
(n + 1)(n + 2) | n + 2 〉 = (a † ) 2 | n 〉 ,
√
n(n − 1) | n − 2 〉 = a 2 | n 〉
|ψ 1 〉 = b (
8mω 2 − (a † ) 2 + a 2) | n 〉 = b (2q d )
8k dq + 1 | n 〉 (7.17)
H = H 0 + H ′
d 2
H = − 2
2m dx 2 + k + b
2
H 0 k k + b , E n ′
(
E n ′ = ω ′ n + 1 )
√
k + b
, ω ′ =
√1
2
m = ω + b k
ω ′ b
x 2
(
ω ′ = ω 1 + b
)
2k − b2
8k 2 +
b3
16k 3 + · · ·
, (7.16) E ′ n b 2
H 0 u n
( mω
) 1/4
u n (q) = fn (q) , f n (q) = √ 1 /2 H
π
n!2
n e−q2 n (q)

7 140
H
( ) mω
′ 1/4
√
mω
ϕ n (q) =
f n (q ′ ) , q ′ ′
=
π
x
(
q ′ = q 1 + b )
4k + · · · ,
( mω
ϕ n (q) =
π
=
( ) mω
′ 1/4
=
π
( mω
) ( 1/4
1 + b )
π 8k + · · ·
) 1/4
(
1 + b
8k + · · · ) (
f n (q) + b
4k q df n
dq + · · · )
(
1 + b
8k + b
4k q d dq + · · · )
u n
(7.17) b 1
7.1
H ′ = bx 2
W 1 = 0 ,
W 2 = − b2
2k
)
d
( )
|ψ 1 〉 = q 0
dq u n(q) , |ψ 2 〉 = q2 0
(q 2 + d2
4 dq 2 u n (q) , N = 1 − q2 0
2n + 1
4
q 0 =
b
√
ω mω = b √ mω
k
7.2
H ′ = v 0 x 4
W 1 = ω 3u 0
8
(
2n 2 + 2n + 1 ) , W 2 = − ω u2 0
32 (2n + 1) ( 17n 2 + 17n + 21 )
u 0 = 2v ( ) 2
0
ω mω
q = √ mω/ x
d 2 ψ
( )
dq 2 + ε − U(q) ψ = 0
ε = 2E
ω , U(q) = q2 + u 0 q 4
1.2
ε
1.1
1.0
0.9
37
0.0 0.2 u 0
n = 0 , 1 , 2
u 0 < 0

7 141
7.3
|ψ 0 〉 = | i 〉 , H 0 g
| i, α 〉 , ( α = 1, 2 · · · , g )
H 0 | i, α 〉 = E i | i, α 〉 , 〈 i, α | i, α ′ 〉 = δ αα ′
| i, α 〉
g∑
a α | i, α 〉
α=1
H 0 E i (7.2) ,
(7.2)
g∑
|ψ 0 〉 = a α | i, α 〉 ,
α=1
W 0 = E i
, (7.3)
g∑
(H 0 − E i ) |ψ 1 〉 = W 1 a α ′| i, α ′ 〉 −
α ′ =1
g∑
a α ′H ′ | i, α ′ 〉
| i, α 〉 H 0 | i, α 〉 = E i | i, α 〉 0
g∑
α ′ =1
α ′ =1
(
)
W 1 δ αα ′ − h αα ′ a α ′ = 0 , h αα ′ ≡ 〈 i, α |H ′ | i, α ′ 〉 (7.18)
(7.18) a α = 0 ,
(
)
det W 1 δ αα ′ − h αα ′ = 0 (7.19)
g W 1 , W 1 (7.18) a α (7.19)
|ψ 0 〉
|ψ 1 〉 = ∑ n
c n | n 〉
(7.3)
| m 〉
∑
c n (E n − E i ) | n 〉 = (W 1 − H ′ ) |ψ 0 〉
n
(E m − E i ) c m = W 1 〈 m |ψ 0 〉 − 〈 m |H ′ |ψ 0 〉
E m ≠ E i 〈 m |ψ 0 〉 = 0
c i, α = 0 (7.6) 〈ψ 0 |ψ 1 〉 = 0
|ψ 1 〉 = ∑
m≠{i,α}
| m 〉 〈 m |H′ |ψ 0 〉
=
E i − E m
c m = 〈 m |H′ |ψ 0 〉
E i − E m
(7.20)
1
E i − H 0
Q i H ′ |ψ 0 〉 , Q i | n 〉 =
⎧
⎨
⎩
| n 〉 , E n ≠ E i
(7.21)
0 , E n = E i

7 142
〈 m |H ′ |ψ 0 〉 = 〈 m |H ′ | i 〉 ,
〈 m |H ′ |ψ 0 〉 = ∑ α
a α 〈 m |H ′ | i, α 〉
2 W 2 〈ψ 0 |ψ 0 〉 = 1
W 2 = 〈ψ 0 |H ′ |ψ 1 〉 = 〈ψ 0 |H ′ 1
E i − H 0
Q i H ′ |ψ 0 〉 =
∑
m≠{i,α}
|〈 m |H ′ |ψ 0 〉| 2
E i − E m
7.3 2 ( g = 2 )
√
W 1 = h (
11 + h 22 ± ∆h
1
, a 1 = 1 ± h )
√
11 − h 22
, a 2 = ∓ |h 12| 1
2
2 ∆h
h 12 2
(
1 ∓ h )
11 − h 22
∆h
∆h =
√
(h 11 − h 22 ) 2 + 4|h 12 | 2
, H 0 H ′
F , H 0 F , | i, α 〉 F
F f iα
F | i, α 〉 = f iα | i, α 〉 ,
α = 1, 2 · · · , g
H ′ F − F H ′ = 0
0 = 〈 i, α | (H ′ F − F H ′ ) | i, α ′ 〉 = (f iα ′ − f iα ) 〈 i, α |H ′ | i, α ′ 〉
f iα ≠ f iα ′
h αα ′ = 〈 i, α |H ′ | i, α ′ 〉 = 0
g f iα h αα ′ = δ αα ′h αα , (7.18)
)
(W 1 − h αα a α = 0 , ∴ W 1 = 〈 i, α |H ′ | i, α 〉 , |ψ 0 〉 = | i, α 〉 (7.22)
, H 0 F | i, α 〉
f iα , (7.18) ,
f 1 f 2 2 , g 1 g 2 = g − g 1
1 ≤ α ≤ g 1 F | i, α 〉 = f 1 | i, α 〉 , g 1 + 1 ≤ α ≤ g F | i, α 〉 = f 2 | i, α 〉
α ≤ g 1 , α ′ > g 1 h αα ′ = 0 (7.18)
∑g 1
α ′ =1
(
W 1 δ αα ′ − h αα ′
)
a α ′ = 0

7 143
α ≥ g 1 + 1
g∑
α ′ =g 1 +1
(
)
W 1 δ αα ′ − h αα ′ a α ′ = 0
, g 1 ×g 1 g 2 ×g 2 H 0 F
g 1
∑
α=1
a α | i, α 〉 ,
g∑
α=g 1+1
a α | i, α 〉 (7.23)
H = H 0 + H ′ F ,
(7.23) |ψ 0 〉 , F | i, α 〉
, , f iα | i, α 〉 1 ,
z E ,
H 0 = − 2
2m ∇2 − Zαc , α =
r
H ′ = eEz = eEr cos θ
H 0
,
E n = − (Zα)2 mc 2
2n 2 , n = 1, 2, 3, · · ·
u nlml (r) = χ nl(r)
Y l ml (θ, φ) (7.24)
r
χ nl (r) (6.21) E n n 2 ,
( n = 2 ) n = 2 (n, l, m l ) =
(2, 0, 0), (2, 1, 1), (2, 1, 0), (2, 1, −1) 4
H 0 L , H 0 , L 2 , L z ,
u nlml
[ z , L x ] = [ z , yp z − zp y ] = iy , [ z , L y ] = − ix , [ z , L z ] = 0
, H ′ L x , L y L z , L z m l
n = 2 4 m l
m l = 0 : (2, 0, 0) (2, 1, 0) , m l = 1 : (2, 1, 1) , m l = −1 : (2, 1, −1)
m l = 0
|ψ 0 〉 = a 1 | 2, 0, 0 〉 + a 2 | 2, 1, 0 〉
(7.18)
⎛
⎝ W 1 − 〈 2, 0, 0 |H ′ | 2, 0, 0 〉 − 〈 2, 0, 0 |H ′ | 2, 1, 0 〉
− 〈 2, 0, 0 |H ′ | 2, 1, 0 〉 W 1 − 〈 2, 1, 0 |H ′ | 2, 1, 0 〉
⎞ ⎛
⎠ ⎝ a 1
⎠ = 0
a 2
⎞

7 144
〈 n, l, m l |H ′ | n ′ , l ′ , m l 〉 = eE
∫ ∞
0
∫
dr r χ 2l (r) χ 2l ′(r) d cos θ dφ cos θ (Y lml ) ∗ Y l′ m l
Y l ml (θ, φ) (5.30) Y l ml (θ, φ) = f lml (cos θ) e im lφ , ( f lml
〈 n, l, m l |H ′ | n ′ , l ′ , m l 〉 = 2πeE
∫ ∞
0
dr r χ 2l (r) χ 2l ′(r)
∫ 1
−1
)
dt t f lml (t) f l ′ m l
(t)
f lml (t) t , l = l ′ t f lml (t) f l′ m l
(t) = t f lml (t) 2 t
(6.26), (6.27)
〈 2, 0, 0 |H ′ | 2, 1, 0 〉 = eE
24
(
〈 n, l, m l |H ′ | n ′ , l, m l 〉 = 0 (7.25)
f 00 (t) = √ 1
√
3
, f 10 (t) =
4π 4π t
〈 2, 0, 0 |H ′ | 2, 1, 0 〉 = √ eE ∫ ∞
dr r χ 20 (r) χ 21 (r)
3
( ) 4 ∫ Z ∞
dr r
(2 4 − Z )
r
a B a B
) (
W 1 3eEa B /Z
W 1
W 1 = ± 3eEa B
Z
0
, |ψ 0 〉 =
0
e −Zr/aB
= − 3eEa B
Z
)
a 1
= 0 (7.26)
a 2
| 2, 0, 0 〉 ∓ | 2, 1, 0 〉
√
2
m l = ± 1 1 , ,
(7.25)
W 1 = 〈 2, 1, ±1 |H ′ | 2, 1, ±1 〉 = 0
m l = 1 m l = −1
7.4
n = 2 4 , (7.18) 4×4
10 −14 m , 10 −10 m ,
, Ze ,
Ze R
, V (r)
⎧
⎪⎨ − Zαc ) (3 − r2
2R R
V (r) =
2 , r < R
⎪⎩ − Zαc
, α = e2
4πε 0 c = ≈ 1
137
, r > R
r
(7.27)

7 145
⎧
H = H 0 + H ′ (r) , H 0 = − 2
2m ∇2 − Zαc
⎪⎨
, H ′ (r) =
r
⎪⎩
− Zαc
2R
(3 − r2
R 2 − 2R r
)
, r < R
0 , r > R
H ′ H 0 | n, l, m l 〉 n 2 ,
(7.19) (7.24)
∫ R
∫
〈 n, l, m l |H ′ | n, l ′ , m ′ l 〉 = dr H ′ (r)χ nl (r)χ nl ′(r)
0
= δ ll ′ δ ml m ′ l
∫ R
0
dr H ′ (r)χ 2 nl(r)
dΩ Ylm ∗
l
(θ, φ)Y l′ m ′ (θ, φ)
l
, (7.19) h αα ′ , 1
W 1 = 〈 n, l, m l |H ′ | n, l, m l 〉 =
∫ R
0
dr H ′ (r) χ 2 nl(r)
, H ′ [ H ′ , L ] = 0 ,
H 0 L 2 L z L 2 L z
,
W 1 l l (6.23)
W 1 = |E n | C ∫ ρmax
(
nl
ρ
2
dρ + 2ρ )
max
− 3 ρ 2l+2 F nl (ρ) (7.28)
ρ max ρ
0
ρ 2 max
E n = − (Zα)2 mc 2
2n 2 , ρ max = 2R
na z
= 2ZR
na B
, a B =
mcα
C nl =
(n + l)!
[(2l + 1)!] 2 (n − l − 1)! , F nl(ρ) = e −ρ( M(l + 1 − n, 2l + 2, ρ)
0 ≤ ρ ≤ ρ max ≪ 1 F nl (ρ) ≈ 1
l = 0
3C nl
W 1 ≈ |E n |
(l + 1)(2l + 3)(2l + 5) ρ2l+2 max
W 1 ≈ 4 ( ) 2 ZR
|E n | (7.29)
5n a B
a B a B = 0.53 × 10 −10 m R ∼ 10 −14 m
R/a B ∼ 10 −4 W 1 /|E n | ,
207
µ
a B = 0.53 × 10 −10 /207 = 2.6 × 10 −13 m
, Z , a B /Z R R ≈
7 × 10 −15 m , Z = 82 R/(a B /Z) = 0.45 , µ
,
) 2

7 146
7.5
(7.27) , 131
λ λ = a z = a B /Z
⎧
ε = 2mλ2 E 2E
2 =
mc 2 (Zα) 2 , 2mλ
⎪⎨
2
2 V (r) =
⎪⎩ − 2 q ,
− 1 ( )
3 − q2
R z Rz
2 , q < R z
q > R z
R z = ZR/a B q max 123
n = 1, 2, 3 ε R z R z = 0
E = − (Zα)2 mc 2
2n 2 , ∴ ε = − 1 n 2
(n, l ) = (1, 0 ) R z 123 ,
q = r/a z = 1 , r
1 (7.28) R z , 1
, l > 0 R z 1
l = 0 1 (7.29)
µ ,
, Z , ,
0.0
ε
−0.2
−0.1
ε
(3, 0)
(2, 1)
(1, 2)
R z
−0.4
(1, 0)
0 1 2 3
0 1
R z
2 3
(2, 0)
−0.6
−0.2
−0.8
(1, 1)
−1.0
2
1
g∑
|ψ 0 〉 = a α | i, α 〉
, (7.21) (7.4) | i, α 〉
0 = W 1 〈 i, α |ψ 1 〉 − 〈 i, α |H ′ |ψ 1 〉 + W 2 〈 i, α |ψ 0 〉
α=1

7 147
〈 i, α |ψ 1 〉 = 0 , 〈 i, α |ψ 0 〉 = a α , (7.21)
W 2 a α − 〈 i, α |H ′ |ψ 1 〉 = W 2 a α − 〈 i, α |H ′ 1
E i − H 0
Q i H ′ |ψ 0 〉 = 0
|ψ 0 〉
W 2 a α −
g∑
α ′ =1
h (2)
αα ′ = 〈 i, α |H′ 1
E i − H 0
Q i H ′ | i, α ′ 〉 =
h (2)
αα ′a α ′ = 0 (7.30)
∑
m≠{i,α}
〈 i, α |H ′ | m 〉〈 m |H ′ | i, α ′ 〉
E i − E m
(7.30) (7.18) , (7.18)
, (7.30) W 2 = h (2)
αα , (7.13) (7.14)

8 WKB () 148
8 WKB ()
8.1 WKB
ψ(r, t) = C exp (iW (r, t)/)
i ∂ψ
∂t = − ∂W ∂t ψ , ∇2 ψ = − 1 (
)
2 (∇W ) 2 − i∇ 2 W ψ
− ∂W ∂t
= 1 (
)
(∇W ) 2 − i∇ 2 W + V (r)
2m
→ 0 ,
− ∂W ∂t
= 1
2m (∇W )2 + V (r)
ψ(r, t) = e −iEt/ ψ(r)
W (r, t) = S(r) − Et , ψ(r) = C exp (iS(r)/)
1
1
(
)
(∇S) 2 − i∇ 2 S + V (r) − E = 0
2m
( )
(S ′ ) 2 + 2m V (x) − E − iS ′′ = 0 , S ′ = dS
dx ,
S′′ = d2 S
dx 2 (8.1)
S
S = S 0 + S 1 + 2 S 2 + · · ·
S 0 S 1 WKB( Wentzel–Kramers–Brillouin )
1
(S ′ ) 2 = ( S ′ 0 + S ′ 1 + 2 S ′ 2 + · · · )2
= (S
′
0 ) 2 + 2S ′ 0S ′ 1 + 2 ( 2S ′ 0S ′ 2 + (S ′ 1) 2) + · · ·
( )
(S 0) ′ 2 + 2m V (x) − E = 0 , 2S 0S ′ 1 ′ − iS 0 ′′ = 0 , 2S 0S ′ 2 ′ + (S 1) ′ 2 − iS 1 ′′ = 0 , · · ·
k(x) =
√
2m ∣ ∣V (x) − E ∣ ∣
, E > V (x)
dS 0
dx = ± k(x) , ∴ ∫
S 0(x) = ±
2
dx k(x)
S ′ 1 = i 2
S ′′
0
S ′ 0
= i k ′ (x)
2 k(x) , ∴ S 1(x) = i log k(x) +
2

8 WKB () 149
( ∫
exp (iS/) ≈ exp (iS 0 / + iS 1 ) = exp ± i dx k(x) − 1 )
2 log k(x)
=
( ∫ )
1
√ exp ± i dx k(x)
k(x)
, E > V (x) C ±
ψ(x) = ψ 1 (x) = C ( ∫ )
+
√ exp i dx k(x)
k(x)
+ C −
√ exp
k(x)
( ∫ )
− i dx k(x)
(8.2)
, E < V (x) ,
ψ(x) = ψ 2 (x) = D (∫ )
+
√ exp dx k(x) + D ( ∫ )
−
√ exp − dx k(x)
k(x) k(x)
(8.3)
WKB , (8.1) 3 iS ′′ 1 (S ′ ) 2
∣ S ′′ ∣∣∣ ∣(S ′ ) 2 =
∣ d ∣ 1 ∣∣∣ dx S ′ ≈
∣ d
∣ ∣
1
∣∣∣ dx ∣ = d 1
∣∣∣ dx k(x) ∣ = 1
2π
S ′ 0
dλ(x)
dx
∣ ≪ 1
λ(x) = 2π/k(x) WKB
d 1
∣dx
k(x) ∣ = 1 √
2
|V ′ (x)|
≪ 1 (8.4)
2 2m |V (x) − E|
3/2
V (x) , V (x) − E = 0, ,
WKB k(x) ≈ 0 (8.2), (8.3)
E = V (a) V (x) , x = a
V (x) ≈ V (a) + V ′ (a)(x − a) = E + V ′ (a)(x − a) (8.5)
(8.4)
|v 1/3 (x − a)| ≫ 1
2m
∼ 1 , v =
22/3 2 V ′ (a) (8.6)
x WKB
√
E > V (x) p(x) = k(x) = 2m ( E − V (x) )
(8.2) 2
ψ 1 = ψ + + ψ − , ψ ± (x) = C ( ∫ )
±
√ exp ± i dx k(x)
k(x)
|ψ ± (x)| 2 = |C ±| 2
k(x) = |C ±| 2
p(x)
p(x) [x, x + dx] P cl dx ,
dt p = m dx/dt
P cl dx ∝ dt = m p dx , ∴ P cl ∝ 1 p

8 WKB () 150
, WKB ψ ± (x)
J ± (x) = (
m Im ψ±(x) ∗ dψ )
±(x)
dx
dψ ± (x)
dx
=
(
) ( ∫ )
− k′ (x)
2k(x) ± ik(x) C±
√ exp ± i dx k(x)
k(x)
J ± (x) = ± m |C ±| 2 = (8.7)
= × = ± p(x)
m × |C ±| 2
p(x)
, ψ 1 = ψ + + ψ − 2
Re(ψ + ψ−) ∗ ∝ 1 ( ∫ )
p(x) cos 2 dx k(x)
,
8.2
E = V (x) x = a
η(x) =
∫ x
a
dx ′ k(x ′ ) , κ(x) =
1
√
k(x)
, (8.2), (8.3)
(
ψ 1 (x) = κ(x) C + e iη(x) + C − e −iη(x)) ,
(
ψ 2 (x) = κ(x) D + e η(x) + D − e −η(x)) (8.8)
WKB , ψ 1 (x) ψ 2 (x)
ψ 1 (x) ψ 2 (x) x a ,
x = a x = a , ψ 1 (x) ψ 2 (x)
, x = a
, ψ 1 (x) ψ 2 (x)
WKB , V (x)
, x = a (8.5) ,
(− 2 d 2 )
( )
d
2
2m dx 2 + V (x) ψ(x) = E ψ(x) ,
dx 2 − v(x − a) ψ(x) = 0
z = v 1/3 (x − a)
d 2 ψ
dz 2 − zψ = 0
(15.100) , ψ = F Ai(z) + GBi(z)
F G (15.102), (15.103) z → ∞
[
1
ψ(x) →
2 √ F exp
(− 2 ) ( )] 2
π z 1/4 3 z3/2 + 2G exp
3 z3/2
(8.9)

8 WKB () 151
z → − ∞
ψ(x) →
=
1
√ π |z|
1/4
[ ( 2
F sin
3 |z|3/2 + π ) ( 2
+ G cos
4 3 |z|3/2 + π )]
4
[
(
1
2i
2 √ (G − iF ) exp
π |z| 1/4 3 |z|3/2 + iπ )
(
+ (G + iF ) exp − 2i
4
3 |z|3/2 − iπ )]
4
(8.10)
277 z → ± ∞ |z| 2
,
WKB ,
, WKB
(8.6) z |z| ≫ 1/2 2/3
x < a
2
3 |z|3/2 = 2 ∫ x
3 |v|1/2 (a − x) 3/2 = − |v| 1/2 dx ′ (a − x ′ ) 1/2
(8.5)
x > a
v > 0
v(x − a) ≈ 2m ( )
2 V (x) − E
2
3 |z|3/2 ≈ −
a
∫ x
a
⎧
2
⎨
3 |z|3/2 ≈
⎩
E
dx ′ √
2m
2 ∣ ∣∣V (x) − E
∣ ∣∣ = − η(x)
− η(x) ,
η(x) ,
x < a
x > a
V (x)
a
WKB ψ 1 WKB ψ 2
(8.11)
x − a z , x > a (8.9), x < a (8.10)
(8.11)
⎧
v 1/6 (
⎪⎨ 2 √ F e −η(x) + 2Ge η(x)) ,
x > a
πk(x)
ψ(x) =
v 1/6 [
⎪⎩
2 √ (G − iF ) e −iη(x)+iπ/4 + (G + iF ) e iη(x)−iπ/4] , x < a
πk(x)
(8.8)
D + = v1/6
√ G , D − = v1/6
π 2 √ π F , C + = v1/6
2 √ π (G + iF ) e−iπ/4 , C − = v1/6
2 √ (G − iF ) eiπ/4
π
C + =
( )
( )
1 1
2 D + + iD − e −iπ/4 , C − =
2 D + − iD − e iπ/4 (8.12)
(8.8)
⎧ (
⎪⎨ κ(x) D + e η(x) + D − e −η(x)) ,
ψ(x) = [ (
⎪⎩ κ(x) D + sin η(x) + π 4
)
+ 2D − cos
x > a
(
η(x) + π ) ]
, x < a
4

8 WKB () 152
D + D + ≠ 0 , x > a e η(x) WKB
η(x) ≫ 1 e η(x) ≫ e −η(x) e −η(x) ,
D + ≠ 0
⎧
⎪⎨ D + κ(x)e η(x) ,
ψ(x) = [
⎪⎩ κ(x) D + sin
(
η(x) + π 4
x > a
) (
+ 2D − cos η(x) + π ) ]
, x < a
4
x > a D + = 0
⎧
⎨ D − κ(x) e −η(x) ,
x > a
ψ(x) =
(
⎩ 2D − κ(x) cos η(x) + π )
, x < a
4
(8.13)
8.1
x < a ψ(x) = Cκ(x)cos
(η(x) + π )
4 − θ
, sin θ ≠ 0 x > a
ψ(x) = Cκ(x) e η(x) sin θ
v < 0
x − a z , x > a
(8.10), x < a (8.9)
⎧
|v| 1/6 [
2 √ (G − iF ) e iη(x)+iπ/4
πk(x)
⎪⎨
ψ(x) =
+ (G + iF ) e −iη(x)−iπ/4] , x > a
⎪⎩
(8.8)
|v| 1/6
2 √ πk(x)
C + =
(
F e η(x) + 2Ge −η(x)) ,
x < a
E
a
V (x)
( )
( )
1 1
2 D − − iD + e iπ/4 , C − =
2 D − + iD + e −iπ/4 (8.14)
, (8.8)
⎧ (
⎪⎨ κ(x) D + e η(x) + D − e −η(x)) ,
ψ(x) = [ (
⎪⎩ κ(x) 2D + cos η(x) − π 4
D ± C ±
⎧
⎪⎨ e −iπ/4 κ(x)
ψ(x) =
⎪⎩
)
− D − sin
x < a
(
η(x) − π ) ]
, x > a
4
[
(C + + iC − ) e −η(x) + iC ]
+ + C −
e η(x) , x < a
2
(
κ(x) C + e iη(x) + C − e −iη(x)) ,
x > a
η(x) x < a η(x) < 0 , x e −η(x)
, D − = 0 x
⎧
⎨ D + κ(x) e η(x) ,
ψ(x) =
⎩ 2D + κ(x) cos
(
η(x) − π 4
x < a
)
, x > a
(8.15)

8 WKB () 153
D − ≠ 0 ,
x > a x C − = 0
⎧
⎪⎨ e −iπ/4 C + κ(x)
(e −η(x) + i )
ψ(x) =
2 eη(x) , x < a
⎪⎩
C + κ(x) e iη(x) ,
x > a
(8.16)
WKB e −η(x) ≫ e η(x) e η(x)
8.3
1 WKB
x = a x = b V (x) = E
x = a v < 0 , x = b v > 0
x → − ∞ ψ(x) → 0
WKB (8.15)
⎧
⎨ D + κ(x) e ηa(x) ,
x < a
ψ(x) =
(
⎩ 2D + κ(x) cos η a (x) − π )
, x > a
4
E
a
η a (x) =
, x → ∞ ψ(x) → 0 (8.13)
⎧
(
⎨ 2D − κ(x) cos η b (x) + π )
, x < b
ψ(x) =
4
η
⎩
b (x) =
D − κ(x) e −ηb(x) ,
x > b
∫ x
a
∫ x
b
b
V (x)
dx ′ k(x ′ ) (8.17)
dx ′ k(x ′ ) (8.18)
a < x < b 2 ,
η a (x) =
∫ x
a
dx ′ k(x ′ ) =
∫ b
a
dx ′ k(x ′ ) +
∫ x
b
dx ′ k(x ′ ) =
∫ b
a
dx ′ k(x ′ ) + η b (x) (8.19)
(
cos η a (x) − π )
= cos
(η b (x) + π )
∫ b
4
4 + θ , θ = dx k(x) − π
a 2
, 2 , n θ = nπ , D − = (−1) n D +
∫ b
a
dx k(x) =
∫ b
a
√ ( )
dx 2m E − V (x)
(
= π n + 1 )
, n = 0, 1, 2, · · · (8.20)
2
n + 1/2 n WKB
⎧
e
ψ(x) = √ D ⎪⎨
ηa(x) ,
x < a
(
× 2 cos η a (x) − π )
, a < x < b
k(x) 4
⎪⎩
(−1) n e −ηb(x) , x > b
(8.21)

8 WKB () 154
(8.20) η a (b) = π (n + 1/2) dη a /dx = k(x) > 0 η a (x) η a (a) = 0
η a (b) = π (n + 1/2) , a ≤ x ≤ b 0 ,
cos (η a (x) − π/4) = 0 x
(
η a (x) = π n ′ − 1 )
, n ′ = 1, 2, 3, · · ·
4
n , n WKB η a (x) ≫ 1
, (8.20) n ≫ 1 ,
a ≤ x ≤ b , , |ψ| 2 ,
cos 2 (· · · ) 1/2
(
|ψ| 2 = 4|D|2
k(x) cos2 η a (x) − π )
=⇒ 2|D|2
4 k(x) = 2|D|2
p(x)
(8.22)
p(x) 1
WKB
V (x) V (x) = V (−x) , k(x) a = − b , b > 0
η a (x) =
∫ x
dx ′ k(x ′ ) = −
∫ −x
dx ′ k(−x ′ ) = −
∫ −x
−b
b
b
dx ′ k(x ′ ) = − η b (−x)
η a (−x) =
∫ −x
−b
dx ′ k(x ′ ) =
∫ b
−b
dx ′ k(x ′ ) +
∫ −x
b
dx ′ k(x ′ ) = π
η a (−x) − π (
4 = − η a (x) − π )
+ nπ
4
(
n + 1 )
− η a (x)
2
, (8.21) n ,
, WKB
1
, V (x) = mω 2 x 2 /2 , (8.20)
∫ x0 √
(
mω dx x 2 0 − x2 = π n + 1 )
√
2E
, x 0 =
−x 0
2
mω 2
x = x 0 sin θ
∫ x0
−x 0
dx
√
x 2 0 − x2 = 2
∫ x0
0
dx
√
x 2 0 − x2 = 2x 2 0
∫ π/2
0
dθ cos 2 θ = π 2 x2 0 = πE
mω 2
E = ω (n + 1/2) ,
V (x) = V (−x) ψ(x) n , x ≥ 0
ψ(x) 1 α = √ mω/ , q = αx
z =
q αx
√ = √ 2n + 1 2n + 1

8 WKB () 155
x 0 = √ 2n + 1/α , WKB
⎧
ψ(z) = √ D ⎨ e −η +(z) , z > 1
× (
k(z) ⎩ 2 cos η + (z) + π )
, 0 ≤ z < 1
4
⎧
⎪⎨
ψ(z) =
⎪⎩
√
2m
k(z) =
2 |V (x) − E| = α√ (2n + 1)|z 2 − 1|
η ± (z) =
∫ x
±x 0
dx ′ k(x ′ ) = (2n + 1)
∫ z
±1
dz ′√ |z ′2 − 1|
∫
dx √ x 2 − 1 = 1 (
x √ x
2
2 ∣
− 1 − log ∣x + √ )
x 2 − 1∣
∫
dx √ 1 − x 2 = 1 (
x √ )
1 − x
2
2 + sin −1 x
2D ′ ( 2n + 1
(
(1 − z 2 ) cos z √ )
1 − z 1/4 2
2 + sin −1 z − nπ 2
D ′ (
z + √ z 2 − 1
(z 2 − 1) 1/4
)
, 0 < z < 1
) (
n+1/2
exp − 2n + 1 z √ )
z
2
2 − 1 , z > 1
n → ∞ H n (q) (15.105)
ψ(q)/ √ α (4.22)
ϕ n (q) 1
√ = √ H α π
1/2
n! 2 n e−q2 n (q)
WKB ψ(q) z 2 = 1 , ( q 2 = 2n + 1 )
|1 − z 2 | −1/4 |ψ| 2 D ′
, ε q = ε ψ(q) = ϕ n (q) D ′
, WKB n , n ,
, ,
| q | < √ 2n + 1 |ψ(q)| 2 , (8.22)
,
1.0
n = 0
n = 3
n = 50
0.6
0.4
0.2
−0.2
1 2 3 4 q
0.2
0.0
−0.2
2 4 6 8 10 q
−0.6
−0.4

8 WKB () 156
8.2 V (a) = ∞ , x = b (8.18) , x = a
(8.17) (8.17) ψ(a) = 0
V (x)
E
a
b
1.
∫ b
a
dx √ 2m (E − V (x)) = π
(
n + 3 )
, n = 0, 1, 2, · · · (8.23)
4
2. u(x) x V (x) = u(x) (8.20) n
, (8.23)
{
u(x) , x > 0
V (x) =
∞ , x ≤ 0
3. (2.32) (8.23) E , (2.36)
8.4
x < a
, x > b
x < a
x > b ,
, x > b
( )
x > b x
a
WKB (8.16)
⎧
(
⎪⎨ e −iπ/4 C + κ(x) e −ηb(x) + i )
ψ(x) =
2 eη b(x)
, x < b
⎪⎩
C + κ(x) e iηb(x) ,
x > b
E
b
V (x)
e ηb(x) (8.19)
∫ b
η b (x) = η a (x) − P , P = dx k(x) = 1 ∫ b
√ ( )
dx 2m V (x) − E
a
x < b
(
ψ(x) = κ(x) D + e ηa(x) + D − e −η a(x)
, D − = e −iπ/4+P C + , D + = i 2 e−iπ/4−P C +
x = a x > a (8.12) x < a
a
(8.24)
ψ(x) = ψ + (x) + ψ − (x) ,
ψ ± = C ′ ±κ(x)e ±iηa(x)

8 WKB () 157
C ′ + =
C ′ − =
( )
1
2 D + + iD − e −iπ/4 =
(e P + e−P
4
( 1
2 D + − iD −
)
e iπ/4 = i
(e P − e−P
4
)
C +
)
C +
ψ + , ψ − (8.7) J I J R
J I = (
m Im ψ+(x) ∗ dψ )
+(x)
dx
J R = (
m Im ψ−(x) ∗ dψ )
−(x)
dx
= m |C′ +| 2 = m
= − m |C′ −| 2 = − m
, x > a J T
J T = (
m Im ψ ∗ (x) dψ(x) )
dx
) 2 (e P + e−P
|C + | 2
= m |C +| 2
4
) 2 (e P − e−P
|C + | 2
, T R
T = |J T|
|J I | = e −2P
(1 + e −2P /4) 2 , R = |J (
R| 1 − e −2P ) 2
|J I | = /4
1 + e −2P /4
R + T = 1 WKB P = η a (b) ≫ 1
T ≈ e −2P = exp
(
− 2
T ≈ e −2P
∫ b
a
√ (
dx 2m V (x) − E) ) , R ≈ 1
4
8.5
, α A ( Z
N ) A > 200 , α ( 2, 2 ,
4 He ) (, ) = (Z − 2, N − 2) α
, 1 fm = 1×10 −15 m
, , α
V (r)
⎧
⎪⎨ − V 0 , r < R
V (r) = Z α Z D c α
⎪⎩
,
r
r > R
− V 0 < 0
, Z α = 2 α , Z D = Z − 2 α
, α
r , R α
V C
V C = Z αZ D c α
R
V C
E α
O
− V 0
R
b
r

8 WKB () 158
r 0 A 1/3 , ( r 0 ≈ 1.2 fm ) , α
R = r 0 A 1/3 + ∆R , ∆R ≈ 1.6 fm ( U )
A ≈ 230 R ≈ 9 fm Z D = 90 (6.30), (6.31)
V C ≈
2 × 90 × 200
9 × 137
MeV ≈ 29 MeV
α E α 4 ∼ 9 MeV , V C
, r < R α , r > R
,
1 (8.24) P 3 ,
V (x) (6.6) V eff (r) , , α l = 0
(8.24)
Z α Z D c α
b
− E α = 0 , b = Z αZ D c α
E α
= V C
E α
R
P = 1
∫ b
R
√ ( ) √
Zα Z D c α
2mEα
dr 2m
− E α =
r
∫ b
R
dr
√
b
r − 1
, m α R/b ≤ r/b ≤ 1
r = b cos 2 θ , 0 ≤ θ ≤ θ 0 = cos −1 ( √
R/b
)
√ 2mEα
P = 2b
∫ θ0
0
√
dθ sin 2 2mEα
θ =
√ (
2mEα
= b
[
b θ −
] θ0
sin 2θ
2
0
cos −1 √
R
b − √
R
b − R2
b 2 )
E α ≪ V C √ R/b ≪ 1 cos −1 x = π/2 − x − x 3 /6 + · · ·
√ ( √
2mEα π R
P ≈ b
2 − 2 b + 1 ( ) ) 3/2 R
= 1 ( )
C1
√ − C 2 + C 3 E α
3 b 2 Eα
(8.25)
(8.26)
√ √
√ 2αZα Z D mc
C 1 = παZ α Z D 2mc2 , C 2 = 4
2 R
, C 3 = 2 2mc 2 R 3
c
3 αZ α Z D (c) 3
α R
α v in , v in /(2R)
e −2P
,
α λ λ = e −2P v in /(2R) t K(t)
, dt α K(t)λ dt
K(t + dt) = K(t) − K(t)λ dt , dK dt = − λK , ∴ K(t) = K 0 e −λt

8 WKB () 159
, T 1/2 ( K(t) K 0 K(T 1/2 ) = K 0 /2 )
(
T 1/2 = log 2
λ = C 0 2 √ 2mE
√ α
exp
b
Eα
(
cos −1 √
R
b − √
R
b − R2
b 2 ))
(8.27)
≈ C ( )
0 C1
√ exp √ − C 2 + C 3 E α , C 0 = R √ 2m log 2 (8.28)
Eα Eα
v in √ 2E α /m , C 3 = 0 (8.28)
( Geiger-Nuttall )
( Po, Z = 84 ), ( U, Z = 92 ), ( Fm, Z = 100 )
, T 1/2 α m mc 2 = 3727 MeV ,
Z D
R C 0 C 1
C 2
C 3
fm sec MeV 1/2 MeV 1/2 MeV −1
Po 82 8.78 1.75×10 −21 325 79.7 0.494
U 90 8.95 1.79×10 −21 356 84.3 0.485
Fm 98 9.16 1.83×10 −21 388 89.0 0.481
(8.27) , (8.28) •
( http://www.nndc.bnl.gov/ensdf/ ), • A
(8.27) , , T 1/2
E α 2 T 1/2 20 , T 1/2
e C1/√ E α
, ( C 1 ≫ √ E α )
238
100
10 15
236
1
T 1/2 ( sec )
10 12
10 9
10 6
10 3
10 0
234
Po
U
232
208
210
206
Fm
230
218
10 18 4 5 6 7 8 9
252
254
250
228
216
226
248
246
1
1
1
1
10 −3
214
224
10 −6
212
222
E α ( MeV )

9 160
9
9.1
H H 0 H ′ (t)
H = H 0 + H ′ (t)
H 0 E n | n 〉
H 0 | n 〉 = E n | n 〉 ,
〈 m | n 〉 = δ mn
H H , E n
,
t | t 〉 ,
i ∂ | t 〉 = H| t 〉 (9.1)
∂t
H 0 , e −iE nt/ | n 〉
, | t 〉
| t 〉 = ∑ n
c n (t) e −iE nt/ | n 〉 (9.2)
(9.1)
i ∂ ∂t | t 〉 = ∑ (
i dc )
n
dt + E nc n e −iEnt/ | n 〉
n
H| t 〉 = ∑ ( )
c n (t) e −iE nt/
H 0 + H ′ (t) | n 〉 = ∑
n
n
( )
c n e −iE nt/
E n + H ′ (t) | n 〉
∑
| m 〉
n
i dc n
dt e−iE nt/ | n 〉 = ∑ n
c n e −iE nt/ H ′ (t)| n 〉
i dc m
dt e−iEmt/ = ∑ n
c n e −iEnt/ 〈 m |H ′ (t)| n 〉
H ′ mn(t) = 〈 m |H ′ (t)| n 〉 e i(Em−En)t/ = 〈 m |H ′ I(t)| n 〉 , H ′ I(t) = e iH0t/ H ′ (t)e −iH0t/ (9.3)
dc m
dt
= 1
i
∑
H mn(t) ′ c n (t) (9.4)
(9.1)
c m (t) = c m (t 0 ) + 1
i
n
∑
n 1
∫ t
t 0
dt 1 H ′ mn 1
(t 1 ) c n1 (t 1 )

9 161
t = t 0 c m (t 0 ) c n1 (t 1 )
c n1 (t 1 ) = c n1 (t 0 ) + 1
i
∑
n 2
∫ t1
t 0
dt 2 H ′ n 1n 2
(t 2 ) c n2 (t 2 )
c m (t) = c m (t 0 ) + 1
i
∑
n 1
∫ t
t 0
dt 1 H ′ mn 1
(t 1 ) c n1 (t 0 )
+ 1 ∑
∫ t ∫ t1
(i) 2 dt 1
n 1n 2
t 0
t 0
dt 2 H ′ mn 1
(t 1 )H ′ n 1 n 2
(t 2 ) c n2 (t 2 )
c (1)
m (t) = 1
i
∑
n 1
∫ t
c m (t) = c m (t 0 ) + c (1)
m (t) + c (2)
m (t) + · · · + c (k)
m (t) + · · · (9.5)
t 0
dt 1 H ′ mn 1
(t 1 ) c n1 (t 0 )
c (2)
m (t) = 1
(i) 2 ∑
n 1 n 2
∫ t
t 0
dt 1
∫ t1
c (k)
m (t) = 1 ∑
∫ t
(i) k
n 1···n k
t 0
dt 1
∫ t1
t 0
dt 2 H ′ mn 1
(t 1 )H ′ n 1n 2
(t 2 ) c n2 (t 0 )
∫ tk−1
t 0 t 0
dt 2 · · · dt k H mn ′ 1
(t 1 )H n ′ 1 n 2
(t 2 ) · · · H n ′ k−1 n k
(t k ) c nk (t 0 )
t = t 0 , , n c n (t 0 ) , t
c n (t) , c (k)
n (t, t 0 ) , (9.5)
H ′ , H ′ mn(t)
c m (t) ≈ c m (t 0 ) + c (1)
m (t)
t = t 0 H 0 | i 〉 c m (t 0 ) = δ mi
, m ≠ i
c m (t) ≈ c (1)
m (t) = 1
i
∫ t
t 0
dt 1 H ′ mi(t 1 ) = 1
i
∫ t
t 0
dt 1 e i(E m−E i )t 1 / 〈 m |H ′ (t 1 )| i 〉 (9.6)
9.2
, (9.5)
(9.2) 〈 n | t 〉 = c n (t)e −iEnt/
c n (t) = e iEnt/ 〈 n | t 〉 = 〈 n |e iH0t/ | t 〉
(9.3) ,
∑
H mn ′ 1
(t 1 )H n ′ 1 n 2
(t 2 ) c n2 (t 0 ) = ∑ 〈 m |H I(t ′ 1 )| n 1 〉〈 n 1 |H I(t ′ 2 )| n 2 〉〈 n 2 |e iH 0t 0 / | t 0 〉
n 1 n 2 n 1 n 2
= 〈 m |H I(t ′ 1 )H I(t ′ 2 )e iH0t0/ | t 0 〉

9 162
U(t, t 0 ) = 1 + 1 ∫ t
i
+ 1
(i) k ∫ t
c m (t) = 〈 m |U(t, t 0 ) e iH 0t 0 / | t 0 〉
t 0
dt 1 H ′ I(t 1 ) + 1
(i) 2 ∫ t
t 0
dt 1
∫ t1
t 0
dt 1
∫ t1
t 0
dt 2 H ′ I(t 1 )H ′ I(t 2 ) + · · ·
∫ tk−1
t 0 t 0
dt 2 · · · dt k H I(t ′ 1 )H I(t ′ 2 ) · · · H I(t ′ k ) + · · · (9.7)
| t 〉 = ∑ n
c n (t) e −iE nt/ | n 〉 = e −iH 0t/ ∑ n
| n 〉 c n (t)
= e −iH 0t/ ∑ n
| n 〉〈 n |U(t, t 0 ) e iH 0t 0 / | t 0 〉
= e −iH 0t/ U(t, t 0 ) e iH 0t 0 / | t 0 〉 (9.8)
U(t, t 0 ) 2
U (2) (t, t 0 ) =
∫ t
∫ t1
dt 1 dt 2 H I(t ′ 1 )H I(t ′ 2 )
t 0 t 0
t 2
t
t 0 ≤ t 2 ≤ t ,
t 2 ≤ t 1 ≤ t
U (2) (t, t 0 ) =
∫ t
∫ t
dt 2 dt 1 H I(t ′ 1 )H I(t ′ 2 )
t 0 t 2
t 0
t 0
t
t 1
U (2) (t, t 0 ) = 1 2
∫ t
U (2) (t, t 0 ) =
t 0
dt 1
∫ t
t 0
dt 2
(
∫ t
t 0
dt 1
∫ t
t 1
dt 2 H ′ I(t 2 )H ′ I(t 1 )
)
θ(t 1 − t 2 )H I(t ′ 1 )H I(t ′ 2 ) + θ(t 2 − t 1 )H I(t ′ 2 )H I(t ′ 1 )
θ(t) =
H ′ I (t 1) H ′ I (t 2)
U (2) (t, t 0 ) = 1 2
∫ t
t 0
dt 1
∫ t
{
1 , t > 0
0 , t < 0
t 0
dt 2 H ′ I(t 1 )H ′ I(t 2 ) = 1 2
(∫ t
t 0
dt 1 H ′ I(t 1 )
, t 1 ≠ t 2 H ′ I (t 1) H ′ I (t 2) T
T[F (t 1 )G(t 2 )] = θ(t 1 − t 2 )F (t 1 )G(t 2 ) + θ(t 2 − t 1 )G(t 2 )F (t 1 )
) 2

9 163
U (2) (t, t 0 ) = 1 2!
∫ t
t 0
dt 1
∫ t
t 0
dt 2 T[H ′ I(t 1 )H ′ I(t 2 )]
3 , T ,
∫ t
t 0
dt 1 · · ·
U(t, t 0 ) =
k=0
∫ tk−1
t 0
dt k H ′ I(t 1 ) · · · H ′ I(t k ) = 1 k!
∫ t
t 0
dt 1 · · ·
∫ t
t 0
dt k T[H ′ I(t 1 ) · · · H ′ I(t k )]
∞∑
∫
1 1 t ∫ t
( ∫ 1 t
)
k! (i) k dt 1 · · · dt k T[H I(t ′ 1 ) · · · H I(t ′ k )] = T exp dt ′ H I (t ′ )
t 0 t 0
i t 0
(9.8)
| t 〉 = V (t, t 0 )| t 0 〉 , V (t, t 0 ) = e −iH 0t/ U(t, t 0 ) e iH 0t 0 /
(9.9)
i ∂ ∂t V (t, t 0) = i ∂e−iH0t/
U(t, t 0 ) e iH0t0/ + e −iH0t/ i ∂U(t, t 0)
e iH0t0/
∂t
∂t
= H 0 e −iH 0t/ U(t, t 0 ) e iH 0t 0 / + e −iH 0t/ i ∂U(t, t 0)
∂t
e iH 0t 0 /
(9.7)
i ∂U(t, t 0)
∂t
= H I(t) ′ + 1 ∫ t
dt 2 H
i
I(t)H ′ I(t ′ 2 ) + · · ·
t 0
+
∫
1 t
(i) k−1
t 0
dt 2 · · ·
∫ tk−1
t 0
dt k H ′ I(t)H ′ I(t 2 ) · · · H ′ I(t k ) + · · ·
= H ′ I(t)U(t, t 0 ) = e iH0t/ H ′ (t)e −iH0t/ U(t, t 0 ) (9.10)
i ∂ ∂t V (t, t 0) = H 0 V (t, t 0 ) + H ′ (t)V (t, t 0 ) = H(t)V (t, t 0 )
, , | t 〉 = V (t, t 0 )| t 0 〉
(9.1) H , V (t 0 , t 0 ) = 1
V (t, t 0 ) = e −i(t−t 0)H/
t 2
| α(t) 〉 = V (t, t 0 )| α 0 〉 , | β(t) 〉 = V (t, t 0 )| β 0 〉
F f αβ (t) = 〈 α(t) | F | β(t) 〉
(9.9)
f αβ (t) = 〈 α 0 | F H (t) | β 0 〉 , F H (t) = V † (t, t 0 )F V (t, t 0 )

9 164
t 0 f αβ (t) F H (t)
F H (t)
( , F )
i ∂F H
∂t
F H 1 = V V †
i ∂F H
∂t
= i ∂V † (t, t 0 )
F V (t, t 0 ) + V † (t, t 0 )F i ∂V (t, t 0)
∂t
∂t
= − V † (t, t 0 )HF V (t, t 0 ) + V † (t, t 0 )F HV (t, t 0 )
= [ F H (t) , H H (t) ] , H H (t) ≡ V † (t, t 0 )H(t)V (t, t 0 ) (9.11)
, (9.1) (9.11)
H H H (t) (9.11)
i ∂H H
∂t
= [ H H (t) , H H (t) ] = 0 , H H (t) = = H
V (t, t 0 ) = e −i(t−t0)H/
H H (t) = e i(t−t0)H/ He −i(t−t0)H/ = H
( )
(9.8)
| α(t) 〉 = e −iH 0t/ | α(t) 〉 I , | α(t) 〉 I = U(t, t 0 ) e iH 0t 0 / | α 0 〉
f αβ (t) = I 〈 α(t) | F I (t) | β(t) 〉 I ,
F I (t) = e iH 0t 0 / F e −iH 0t/
(9.10)
i ∂ ∂t | α(t) 〉 I = i ∂U(t, t 0)
∂t
(9.10)
e iH0t0/ | α 0 〉 = H ′ I(t)U(t, t 0 ) e iH0t0/ | α 0 〉 = H ′ I(t)| α(t) 〉 I
i ∂ ∂t | α(t) 〉 I = i ∂ ∂t eiH0t/ | α(t) 〉 = i ∂eiH 0t/
| α(t) 〉 + e iH0t/ i ∂ ∂t
∂t | α(t) 〉
( )
= − H 0 e iH0t/ | α(t) 〉 + e iH0t/ H 0 + H ′ (t) | α(t) 〉
= e iH 0t/ H ′ (t)e −iH 0t/ e iH 0t/ | α(t) 〉
, F I (t)
= H ′ I(t)| α(t) 〉 I
i ∂ ∂t F I(t) = e iH0t/( − H 0 F + F H 0
)e −iH0t/ = [ F I (t) , H 0 ]
H H ′ I (t) H 0
( )
H I ′(t) , H′ I (t)

9 165
H H I ′ (t)
H 0 H
| α(t) 〉 I H 0 | n 〉
| α(t) 〉 I = ∑ n
c n (t)| n 〉
i ∂ ∂t | α(t) 〉 I = i ∑ n
dc n
dt | n 〉 = ∑ n
c n H ′ I(t)| n 〉
| m 〉
i ∑ n
dc m
dt
= ∑ n
H ′ mn(t)c n (t) , H ′ mn(t) = 〈 m |H ′ I(t)| n 〉
(9.4)
9.1
i d dt | t 〉 I = H ′ I(t) | t 〉 I (9.8)
9.3
H ′ (t)
H ′ (t) =
{
V , t > 0
0 , t < 0
V ω mn ≡ (E m − E n )/ (9.6)
c (1)
m (t) = 1 ∫ t
i 〈 m |V | i 〉 dt 1 e iωmit1 = 1
i 〈 m |V | i 〉sin(ω mit/2)
ω mi /2
0
e iωmit/2 (9.12)
, | i 〉 t > 0 | m 〉 , 1
|c (1)
m (t)| 2 = 1 ( ) 2
2 |〈 m |V | i sin(ωmi t/2)
〉|2 = π t
ω mi /2 2 |〈 m |V | i 〉|2 f(ω mi /2, t) (9.13)
ω mi ≠ 0 |c (1)
m (t)| 2
f(x, t) = sin2 tx
πtx 2
0 ≤ |c (1)
m (t)| 2 4|〈 m |V | i 〉|2
≤
(ω mi ) 2
2π/ω mi , ω mi → 0
|c (1)
m (t)| 2 = t2
|〈 m |V | i 〉|2
2 t 2 , |〈 m |V | i 〉| 2 |c (1)
m (t)| 2 > 1
, 1 , ω mi ≈ 0 t
t
t 2
2 |〈 m |V | i 〉|2 ≪ 1

9 166
f(x, t) x
t , f(x, t) x =
0 t/π , 2π/t
, t (
|c (1)
m (t)| 2 ≪ 1 )
|ω mi | =
E m − E i
∣ ∣ < 2π (9.14)
t
t/π
| m 〉 |c (1)
m (t)| 2
−π/t
, | m 〉 , E m ≈
E i | m 〉
E i − ∆E ≤ E m ≤ E i + ∆E
| m 〉 ,
E E +dE ρ(E) dE ρ(E)
ρ(E) ρ(E) , P
P =
∫ Ei +∆E
E i −∆E
t
dE m ρ(E m ) |c (1)
m (t)| 2 = πt ∫ Ei +∆E
2 dE m ρ(E m ) |〈 m |V | i 〉| 2 f(ω mi /2, t)
∆E
E i −∆E
≫ 2π
t
π/t
(9.15)
, (9.14) , |〈 m |V | i 〉| 2
ρ(E m ) E m , (9.14) ,
E m = E i
P = πt (
)
2 |〈 m |H ′ | i 〉| 2 ρ(E m ) I , I =
E m =E i
x = t ω mi /2 = t (E m − E i )/2
∫ Ei +∆E
E i−∆E
dE m f(ω mi /2, t)
I = 2 π
∫ t∆E/2
−t∆E/2
dx sin2 x
x 2
(9.15) t∆E/2 ≫ π −∞ ∞
I = 2 π
∫ ∞
−∞
dx sin2 x
= 2
x 2
P = t 2π
w = P t = 2π
(
)
|〈 m |H ′ | i 〉| 2 ρ(E m )
E m=E i
(
)
|〈 m |H ′ | i 〉| 2 ρ(E m )
(9.16)
E m=E i
f(x, t → ∞) = δ(x) (9.17)

9 167
(9.13) t → ∞
|c (1)
m (t)| 2 = t π 2 |〈 m |V | i 〉|2 δ(ω mi /2) = t 2π |〈 m |V | i 〉|2 δ(E m − E i )
δ(ax) = δ(x)/|a| 1 | m 〉 w i→m
w i→m = |c(1) m (t)| 2
= 2π t |〈 m |V | i 〉|2 δ(E m − E i ) (9.18)
∫
w =
dE m ρ(E m ) w i→m = 2π
= 2π
∫
dE m ρ(E m ) |〈 m |V | i 〉| 2 δ(E m − E i )
(
)
|〈 m |V | i 〉| 2 ρ(E m )
E m =E i
(9.16)
t = 0 V , t = − ∞
H ′ (t) = V e −ε|t| , ε → + 0
ε e −ε|t| t H ′ (t) = V
, t → ± ∞ e −ε|t| → 0 H ′ (t) → 0 , ( t 0 → − ∞ )
( t → + ∞ ) H 0 t 0 → − ∞ , t → + ∞
c (1)
m (∞) = 1 ∫ ∞
i 〈 m |V | i 〉 dt e iωmit−ε|t| = 2π
i 〈 m |V | i 〉 δ(ω mi)
(14.10)
−∞
)
|c (1)
m (∞)| 2 = 4π2
2
2 |〈 m |V | i 〉|2( δ(ω mi )
2 , t 0 → − ∞ , t → + ∞
w i→m
(9.18)
δ(ω) = 1 ∫ t
dt e iωt
2π t 0
δ(0) = 1 ∫ t
dt = t − t 0
2π t 0
2π
( ) 2 t − t 0
δ(ω) = δ(0) δ(ω) =
2π
δ(ω)
w i→m = |c(1) m (∞)| 2
= 2π
t − t 0 2 |〈 m |V | i 〉|2 δ(ω mi ) = 2π |〈 m |V | i 〉|2 δ(E m − E i )
9.2
(9.17)

9 168
9.4
H 0 = − 2 ∇ 2 /2m , exp(ik·r)
,
1. 1
2.
1 L 1 ,
u(x, y, z) = u(x + L, y, z) = u(x, y + L, z) = u(x, y, z + L) ,
u(r) = C exp(ik·r)
exp(ik x L) = exp(ik y L) = exp(ik z L) = 1
, n x , n y , n z
k x = 2π L n x , k y = 2π L n y , k z = 2π L n z (9.19)
, k L → ∞ k 1
u k (r) = 1 √
V
exp(ik·r) , V = L 3 (9.20)
V = L 3
u k 2 k 2 /2m ,
E 0
F (k x )
N = ∑ k
1 ,
,
|k| ≤ k 0 , k 0 =
√
2mE0
k k
k x ∆k x = 2π/L
2
∑
F (k x ) = L ∑
∆k x F (k x )
2π
k x
k x
∆k x
k x
, ∆k x L → ∞ ∆k x → 0
,
∑
F (k x ) −−−−→ L→∞ L ∫
dk x F (k x ) , 3 ∑
2π
k
k x
F (k) −−−−→
L→∞ V ∫
(2π) 3 d 3 k F (k)
N =
V ∫
(2π) 3 d 3 k =
V ∫ k0
∫
|k|≤k 0
(2π) 3 dk k 2 dΩ
0

9 169
k E = 2 k 2 /2m
N =
∫ E0
0
∫
dE dΩ
V
√
m 2mE
(2π) 3 2 2
E E + dE , k Ω Ω + dΩ
ρ box (E) dE dΩ
ρ box (E) =
V
√
m 2mE
(2π) 3 2 2 (9.21)
(9.21) ( ρ box (E)
4π ) ,
,
k
∫
∫
d 3 x u ∗ k(r) u k ′(r) = Ck ∗ C k ′
u k (r) = C k exp(ik·r)
d 3 x exp(i(k ′ − k)·r) = (2π) 3 C ∗ k C k ′δ(k − k ′ ) = δ(k − k ′ )
C k C k = 1/(2π) 3/2 ,
u k (r) =
1
exp(ik·r)
(2π)
3/2
(9.20) V = (2π) 3
ρ(E) = m 2 √
2mE
2
, ,
9.5
H ′ (t) = V e −iωt + V † e iωt
V t = 0 | i 〉
t
c (1)
m (t) = 1
i
∫ t0
0
dt 1 〈 m | ( V e −iωt 1
+ V † e iωt 1 ) | i 〉
(9.12) ω mi → ω mi ± ω
c (1)
m (t) = − i 〈 m |V | i 〉 sin((ω mi − ω)t/2)
e i(ω mi−ω)t/2
(ω mi − ω)/2
− i 〈 m |V † | i 〉 sin((ω mi + ω)t/2)
(ω mi + ω)/2
e i(ωmi+ω)t/2 (9.22)

9 170
9.2 , c (1)
m (t) ≈ 0
1 E m ≈ E i + ω , 2
ω | i 〉 | m 〉 2
E m ≈ E i − ω , 1 ω
| i 〉 | m 〉
E m ≈ E i + ω
|c (1)
m (t)| 2 = |〈 m |V | i 〉| 2 πt
2 f((ω mi − ω)/2, t)
E i + ω − ∆E ≤ E m ≤ E i + ω + ∆E
| m 〉 w (9.16)
E m = E i E m = E i + ω
w = 2π
(
)
|〈 m |H ′ | i 〉| 2 ρ(E m )
E m=E i+ω
9.3
(5.66) H
)
H = H 0 + H ′ (t) , H 0 = ω 0 σ z , H ′ (t) = ω 1
(σ x cos ωt + σ y sin ωt
σ ± = σ x ± i σ y
H ′ (t) = ω (
1
σ + e −iωt + σ − e iωt) , σ − = (σ + ) †
2
ω 0 ≫ ω 1 H ′ (t) (9.22)
P − (t) (5.70)
9.6
H ′ (t) , H ′ (t)
t , H(t) | n(t) 〉 E n (t)
H(t) | n(t) 〉 = E n (t) | n(t) 〉 (9.23)
t = 0 | n(0) 〉 , H(t) , t
| n(t) 〉 ,
t | t 〉 | n(t) 〉
| t 〉 = ∑ n
c n (t) e −iθn(t) | n(t) 〉 ,
θ n (t) = 1
∫ t
0
dt ′ E n (t ′ )
| ṅ(t) 〉 = ∂ ∂H
| n(t) 〉 , Ḣ =
∂t ∂t

9 171
i ∂ ∂t | t 〉 = ∑ n
(
i dc )
n
dt | n(t) 〉 + ic n(t)| ṅ(t) 〉 + E n (t)c n (t)| n(t) 〉 e −iθn(t)
,
i ∂ ∂t | t 〉 = H(t)| t 〉
∑
n
( )
dcn
dt | n(t) 〉 + c n(t)| ṅ(t) 〉 e −iθn(t) = 0
| m(t) 〉 〈 m(t) | n(t) 〉 = δ mn
dc m
dt
= − ∑ n
c n (t)〈 m(t) | ṅ(t) 〉 e iθ m(t)−iθ n (t)
(9.24)
(9.23) t
Ḣ| n(t) 〉 + H| ṅ(t) 〉 = Ėn| n(t) 〉 + E n | ṅ(t) 〉
| m(t) 〉
〈 m(t) |Ḣ| n(t) 〉 + E m(t)〈 m(t) | ṅ(t) 〉 = δ mn Ė n + E n (t)〈 m(t) | ṅ(t) 〉
m ≠ n
〈 m(t) | ṅ(t) 〉 = −
〈 m(t) |Ḣ| n(t) 〉
E m (t) − E n (t)
(9.25)
m = n , 〈 n(t) | n(t) 〉 = 1 t
〈 ṅ(t) | n(t) 〉 + 〈 n(t) | ṅ(t) 〉 = 〈 n(t) | ṅ(t) 〉 ∗ + 〈 n(t) | ṅ(t) 〉 = 0
〈 n(t) | ṅ(t) 〉 , γ n (t)
〈 n(t) | ṅ(t) 〉 = − i dγ n(t)
dt
(9.25), (9.26) (9.24)
dc m
dt
= i dγ m(t)
dt
c m (t) + ∑ n≠m
c n (t)
〈 m(t) |Ḣ| n(t) 〉
ω mn (t)
(
exp i
∫ t
0
)
dt ′ ω mn (t ′ )
(9.26)
(9.27)
ω mn (t) = E m (t) − E n (t)
c n (t), Ḣ, ω mn (t) , (9.27)
, t = 0 | i(0) 〉 c n (t) = δ ni m ≠ i
dc m
dt
≈
〈 m(t) |Ḣ| i(t) 〉
ω mi (t)
e iω mi(t)t
c m (t) ≈
〈 m(t) |Ḣ| i(t) 〉
ω mi (t)
∫ t
0
dt ′ e iωmi(t)t′ ≈
〈 m(t) |Ḣ| i(t) 〉
(
)
iω mi (t) 2 e iωmi(t)t − 1
(9.28)

9 172
H ′ (t) = V e −iωt + V † e iωt
H ′ (t) H 0 | n(t) 〉
E n (t) H 0
H 0 | n 〉 = E n | n 〉
(9.27) m ≠ i
dc m
dt
≈ 〈 m |Ḣ| i 〉
ω mi
e iω mit = −iω
ω mi
〈 m | ( V e −iωt − V † e iωt) | i 〉 e iω mit
c m = −
ω (
〈 m |V | i 〉 ei(ωmi−ω)t − 1
− 〈 m |V † | i 〉 ei(ωmi+ω)t )
− 1
ω mi ω mi − ω
ω mi + ω
(9.29)
H ′ (t) , , ω , |ω mi | ≫ ω
(9.29)
c m ≈ −
ω (
)
〈 m |V | i 〉 ei(ωmi−ω)t − 1
− 〈 m |V † | i 〉 ei(ωmi+ω)t − 1
≈ (9.28)
ω mi ω mi ω mi
, ω mi ≈ ω ( ) (9.28)
9.7
H(t) Ḣ ≈ 0 , , (9.27) 2
dc m
dt
= i dγ m(t)
dt
c m (t)
c m (t) = c m (0) e iγm(t)
| t 〉 = ∑ m
c m (0) e iγ m(t)−iθ m (t) | m(t) 〉
(
e −iθm(t) = exp − i
∫ t
0
)
dt ′ E m (t ′ )
e −iEt/
, γ m (t) t
| m(t) 〉
|〈 m(t) | t 〉| 2 = |c m (0) e iγm(t)−iθm(t) | 2 = |c m (0)| 2
, ,
B(t) ω (5.66)
H = µ B B(t)·σ , B x = B 1 cos ωt , B y = B 1 sin ωt , B z = B 0

9 173
B 0 B 1 B = |B(t)| = √ B 2 0 + B2 1
E m (t) = mµ B B
H(t) | m(t) 〉 = mµ B B | m(t) 〉 , m = ± 1
θ m (t) = 1
∫ t
0
dt ′ E m (t ′ ) = mµ BB
σ | m(t) 〉 2 χ m (5.53)
⎛
⎞
√ 1
B + mB0 ⎜
χ m (t) =
⎝
2B m B ⎟
x + iB y
⎠ (9.30)
B + mB 0
t
(9.26) γ m
⎛
√
d B +
dt χ mB0 ⎜
m =
⎝
2B
0
mω −B y + iB x
B + mB 0
⎞
⎟
⎠
dγ m
dt
= i χ † d
m
dt χ ωB1
2
m = −
2B(B + mB 0 ) = mB 0 − B
ω
2B
, t ψ(t)
ψ(t) = ∑
γ m (t) = mB 0 − B
ωt (9.31)
2B
γ m − θ m = − ωt
2 + mϕ(t) , ϕ(t) = − µ BB
t + B 0
2B ωt
m=±1
c m (0) e i(γm−θm) χ m (t) = e −iωt/2
∑
m=±1
t = 0 σ z +1
( ) √
c m (0) = χ † 1 B + mB0
m(0) =
0
2B
c m (0) e imϕ(t) χ m (t)
ψ(t) = e−iωt/2
2B
∑
m=±1
e imϕ ⎛
⎝ B + mB 0
m(B x + iB y )
⎞
⎠ = e−iωt/2
B
⎛
⎝ B cos ϕ + iB 0 sin ϕ
i (B x + iB y ) sin ϕ
⎞
⎠
, t σ z −1 P −
P − (t) = | ( 0 1 )ψ(t) | 2 =
(B x + iB y ) sin ϕ
∣ B ∣
2
= B2 1
B 2 sin2 (
µB B
t − B 0
2B ωt )
(9.32)
2 , z
B(t) , t σ z ±1 N ±
P − = N − /(N + + N − ) (9.32) ,

9 174
(5.70)
ω 2 1
P − =
(ω 0 − ω/2) 2 + ω1
2
sin 2 (√
(ω 0 − ω/2) 2 + ω 2 1 t )
ω 0 = µ B B 0 / , ω 1 = µ B B 1 / ω ≪ ω 0 , ω 1
√ (
)
√(ω 0 − ω/2) 2 + ω1 2 ≈ ω0 2 + ω 0 ω
ω2 1 1 −
2(ω0 2 + ω2 1 ) = µ BB
− B 0
2B ω
, (9.32)
H B(t) , ( ) B(t)
, (9.30) | n(t) 〉 B | n(B) 〉
(
∂
∂t | n(B) 〉 = dBx ∂
+ dB y ∂
+ dB )
z ∂
| n(B) 〉 = dB dt ∂B x dt ∂B y dt ∂B z dt ·∇ B| n(B) 〉
(9.26)
dγ n
dt
γ n B
γ n (t) = i
= i 〈 n(B) |∇ B | n(B) 〉· dB dt
∫ B(t)
B(0)
〈 n(B) |∇ B | n(B) 〉·dB
B(t) T , B(T ) = B(0) t = 0 t = T
B C
∮
γ n (T ) = i 〈 n(B) |∇ B | n(B) 〉·dB (9.33)
∫
γ n (T ) = dS·V (B) , V (B) = i ∇ B ×〈 n(B) |∇ B | n(B) 〉
S
C
S C , dS
9.4
B
B = B( sin θ cos φ , sin θ sin φ , cos θ )
| m(B) 〉 (5.53) , ∇ B
∂
∇ B = e r
∂B + e 1 ∂
θ
B ∂θ + e 1 ∂
φ
B sin θ ∂φ ,
dB = dB e r + B dθ e θ + B sin θ dφ e φ
i 〈 m(B) |∇ B | m(B) 〉 = m cos θ − 1
2B sin θ
(9.33) γ m (T ) , B z = B cos θ 1
(9.31) t = 2π/ω
e φ

10 175
10
10.1
N in , dΩ dN N in dΩ
dσ
dN = N in dΩ (10.1)
dΩ
σ , dσ/dΩ dN
[] −1 , N in [] −1 [] −1 , σ
, σ
1
(
)
− 2
2m ∇2 + V (r) ψ(r) = E ψ(r)
,
z r , ψ(r)
:
(
ψ(r) −−−−→ r→∞
A e ikz + f(θ, φ) χ(r) )
, k > 0
r
1 , 2 z k > 0 r → ∞
V (r) → 0 ,
2 k 2
2m eikz − 2
2m ∇2 f(θ, φ) χ(r) (
= E e ikz + f(θ, φ) χ(r) )
r
r
(6.1)
− 2
2m ∇2 f(θ, φ) χ(r) (
= − 2 d 2 χ
r 2mr dr 2 − χ )
r 2 L2 f(θ, φ)
r → ∞ 2 1
( 2 k 2 )
(
2m − E e ikz f(θ, φ)
2
d 2 )
χ
−
r 2m dr 2 + Eχ = 0
E = 2 k 2
2m ,
d 2 χ
dr 2 + k2 χ = 0
2 χ(r) e ikr e −ikr
, ,
, χ(r) ∝ e ikr
) √
ψ(r) −−−−→ r→∞
A
(e ikz + f(θ, φ) eikr
2mE
, k =
r
2 (10.2)
ϕ in , ϕ out
ϕ in (r) = A e ikz ,
ϕ out (r) = A f(θ, φ) eikr
r

10 176
f(θ, φ) E > 0 , (10.2)
1 E < 0
, ψ(r) r→∞
−−−−→ 0 , E < 0
1 , ∇
∇ϕ out (r) = A
∂
∇ = e r
∂r + 1 u(θ, φ) ,
r u(θ,
φ) = e ∂
θ
∂θ + e φ ∂
sin θ ∂φ
) (ikf eikr
r e r + eikr
r 2 (u − e r) f ≈ ikAf eikr
r e r = ik ϕ out (r) e r (10.3)
, J = )
(ψ
m Im ∗ (r)∇ψ(r) r → ∞
J → m Im [(ϕ ∗ in + ϕ ∗ out
= k m
)
)]
ik
(ϕ in e z + ϕ out e r
(
)
|ϕ in | 2 e z + |ϕ out | 2 e r + Re (ϕ ∗ inϕ out ) (e r + e z )
(10.4)
1 k/m z ,
, , N in
N in = k m |ϕ in| 2 = k m |A|2
2 k/m r
dΩ r 2 dΩ ,
k
m |ϕ out| 2 r 2 dΩ = k m |A|2 |f(θ, φ)| 2 dΩ
dΩ dN , (10.1)
2 ,
dσ
dΩ = |f(θ, φ)|2 (10.5)
r → ∞ f(θ, φ) ,
(10.4) 3 ,
e −ik·r kr→∞
−−−−−→ 2πi (
δ(Ω r − Ω k )e −ikr − δ(Ω r + Ω k )e ikr) + O(1/(kr) 2 ) (10.6)
kr
Ω r , Ω k r , k
δ(Ω − Ω ′ ) = δ(θ − θ′ ) δ(φ − φ ′ )
sin θ
, Ω F (Ω) = F (θ, φ)
∫
∫ π ∫ 2π
dΩ ′ δ(Ω − Ω ′ )F (Ω ′ ) = dθ ′ dφ ′ δ(θ − θ ′ ) δ(φ − φ ′ )F (θ ′ , φ ′ ) = F (Ω)
k z
∫
I = dΩ r e −ik·r F (Ω r ) =
0
=
0
∫ 2π
0
∫ 2π
0
∫ π
dφ r dθ r sin θ r e −ikr cos θ r
F (θ r , φ r )
0
∫ 1
dφ r dt e −ikrt F (θ r , φ r )
−1

10 177
t = cos θ r t
I =
∫ 2π
0
dφ r
( i
kr
[
e −ikrt F (θ r , φ r ) ] t=1
t=−1 − i
kr
∫ 1
−1
dt e −ikrt ∂ )
∂t F (θ r, φ r )
t = 1 , θ r = 0 r k , t = −1 r k
I = i
kr
= 2πi
kr
∫ 2π
,
0
)
dφ r
(e −ikr F (Ω k ) − e ikr F (−Ω k )
(
e −ikr F (Ω k ) − e ikr F (−Ω k )
)
− i
kr
I = 2πi (
)
e −ikr F (Ω k ) − e ikr F (−Ω k )
kr
+ 1 ∫ 2π
(kr) 2 dφ r
[e −ikrt ∂ ] t=1
∂t F (θ r, φ r )
0
t=−1
− i
kr
∫ 2π
0
∫ 2π
0
∫ 1
dφ r dt e −ikrt ∂ ∂t F (θ r, φ r )
−1
∫ 1
dφ r dt e −ikrt ∂ ∂t F (θ r, φ r )
−1
− 1 ∫ 2π ∫ 1
−ikrt ∂2
(kr) 2 dφ r dt e
∂t 2 F (θ r, φ r )
kr → ∞ 2, 3 1 1/r
∫
I = dΩ r e −ik·r F (Ω r ) = 2πi (
)
e −ikr F (Ω k ) − e ikr F (−Ω k ) + O(1/(kr) 2 )
kr
(10.6)
(10.4) 3
J inter (r) = k m Re (ϕ∗ inϕ out ) (e r + e z ) = k (
)
mr |A|2 Re e ikr−ikz f(θ, φ) (e r + e z )
k z kz = k·r (10.6)
e −ikz r→∞
−−−−→ 2πi
kr
0
−1
(
)
e −ikr δ(Ω − Ω z ) − e ikr δ(Ω + Ω z )
, Ω r , Ω z k , , z ( Re(if) = − Imf )
J inter (r) = − 2π [ (
) ]
mr 2 Im f(θ, φ) δ(Ω − Ω z ) − e 2ikr δ(Ω + Ω z ) (e r + e z )
Ω = − Ω z r z e r = − e z δ(Ω + Ω z ) 0
Ω = Ω z r z θ = φ = 0 , e r = e z
J inter (r) = − 4π
mr 2 Imf(0) δ(Ω − Ω z) e z
( θ = 0 ) θ = π
,
J inter
, ∂ρ/∂t + ∇·J = 0 ∇·J = 0 , r
,
∫
4πr 2 dΩ J ·e r = 0

10 178
r → ∞ J (10.4)
∫
k
dΩ |ϕ in | 2 e z ·e r = k ∫
m
m |A|2 dΩ cos θ = 0
∫
k
dΩ |ϕ out | 2 e r ·e r = k |A| 2 ∫
m
m r 2 dΩ |f(θ, φ)| 2 = k m
∫
dΩ J inter (r)·e r = − 4π
mr 2 |A|2 Imf(0)
∫
∫
σ tot = dΩ |f(θ, φ)| 2 =
dΩ dσ
dΩ
∫
dΩ J ·e r = k |A| 2 (
m r 2 σ tot − 4π )
k Im f(0) = 0
|A| 2
r 2
σ tot
σ tot = 4π k
Im f(0) (10.7)
10.2
V (r) 1 , f(θ, φ)
k =
√
2mE
2 , U(r) = 2m
2 V (r)
(
∇ 2 + k 2) ψ(r) = U(r)ψ(r) (10.8)
(
∇ 2 + k 2) G(r) = δ(r) (10.9)
G(r) (10.8)
∫
ψ(r) = ϕ(r) + d 3 r ′ G(r − r ′ )U(r ′ )ψ(r ′ ) , ( ∇ 2 + k 2) ϕ(r) = 0 (10.10)
(
∇ 2 + k 2) ψ(r) = ( ∇ 2 + k 2) ∫
ϕ(r) + d 3 r ′ ( ∇ 2 + k 2) G(r − r ′ )U(r ′ )ψ(r ′ )
∫
= d 3 r ′ δ(r − r ′ ) U(r ′ )ψ(r ′ ) = U(r)ψ(r)
(10.10) (10.8) (10.10) ϕ(r) G(r) , (10.2)
, (10.8) (10.10)
∇ 2 e±ikr
r
= e ±ikr ∇ 2 1 (
r + 2 ∇ 1 )
·∇e ±ikr + 1 r
r ∇2 e ±ikr

10 179
∇ 2 1 r = − 4π δ(r) , ∇1 r = − r r 3 , ∇e±ikr = ± ikr
(
r e±ikr , ∇ 2 e ±ikr = − k 2 ± 2ik )
e ±ikr
r
∇ 2 e±ikr
r
= − 4π δ(r) − k 2 e±ikr
r
G (±) (r) = − 1 e ±ikr
4π r
(10.9) (10.10)
ψ(r) = ϕ(r) − 1 ∫
d 3 r ′ exp (± ik|r − r′ |)
4π
|r − r ′ |
r r ′ α
√
|r − r ′ | = r
1 + r′2
r 2 − 2r′
r→∞
cos α −−−−→ r
r
U(r ′ )ψ(r ′ ) (10.11)
(1 − r′
r cos α + · · · )
= r − e r ·r ′ + O (r ′ /r)
exp (± ik|r − r ′ |)
|r − r ′ |
= exp (± ikr ∓ ik f ·r ′ )
r
+ O (r ′ /r) , k f = ke r
ψ(r) −−−−→ r→∞
ϕ(r) − 1 e ±ikr ∫
d 3 r ′ e ∓ik f ·r ′ U(r ′ )ψ(r ′ )
4π r
G(r) G (+) (r) , ( ∇ 2 + k 2) ϕ(r) = 0 ϕ(r) ϕ(r) = e ikz
(10.2) ,
f(θ, φ) = − 1 ∫
d 3 r ′ e −ik f ·r ′ U(r ′ )ψ(r ′ ) (10.12)
4π
k f = ke r (θ, φ) , ,
(10.12) , ψ(r) f(θ, φ)
(10.11)
∫
ψ(r) = ϕ(r) + d 3 r ′ G (+) (r − r ′ )U(r ′ )ψ(r ′ )
∫
( ∫
)
= ϕ(r) + d 3 r ′ G (+) (r − r ′ )U(r ′ ) ϕ(r ′ ) + d 3 r ′′ G (+) (r ′ − r ′′ )U(r ′′ )ψ(r ′′ )
∫
= ϕ(r) + d 3 r ′ G (+) (r − r ′ )U(r ′ )ϕ(r ′ )
∫
+ d 3 r ′ d 3 r ′′ G (+) (r − r ′ )U(r ′ )G (+) (r ′ − r ′′ )U(r ′′ )ϕ(r ′′ ) + · · · (10.13)
U ,
(10.12)
f(θ, φ) = − 1 ∫
d 3 r ′ e −ik f ·r ′ U(r ′ )ϕ(r ′ )
4π
− 1 ∫
d 3 r ′ d 3 r ′′ e −ik f ·r ′ U(r ′ )G (+) (r ′ − r ′′ )U(r ′′ )ϕ(r ′′ ) + · · · (10.14)
4π

10 180
2
f(θ, φ) ≈ − 1 ∫
d 3 r ′ e −ik f ·r ′ U(r ′ )ϕ(r ′ ) = −
m ∫
4π
2π 2 d 3 r ′ e −ik f ·r ′ V (r ′ )ϕ(r ′ ) (10.15)
k i = ke z k i ϕ(r) = e ikz =
e ik i·r
f(θ, φ) ≈ −
m
∫
2π 2 V(q) , V(q) = d 3 r e −iq·r V (r) , q = k f − k i (10.16)
, V(q)
dσ
( m
) 2
dΩ = |f(θ, φ)|2 = |V(q)|
2
2π 2
q ,
q
k i =ke z
k f =ke r
θ
q = 2k sin θ 2
( k ≈ 0 ) q ≈ 0
∫
V(q) ≈ V(0) = d 3 r V (r)
f(θ, φ) , ( k → ∞ ) θ ≠ 0 q
e −iq·r r V(q) ≈ 0 f(θ, φ)
θ ≈ 0 , V(0) f(0) ,
(10.7) , f(θ, φ) V 1 , |f(θ, φ)| 2
V 2 , (10.14) 2 Imf(0)
V(q) = 2π
∫ ∞
0
dr r 2 V (r)
∫ 1
−1dt e −iqrt = 4π q
∫ ∞
0
dr rV (r) sin(qr) (10.17)
q , φ ,
q E = 2 k 2 /(2m) θ , E, θ
q σ tot
∫
( m
) 2
∫ π
σ tot = dΩ |f(θ, φ)| 2 = 2π
2π 2 dθ sin θ |V(q)| 2
q = 2k sin(θ/2) 0 ≤ q ≤ 2k , qdq = k 2 sin θ dθ
( k → ∞ )
E
0
∫ 2k
σ tot =
m2
2π 4 k 2 dq q |V(q)| 2 (10.18)
∫ ∞
σ tot →
m2
2π 4 k 2 dq q |V(q)| 2 =
m
4π 2 E
0
0
∫ ∞
0
dq q |V(q)| 2

10 181
(10.12) ψ(r) ϕ(r) (10.13)
∫
|ϕ(r)| = 1 ≫
∣ d 3 r ′ G (+) (r − r ′ )U(r ′ )ϕ(r ′ )
∣
r = 0
∫
∣ d 3 r G (+) (r)U(r)ϕ(r)
∣ = m ∣∫
∣∣∣
2π 2 d 3 r eik(r+z)
V (r)
r ∣ ≪ 1 (10.19)
(10.6) k → ∞
,
m
2 k ∣
e ik(r+z) ≈ 2πi (
)
δ(Ω + Ω z ) − e 2ikr δ(Ω − Ω z )
kr
∫ ∞
0
∫
dr
(
)
dΩ δ(Ω + Ω z ) − e 2ikr δ(Ω − Ω z )
,
V (r)
∣ ≪ 1
V (r)
(10.17)
V (r) =
{
V0 , r < a
0 , r > a
(10.20)
V(q) = 4πV 0
q
∫ a
0
dr r sin(qr) = 4πV 0 a 3 v(aq) ,
v(x) ≡
sin x − x cos x
x 3 (10.21)
,
(
dσ
dΩ = 2mV0 a 2 ) 2 a2
2 v(aq) 2
0.1
v(x) 2 x 4
v(x) 2 ≈ 0 ,
v(x) 2
aq = 2ak sin θ 2 4 , sin θ 2 2
ak
, k
v(x) = 1 )
(1 − x2
3 10 + · · ·
0 2 4 6
( )
dσ dσ
(1
dΩ = − 4 dΩ
0
5 a2 k 2 sin 2 θ )
2 + · · · ,
( ) ( dσ
= a 2 2mV0 a 2 ) 2
dΩ
0
3 2 (10.22)
( θ = 0 ) , k → 0
θ k , θ ≠ 0

10 182
(10.18) (10.21)
σ tot = 8π ( mV0 a 2 ) 2 ∫ 2k
k 2 2 a 2 dq q v(aq) 2 = 8π ( mV0 a 2 ) 2
0
k 2 2 S(2ak) (10.23)
S(x) =
∫ x
0
dy y v(y) 2 =
∫ x
0
dy
(sin y − y cos y)2
y 5 = 1 [
− 1 sin 2y
+
4 y2 y 3
= 1 4
− sin2 y
y 4 ] x
0
(
1 − 1 )
sin 2x
+
x2 x 3 − sin2 x
x 4
S ′ (x) = xv(x) 2 , S ′′ (x) = v(x) 2 + 2xv(x)v ′ (x)
S(x) = S(0) + S ′ (0)x + S′′ (0)
x 2 + · · · = x2
2
18 + · · ·
ak ≪ 1
σ tot = 4πa 2 ( 2mV0 a 2
3 2
) 2 ( ) dσ
= 4π
dΩ
0
(10.24)
, ak ≫ 1
σ tot = 2π
k 2 ( mV0 a 2
2 ) 2
(10.19)
∫
d 3 r eik(r+z)
r
∫ a ∫ 1
V (r) = 2πV 0 dr re ikr dt e ikrt = πV (
0
k 2 1 + 2iak − e 2iak)
e 2iak = 1 + 2iak − 2a 2 k 2 + · · ·
0
−1
m
∣ ( ∣∣V0
2 2 k 2 1 + 2iak − e 2iak)∣ ∣ ≪ 1 (10.25)
ma 2 |V 0 |
2 ≪ 1
3 , 121
√
2m|V0 |
a
2 > π 2 , ∴ ma 2 |V 0 |
2
> π2
8
, ,
(10.25)
ma|V 0 |
2 k
≪ 1
, k → ∞

10 183
10.3
,
, ,
, (10.2)
V (r) , H, L 2 , L z ψ(r)
Y l ml (θ, φ)
ψ(r) = R l (r) Y l ml (θ, φ)
)
(− 2
2m ∇2 + V (r) ψ(r) = E ψ(r) (10.26)
(6.3)
( 1
r
d 2
)
l(l + 1)
r −
dr2 r 2 − U(r) + k 2 R l (r) = 0 (10.27)
U(r) = 2m
2 V (r) ,
R l (r) = χ l(r)
r
k = √
2mE
2
( )
d
2
l(l + 1)
−
dr2 r 2 − U(r) + k 2 χ l (r) = 0 (10.28)
V (r) = 0
( 1
r
d 2
)
l(l + 1)
r −
dr2 r 2 + k 2 R l (r) = 0
15.4 ( 259 ) , j l (kr)
j l (kr) Y l ml (θ, φ) , l = 0, 1, 2, · · · , m l = −l, − l + 1, · · · , l
(
∇ 2 + k 2) ψ(r) = 0 (10.29)
, A lml
∞∑ l∑
ψ(r) = A lml j l (kr) Y l ml (θ, φ)
l=0 m l =−l
(10.29) , , (10.29) e ikz
∞∑
∞∑
e ikz = e ikr cos θ = A l j l (kr) Y l0 (θ, φ) = C l j l (kr) P l (cos θ)
l=0
l=0

10 184
e ikz φ m l = 0 C l
(15.46)
∞∑
n=0
i n
n! (kr cos θ)n =
j l (ρ) =
∞∑
C l j l (kr) P l (cos θ) (10.30)
l=0
ρ l (
ρ 2 )
1 −
(2l + 1)!! 2(2l + 3) + · · ·
, j l (kr) (kr) n l ≤ n , (5.31)
d l
P l (x) = 1
2 l l! dx l (x2 − 1) l =
(2l)! (
2 l (l!) 2 x l −
)
l(l − 1)
2(2l − 1) xl−2 + · · ·
, P l (cos θ) (cos θ) n l ≥ n , (kr cos θ) n
l = n , (10.30)
(2n + 1)!!
n! = C 1 (2n)!
n
(2n + 1)!! 2 n (n!) 2
(2n + 1)!! = (2n + 1)(2n − 1)(2n − 3) · · · 5 · 3 =
C n = i n (2n + 1)
i n
e ikz =
(2n + 1)! (2n + 1)!
=
2n(2n − 2) · · · 4 · 2 2 n n!
∞∑
i l (2l + 1) j l (kr) P l (cos θ) (10.31)
l=0
( 263 (15.50) )r → ∞ (15.49)
e ikz
r→∞
−−−−→ 1
2ikr
∞∑ (
(2l + 1) e ikr + (−1) l+1 e −ikr) P l (cos θ) (10.32)
l=0
(10.2) (10.27) r = 0 (6.7)
R l (r) = χ l(r)
r
= C l r l , C l =
r → ∞ (10.27) R l (r) C l
U(r) 1/r 2 0 ( , U ∝ 1/r
), r ,
( 1 d 2
)
l(l + 1)
r −
r dr2 r 2 + k 2 R l (r) = 0
R l (r)
)
R l (r) = C l
(a l j l (kr) + b l n l (kr)
a l , b l
)
R l (r) = A l
(cos δ l j l (kr) − sin δ l n l (kr)
(10.33)

10 185
A l δ l R l r
, n l (kr) , δ l
k (15.49) r → ∞
R l (r) = A (
l
kr sin kr − lπ )
2 + δ l
(10.34)
, (10.33) δ l = 0 , r → ∞
δ l δ l l ( phase shift )
R l (r)Y l ml (θ, φ) l, m l ,
(10.26) z ,
z φ , m l = 0
ψ(r) =
∞∑
R l (r)P l (cos θ) (10.35)
l=0
, R l (r)P l (cos θ) ,
(10.35) (10.2) r → ∞
∞∑
(
A l
ψ(r) =
kr sin kr − lπ )
2 + δ l P l (cos θ)
l=0
= eikr
2ikr
∞∑
l=0
A l (−i) l e iδ l
P l (cos θ) − e−ikr
2ikr
∞∑
A l i l e −iδ l
P l (cos θ) (10.36)
, (10.2) (10.32)
) ( ∞
)
A
(e ikz + f(θ) eikr
= eikr
r 2ikr A ∑
(2l + 1)P l (cos θ) + 2ikf(θ)
+ e−ikr
l=0
l=0
∞
2ikr A ∑
(2l + 1)(−1) l+1 P l (cos θ) (10.37)
(10.36) (10.37) , e ikr e −ikr
l=0
l=0
∞∑
∞∑
A l (−i) l e iδ l
P l (cos θ) = A (2l + 1)P l (cos θ) + 2ikf(θ)A
l=0
l=0
∞∑
∞∑
A l i l e −iδ l
P l (cos θ) = A (2l + 1)(−1) l P l (cos θ)
l=0
P l (cos θ) 2
l=0
A l = i l e iδ l
(2l + 1)A (10.38)
1
f(θ) = 1 ∞∑ ( )
(2l + 1) e 2iδ l
− 1 P l (cos θ) = 1 ∞∑
(2l + 1) e iδ l
sin δ l P l (cos θ) (10.39)
2ik
k
δ l , f(θ)
l=0
f l (k) = e2iδ l(k) − 1
2ik
= eiδ l
sin δ l
k
= 1 k
1
cot δ l − i
(10.40)

10 186
S l (k) = e 2iδl(k) = cot δ l + i
(10.41)
cot δ l − i
S l S ( )
dσ
dΩ = |f(θ)|2 = 1 ∑
k 2
l,l ′ (2l + 1)(2l ′ + 1) e iδ l−iδ l′
sin δ l sin δ l ′ P l (cos θ)P l ′(cos θ)
, , σ tot t = cos θ
σ tot = 2π
k 2 ∑
l,l ′ (2l + 1)(2l ′ + 1) e iδ l−iδ l′
sin δ l sin δ l ′
∫ 1
−1
dt P l (t)P l ′(t)
= 2π ∑
k 2 (2l + 1)(2l ′ + 1) e iδ l−iδ l′
2 δ ll ′
sin δ l sin δ l ′
2l + 1 = 4π
k 2
l,l ′
l σ l
∞ ∑
l=0
(2l + 1) sin 2 δ l
σ l = 4π
k 2 (2l + 1) sin2 δ l ≤ 4π (2l + 1) (10.42)
k2 σ l , δ l n δ l = (n + 1/2)π
P l (cos 0) = P l (1) = 1
f(0) = 1 ∞∑
(2l + 1) e iδ l
sin δ l , ∴ Imf(0) = 1 ∞∑
(2l + 1) sin 2 δ l = k
k
k
4π σ tot
l=0
(10.38) (10.36)
ψ(r) −−−−→
r→∞ A
2ikr
l=0
∞∑ (
(2l + 1) e 2iδ l
e ikr + (−1) l+1 e −ikr) P l (cos θ)
l=0
e ikz (10.32) ,
,
( )
e 2iδ l
e ikr = e ikr + e 2iδ l
− 1 e ikr
, (10.32) e ikr ,
, (10.39) e 2iδ l
− 1
(10.34) l r → ∞
R l (r) = − il A l e −iδ l
2ikr
(
e −ikr − (−1) l S l (k) e ikr) (10.43)
R l (r) = (−i)l e iδ l
A
(
l
e ikr − (−1) l S −1
l
(k)e −ikr)
2ikr
E κ = √ − 2mE/ 2 k = iκ
R l (r) = (−i)l e iδ l
A
(
l
e −κr − (−1) l S −1
l
(iκ)e κr)
2ikr
, r → ∞ S −1
l
(iκ) = 0
, E l 1
, E

10 187
10.4
V (r) χ l (r) = rR l (r) (10.28)
( )
d
2
l(l + 1)
−
dr2 r 2 − U(r) + k 2 χ l (r) = 0 (10.44)
, r = 0 r → ∞ (10.34)
χ l (r) r→0
−−−−→ C l r l+1 ,
(
χ l (r) −−−−→ r→∞
sin kr − lπ )
2 + δ l
, sin 1
, Ṽ (r) ˜χ l(r) :
( )
d
2
l(l + 1)
−
dr2 r 2 − Ũ(r) + k2 ˜χ l (r) = 0 (10.45)
˜χ l (r) r→0
−−−−→ ˜C l r l+1 ,
χ l × (10.45) − ˜χ l × (10.44)
˜χ l (r) −−−−→ r→∞
sin
(kr − lπ )
2 + ˜δ l
dW
dr = (Ũ(r) − U(r)
)
˜χ l χ l , W (r) = χ l
d˜χ l
dr − dχ l
dr ˜χ l
W (∞) − W (0) =
W (0) = 0 ,
sin
(δ l − ˜δ
)
l = 2m
2 k
∫ ∞
0
)
dr
(Ũ(r) − U(r) ˜χ l χ l
W (∞) = k sin
(δ l − ˜δ
)
l
∫ ∞
, Ṽ (r) = 0 ˜δ l = 0 , ˜χ l (r) = kr j l (kr)
sin δ l = − 2m
2
0
∫ ∞
(10.44) χ l (r) , δ l
∆V (r) = V (r) − Ṽ (r) (10.46)
δ l − ˜δ l ≈ − 2m
2 k
0
)
dr
(Ṽ (r) − V (r) ˜χ l χ l (10.46)
dr rV (r) j l (kr) χ l (r) (10.47)
∫ ∞
0
dr ∆V (r) (˜χ l (r)) 2
, , V (r) = 0
V (r) , V (r) < 0 δ l > 0 ,
V (r) > 0 δ l < 0 , V (r) = 0
r → ∞ χ l (r)
(
χ l (r) = sin kr − lπ )
2 + δ l
δ l > 0 , δ l < 0
, δ l ×π , V (r) = 0 δ l = 0 , δ l

10 188
δ l
sin(kr − lπ/2 + δ l )
sin(kr − lπ/2)
kr
V (r)
∣ |V (r)| ≪
∣ E − 2 l(l + 1) ∣∣∣
2mr 2 (10.48)
, (10.44) χ l χ l , , kr j l (kr) δ l ≈ 0
, (10.47) χ l kr j l (kr)
l=0
sin δ l ≈ − 2mk
2
∫ ∞
0
l=0
dr r 2 V (r) j 2 l (kr)
, (10.39) δ l ≈ 0
f(θ) ≈ 1 ∞∑
(2l + 1) sin δ l P l (cos θ) ≈ − 2m ∑
∞
k
2 (2l + 1)P l (cos θ)
∫ ∞
0
dr r 2 V (r)j 2 l (kr) (10.49)
(10.16) (10.48) , E
l , V (r)
(10.20)
sin δ l ≈ − 2mV 0k
2
ak ≪ 1
j l (x) x→0
−−−−→
f(θ) ≈ − 2mV 0a 3
2
ak ≪ 1 l = 0
∫ a
0
dr r 2 j 2 l (kr) = − 2mV 0
2 k 2
(
xl
1 −
(2l + 1)!!
∫ ak
0
x 2 )
2(2l + 3) + · · ·
sin δ l ≈ − 2mV 0a 2
2 (ak) 2l+1
[(2l + 1)!!] 2 (2l + 3)
∞ ∑
l=0
2l + 1
[(2l + 1)!!] 2 (2l + 3) (ak)2l P l (cos θ)
f(θ) ≈ − 2mV 0a 3
3 2
(
dσ
dΩ = 2mV0 a 2 ) 2 ∫
(
3 2 , σ tot = dΩ |f(θ)| 2 ≈ 4πa 2 2mV0 a 2 ) 2
3 2
(10.24) l = 0
10.1 (15.50)
e ik·r = 4π
∞∑
l∑
l=0 m=−l
(10.16) (10.49)
i l j l (kr)Y ∗
lm(Ω k )Y lm (Ω r )
dx x 2 j 2 l (x) (10.50)
10.2
(10.50) (ak) 2 (10.22)

10 189
10.5
V (r)
V (r) =
{
− V0 , r < a
0 , r > a , V 0 > 0
r < a (10.27)
( 1
r
d 2
)
l(l + 1)
r −
dr2 r 2 + ρ2 1
a 2 R l (r) = 0 ,
ρ 1 ≡ a
√
k 2 + 2m
2 V 0
,
R l (r) = C l j l (ρ 1 r/a) (10.51)
, r > a
)
R l (r) = A l
(cos δ l j l (ρr/a) − sin δ l n l (ρr/a) , ρ ≡ ak (10.52)
r = a
)
(
)
C l j l (ρ 1 ) = A l
(cos δ l j l (ρ) − sin δ l n l (ρ) , C l ρ 1 j l(ρ ′ 1 ) = A l ρ cos δ l j l(ρ) ′ − sin δ l n ′ l(ρ)
tan δ l = (γ l − l − 1) j l (ρ) − ρ j
l ′(ρ)
(γ l − l − 1) n l (ρ) − ρ n ′ l (ρ) , γ l(ρ 1 ) = ρ 1j
l ′(ρ 1)
+ l + 1 (10.53)
j l (ρ 1 )
j ′ l(x) = l x j l(x) − j l+1 (x) ,
n ′ l(x) = l x n l(x) − n l+1 (x)
tan δ l = (γ l − 2l − 1)j l (ρ) + ρ j l+1 (ρ)
(γ l − 2l − 1)n l (ρ) + ρ n l+1 (ρ) , γ l = 2l + 1 − ρ 1j l+1 (ρ 1 )
= ρ 1j l−1 (ρ 1 )
j l (ρ 1 ) j l (ρ 1 )
(10.54)
j −1 (x) ≡ − n 0 (x) = cos x
x
( ρ = ak ≪ 1 ) (15.46), (15.47)
j l (ρ) =
ρ l (
1 −
(2l + 1)!!
ρ 2 )
2(2l + 3) + · · · , n l (ρ) = −
(2l + 1)!!
2l + 1
(
1
ρ l+1 1 +
ρ 2 )
2(2l − 1) + · · ·
ρ j l+1 (ρ)
j l (ρ)
ρ n l+1 (ρ)
n l (ρ)
(
= ρ2
ρ 2
)
1 +
2l + 3 (2l + 3)(2l + 5) + · · ·
= 2l + 1 − ρ2
2l − 1 + · · ·
j l (ρ)
n l (ρ) = − 2l + 1
[(2l + 1)!!] 2 ρ2l+1 (
1 −
)
2l + 1
(2l + 3)(2l − 1) ρ2 + · · ·

10 190
tan δ l ≈ 2l + 1 − γ l(ρ 1 ) − d l ρ 2
γ l (ρ 1 ) − ρ 2 /(2l − 1)
2l + 1
[(2l + 1)!!] 2 ρ2l+1 (10.55)
ρ 1 =
d l = 1 (
2l + 3
1 + 2l + 1
2l − 1
(
) )
2l + 1 − γ l (ρ 1 )
√ √v 0 2 + ρ2 = v 0 + ρ2
2mV0
+ · · · , v 0 = a
2v 0 2
γ l (ρ 1 ) = γ l (v 0 ) + γ′ l (v 0)
2v 0
ρ 2 + · · ·
tan δ l ≈ 2l + 1 − γ l(v 0 ) + O(ρ 2 )
γ l (v 0 ) − c l ρ 2 2l + 1
[(2l + 1)!!] 2 ρ2l+1 , c l = 1
2l − 1 − γ′ l (v 0)
2v 0
(10.56)
γ l (v 0 ) ≈ 0 2l + 1 − γ l (v 0 ) ≈ 0
tan δ l ≈ 2l + 1 − γ l(v 0 )
γ l (v 0 )
f l = 1 k
2l + 1
[(2l + 1)!!] 2 ρ2l+1 =
2l + 1 j l+1 (v 0 )
[ (2l + 1)!! ] 2 j l−1 (v 0 ) (ka)2l+1
k→0
−−−−→ 0
1
cot δ l − i ≈ tan δ l 2l + 1 j l+1 (v 0 )
≈
k [ (2l + 1)!! ] 2 j l−1 (v 0 ) (ka)2l a (10.57)
l ≠ 0 f l ≈ 0 , l = 0 s
f 0 ≈ a j (
1(v 0 )
j −1 (v 0 ) = − a 1 − tan v )
0
v 0
(
dσ
dΩ ≈ |f 0| 2 = a 2 1 − tan v ) 2 (
0
, σ tot ≈ 4πa 2 1 − tan v ) 2
0
(10.58)
v 0 v 0
, ,
l = 0 ,
l = 0
dσ/dΩ ≈ 0
2l + 1 − γ l (v 0 ) = v 0j l+1 (v 0 )
j l (v 0 )
≈ 0
tan δ 0 ≈ d 0 (ka) 3 , ∴ f 0 ≈ d 0 a 3 k 2
10.3
(10.58) k → 0 (10.24)

10 191
(10.55) ρ → 0 ρ ≪ l l
γ l (ρ 1 ) = ρ (
1j l−1 (ρ 1 )
ρ 2 )
1
≈ (2l + 1) 1 −
j l (ρ 1 )
(2l + 1)(2l + 3)
, l
,
2l + 1 − γ l (ρ 1 ) − d l ρ 2 ≈ ρ2 1 − ρ 2
2l + 3 = v2 0
2l + 3
ρ 2l+1
(
tan δ l ≈ v2 0
2l + 3 [(2l + 1)!!] 2 ∼ v2 0 eρ
8e l 2 2l
(2l + 1)!! =
(2l + 1)!
2 l l!
∼ √ 2 e −l (2l) l+1
) 2l+1
tan δ l l 0 ,
l , ,
l , l tan δ l
,
γ l (v 0 ) = v 0j l−1 (v 0 )
j l (v 0 )
≈ 0 , j l−1 (v 0 ) ≈ 0
6.1 j l−1 (v 0 ) = 0 l 0
(10.56)
tan δ l ≈
(2l + 1)2 ρ 2l+1
[(2l + 1)!!] 2 γ l (v 0 ) − c l ρ 2
γ l (v 0 ) c l ρ 2
( ) ′
γ l(v ′ v0
0 ) = j l−1 (v 0 ) + j ′ v 0
j l (v 0 )
l−1(v 0 )
j l (v 0 )
( ) ′ v0
= j l−1 (v 0 ) + (l − 1)j l−1(v 0 ) − v 0 j l (v 0 )
j l (v 0 )
j l (v 0 )
j l−1 (v 0 ) ≈ 0 γ ′ l (v 0) ≈ − v 0
γ l (v 0 )/c l > 0
c l = 1 2 + 1
2l − 1 = 2l + 1
2(2l − 1)
ρ = ρ r ≡
√
γ l (v 0 )
c l
tan δ l , (10.42) σ l k (σ l ) max =
4π(2l + 1)/k 2 ρ = ρ r
cot δ l ≈ d cot δ l(ρ r )
[(2l + 1)!!]2
(ρ − ρ r ) = −
dρ r
(2l + 1) 2 2c lρ −2l
r
(ρ − ρ r ) = − ρ − ρ r
ρ i
(10.59)

10 192
(2l + 1)2 ρ 2l
r
ρ i =
[(2l + 1)!!] 2 (10.60)
2c l
f l = 1 1
k cot δ l − i ≈ − a ρ i
ρ r ρ − ρ r + iρ i
σ l = 4π(2l + 1)|f l | 2 ≈ 4πa2
ρ 2 r
ρ 2 i
(ρ − ρ r ) 2 + ρ 2 i
l ≠ 0 c l > 0 ρ i ∝ (γ l (v 0 )) l γ l (v 0 ) ≈ 0 ρ i
, σ l ρ = ρ l ρ i
c l > 0 γ l (v 0 ) > 0 γ l (v 00 ) = 0, j l−1 (v 00 ) = 0
γ l (v 0 ) = γ ′ l(v 00 )(v 0 − v 00 ) + · · · = − v 00 (v 0 − v 00 ) + · · ·
γ l (v 0 ) > 0 v 0 < v 00 , σ l
ρ ρ r , ρ i
l = 1
σ l = 4πa 2 (2l + 1) sin2 δ l
(2l + 1)2 ρ 2l+1
ρ 2 , tan δ l ≈
[(2l + 1)!!] 2 γ l (v 0 ) − c l ρ 2
ρ = ak j l−1 (x) = j 0 (x) = sin x/x
v 0 = π, 2π, · · · ( 121 ) v 0 = π σ 1
σ 1 (σ l ) max
6
100
σ1/(4πa 2 )
4
2
v 0 = π
v 0 = 0.995π
σ1/(4πa 2 )
v 0 = 0.995π
v 0 = 1.005π
0.0 0.5
ρ = ak
0.15 0.20
ρ = ak
(10.59) ,
(
δ l = n + 1 )
π + ρ − ρ r
2 ρ i
ρ i , ρ = ak δ l (n + 1/2)π
(10.52) A l = k r > a
(
) (
r→∞
rR l (r) = kr cos δ l j l (ρr/a) − sin δ l n l (ρr/a) −−−−→ sin kr − lπ )
2 + δ l
r < a q = r/a
rR l (r) = rC l j l (ρ 1 q) = ρ
(
cos δ l j l (ρ) − sin δ l n l (ρ)
) q jl (ρ 1 q)
j l (ρ 1 )

10 193
sin δ l =
1
√
cot 2 δ l + 1 ≈ ρ
√ i
(ρ − ρr ) 2 + ρ 2 i
, cos δ l = sin δ l cot δ l ≈ −
ρ − ρ r
√
(ρ − ρr ) 2 + ρ 2 i
j l (ρ) =
ρ l
(2l + 1)!! , n l(ρ) = −
(2l + 1)!!
2l + 1
1
ρ l+1
rR l (r) ≈
1
√
(ρ − ρr ) 2 + ρ 2 i
ρ l ( [(2l + 1)!!]
2
(2l + 1)!! 2l + 1
)
ρ i
q
ρ 2l − ρ(ρ − ρ jl (ρ 1 q)
r)
j l (ρ 1 )
√
(ρ − ρr ) 2 + ρ 2 i
ρ ρ r (10.60)
rR l (r) ≈
ρ 2 i
√
(ρ − ρr ) 2 + ρ 2 i
1
√ 2cl ρ i
q j l (ρ 1 q)
j l (ρ 1 )
ρ = ρ r r < a r ≫ a ,
l = 1, v 0 = 0.995π, ρ = ρ r krj 1 (kr)
5
rRl(r)
0
0 1 10 20 30
r/a
, r > a r → ∞
,
, ,
ρ = ρ r
V eff (r) = V (r) + 2 l(l + 1)
2mr 2
, l ≠ 0
, E > 0
V (r)
E ≈ 0 E
2 l(l + 1)/(2ma 2 )
0
V eff (r)
1 2 r/a
E , r = 0 R(r)
E < 0 , R(r) r → ∞ , E r → ∞ R(r) → 0
E > 0 , r → ∞ R(r) ,

10 194
, E R(r)
k = k r cot δ l (k) = 0 k r ≈ 0 (10.59)
f l = 1 1
k cot δ l − i ≈ 1 k r
cot δ l ≈ k − k r
k i
,
1
= d cot δ l(k r )
k i dk r
k i
k − k r − ik i
, ∴ σ l ≈
4π(2l + 1)
k 2 r
k 2 i
(k − k r ) 2 + k 2 i
k 2 i σ l k = k r (10.41)
S l (k) = e 2iδ l
S l (k) = cot δ l + i
cot δ l − i ≈ k − k r + ik i
k − k r − ik i
= k − k∗ res
k − k res
,
k res = k r + ik i
, k S −1
l
(k) = 0, , S l (k) k r , k i
(k) = 0 , Re k Im k , ,
S −1
l
(10.43) k , k 2 /(2m) < 0 S l (k)
(10.54) η(ρ 1 ) = ρ 1 j l+1 (ρ 1 )/j l (ρ 1 )
(
) (
)
ρ n l+1 (ρ) − ij l+1 (ρ) − η(ρ 1 ) n l (ρ) − ij l (ρ)
cot δ l − i =
ρ j l+1 (ρ) − η(ρ 1 )j l (ρ)
S −1
l
(k) = 0
= − i ρ h(1) l+1 (ρ) − η(ρ 1)h (1)
l
(ρ)
ρ j l+1 (ρ) − η(ρ 1 )j l (ρ)
ρ h (1)
l+1 (ρ) − η(ρ 1)h (1)
l
(ρ) = 0 , ρ 1j l+1 (ρ 1 )
j l (ρ 1 )
ρ = ak ρ = iaβ
ρ 1 j l+1 (ρ 1 )
j l (ρ 1 )
=
iaβ h(1)
l+1 (iaβ)
h (1)
l
(iaβ)
, ρ 1 = a
(6.14)
√
k 2 + 2mV 0
2
= ρ h(1) l+1 (ρ)
(ρ)
h (1)
l
√
2mV0
= a
2 − β 2
v 0 , (10.54) δ l (k) σ l
σ tot Born (10.23)
• v 0 = 2.7 l = 1 , v 0 → π ,
• v 0 = 4, 5 l = 2, 3
• k l = 0
• v 0 k , ,
• (10.54) k → ∞ tan δ l (k) → 0 δ l (∞) = 0 (10.56) l = 0
γ 0 (v 0 ) = 0 tan δ l (0) = 0 , n l δ l (0) = n l π
, n l l ( ),
l = 0 v 0 = 2.7 v 0 = 4 1 , v 0 = 5 2 121

10 195
2π
v 0 = 2.7 = 0.859π
l = 0
δ l
π
l = 1
l = 2
l = 3
σl/(4πa 2 )
2
Born
0 2 4 6 8
ak
0 2 4 6 8
ak
2π
v 0 = 4.0 = 1.273π
δ l
π
σl/(4πa 2 )
2
Born
0 2 4 6 8
ak
0 2 4 6 8
ak
2π
v 0 = 5.0 = 1.592π
δ l
π
σl/(4πa 2 )
2
Born
0 2 4 6 8
ak
0 2 4 6 8
ak
10.6
(10.28)
( )
d
2
l(l + 1)
−
dr2 r 2 − U(r) + k 2 χ l (r) = 0 (10.61)
k 2 , r → 0 r 2 U(r) → 0
( )
d
2
l(l + 1)
−
dr2 r 2 χ l (r) = 0
α r l+1 + β r −l , (10.61) 2 Φ l (k, r) ,
Φ (1)
l
(k, r)
Φ l , Φ (1)
l
Φ l (k, r)
r→0
−−−−→ r l+1 , Φ (1) r→0
l
(k, r) −−−−→ r −l (10.62)
r → ∞ , r → ∞ (10.61)
( ) d
2
dr 2 + k2 χ l (r) = 0

10 196
, α e ikr + β e −ikr (10.61)
lim
r→∞ e±ikr F l (± k, r) = 1 (10.63)
F l (k, r) , F l (−k, r) r → 0
(10.61) ϕ 1 , ϕ 2 W (ϕ 1 , ϕ 2 ) = ϕ 1
dϕ 2
dr − dϕ 1
dr ϕ 2
dW
dr = ϕ d 2 ϕ 2
1
dr 2 − d2 ϕ 1
dr 2 ϕ 2
( ) ( )
l(l + 1)
l(l + 1)
= ϕ 1
r 2 + U(r) − k 2 ϕ 2 −
r 2 + U(r) − k 2 ϕ 1 ϕ 2 = 0
(
)
r W F l (k, r), F l (−k, r) r → ∞
(
) (
W F l (k, r), F l (−k, r) = W e −ikr , e ikr) = 2ik
(10.61) F l (k, r) F l (−k, r)
W (ϕ, ϕ) = 0
Φ l (k, r) = AF l (k, r) + BF l (−k, r)
(
) (
)
W F l (k, r), Φ l (k, r) = B W F l (k, r), F l (−k, r) = 2ik B
(
) (
)
W F l (−k, r), Φ l (k, r) = A W F l (−k, r), F l (k, r) = −2ik A
(
)
F l (k) = W F l (k, r), Φ l (k, r)
F l (k) (Jost) Φ l (−k, r) = Φ l (k, r)
A = − F l(−k)
2ik
, B = F l(k)
2ik
r → ∞
Φ l (k, r) = − 1 (
)
F l (−k) F l (k, r) − F l (k) F l (−k, r)
2ik
Φ l (k, r) → − 1 (F l (−k) e −ikr − F l (k) e ikr)
2ik
r → ∞ (10.43)
S l (k) = (−1) l F l(k)
F l (−k)
(10.64)
(10.65)
(10.61) (10.63) k k ∗
( )
d
2
l(l + 1)
−
dr2 r 2 − U(r) + k 2 Fl ∗ (k ∗ , r) = 0 ,
lim
r→∞ e−ikr Fl ∗ (k ∗ , r) = 1
F ∗ l (−k ∗ , r) = F l (k, r) (10.66)

10 197
Φ l (k, r) k Φ ∗ l (k∗ , r) = Φ l (k, r)
(
) (
)
Fl ∗ (k ∗ ) = W Fl ∗ (k ∗ , r), Φ ∗ l (k ∗ , r) = W F l (−k, r), Φ l (k, r) = F l (−k)
(10.65)
S l (−k) = S −1
l
(k) , S ∗ l (k ∗ ) = S l (−k) (10.67)
k , , 1 S l (k) , S l (k)
• k , k = − k ∗ S l (−k ∗ ) = S ∗ l (k) S l(k) = S ∗ l (k)
, S l (k)
• k Sl ∗(k) = S l(−k) = S −1
l
(k) |S l (k)| = 1 , δ l
S l (k) = e 2iδl(k) , δ l (k)
• (10.67) S l (k) = 0
S ∗ l (−k ∗ ) = S l (k) = 0 ,
S l (−k) = Sl ∗ (k ∗ ) = S −1
l
(k) = ∞
k −k ∗ S l , − k
k ∗ S l , ,
k = k n F l (k n ) = 0 (10.64)
r → ∞
Φ l (k n , r) = a n F l (k n , r) , a n = − F l(−k n )
2ik n
Φ l (k n , r) → a n e −ik nr = a n e −ik Rr e k Ir , k R = Re k n , k I = Im k n (10.68)
, k I < 0 r → ∞ Φ l (k n , r) → 0 , Φ l (k n , r) 0
, Φ l (k n , r) , F l (k n ) = 0 k I < 0 k R = 0
, E = − kI 2 /2m
(10.61)
d 2
( )
l(l + 1)
dr 2 Φ l(k, r) =
r 2 + U(r) − k 2 Φ l (k, r)
d 2
dr 2 Φ∗ l (k, r) =
( l(l + 1)
r 2 + U(r) − k ∗ 2 )
Φ ∗ l (k, r)
d
(
)
dr W Φ l (k, r), Φ ∗ l (k, r) = Φ l (k, r) d2
dr 2 Φ∗ l (k, r) − Φ ∗ l (k, r) d2
dr 2 Φ l(k, r) = 2i Im(k 2 ) |Φ l (k, r)| 2
2iIm(k 2 )
∫ ∞
0
dr |Φ l (k, r)| 2 = W
(
(
)∣
Φ l (k, r), Φ ∗ ∣∣r=∞
l (k, r))∣
− W Φ l (k, r), Φ ∗ ∣∣r=0
l (k, r)
(
)∣
= W Φ l (k, r), Φ ∗ ∣∣r=∞
l (k, r)

10 198
(10.68)
(
W Φ l (k n , r), Φ ∗ ∣∣r=∞
l (k n , r))∣
= 2i|a n | 2 k R e ∣ 2k Ir r=∞
( ∫ ∞
k R 2k I dr |Φ l (k n , r)| 2 − |a n | 2 e ∣ ) 2k Ir r=∞
= 0
0
k R = 0 F l (k) = 0 k R = 0, k I < 0
, k R ≠ 0 , k I < 0
∫ ∞
0
dr |Φ l (k n , r)| 2 = 0
Φ l (k n , r) = 0 , k R ≠ 0 , k I < 0 F l (k) = 0 ,
k R ≠ 0, k I > 0 Φ l (k n , r) , F l (k) = 0 k
, S l (k) , S l (k) = S −1 (−k)
, S l (k)
k = k n S l (k) F l (−k n ) = 0
k R = 0 , k I > 0 k R ≠ 0 , k I < 0
|k I | ≪ |k R | ,
E = 2 k 2 n
2m = E R − i 2 Γ ,
E R = 2 (
k
2
2m R − kI 2 ) 2 2
, Γ = −
m k Rk I
, ψ
|ψ(r, t)| 2 ∝ |e −iEt/ | 2 −Γ t/
= e
Γ > 0 ( k R > 0 ) τ e −1
τ = /Γ Γ
F l (−k n ) = 0 k = k n
F l (−k) = a (k − k n ) , a = dF l(−k n )
dk n
= dF ∗ l (k∗ n)
dk n
=
F l (k ∗ n) = F ∗ l (−k n) = 0 k = k ∗ n
F l (k) = b (k − k ∗ n) , b = dF l(k ∗ n)
dk ∗ n
= a ∗
, k I k n ≈ k ∗ n k = k R
( dFl (k ∗ n)
dk ∗ n
) ∗
S l (k) = (−1) l F l(k)
F l (−k) = k − k∗ n
, e2iθ0 e 2iθ0 = (−1) l a∗
k − k n a
(10.69)
S l (k) = e 2iθ0 k R(k − k ∗ n)
k R (k − k n )
k R (k − k n ) = k R (k − k R ) − ik R k I ≈ k2 − k 2 R
2
− ik R k I = m )
(E
2 − E R + iΓ/2

10 199
S l (k) = e 2iθ E − E (
)
0 R − iΓ/2
E − E R + iΓ/2 = iΓ
e2iθ 0
1 −
E − E R + iΓ/2
σ l
(
π(2l + 1)
σ l =
k 2 |1 − S l | 2 4π(2l + 1) Γ e
iθ 0
)
sin θ 0
= σ l, pot + σ l, res −
k 2 Re
E − E R + iΓ/2
(10.70)
(10.71)
σ l, pot =
4π(2l + 1)
k 2 sin 2 θ 0 , σ l, res =
4π(2l + 1)
k 2 (Γ /2) 2
(E − E R ) 2 + (Γ/2) 2
Γ σ l, res E = E R , σ l, pot
E , (10.70)
S l = e 2iδ l
, δ l = θ 0 + θ res , tan θ res = − Γ/2
E − E R
θ res E
π
, θ res nπ , E ≪ E R
θ res = 0, E ≫ E R θ res = π
E ≈ E R
θ res ≈ π 2 + E − E R
Γ/2
π/2
0
E R
Γ , E R θ res
10.7
V (r) r → ∞ 1/r 2 0
m, Ze m ′ , Z ′ e m ′ ≫ m ,
, m
)
(− 2
2m ∇2 + ZZ′ αc
ψ c (r) = Eψ c (r)
r
(
∇ 2 + k 2 − 2kη )
ψ c (r) = 0 , E = 2 k 2
r
2m ,
∂ψ c
∂x
ψ c (r) = e ikz f(q) ,
= eikz
∂q
∂x f ′ (q) = e ikz x r f ′ (q) ,
∂ψ c
∂z = eikz (
ikf + ∂q
∂z f ′ (q)
∂ 2 ψ c
∂z 2 = eikz [
−k 2 f(q) +
)
( x 2 + y 2
r 3
q = r − z = 2r sin 2 (θ/2)
∂ 2 ψ c
∂x 2
(
= e ikz ikf(q) − q )
r f ′ (q)
η = ZZ′ αmc
k
[( )
]
1 = eikz r − x2
r 3 f ′ (q) + x2
r 2 f ′′ (q)
− 2ikq ) ]
f ′ (q) + q2
r
r 2 f ′′ (q)

10 200
∇ 2 ψ c (r) = e ikz ( 2q
r f ′′ (q) +
)
2(1 − ikq)
f ′ (q) − k 2 f(q)
r
(
q d2
dq 2 + (1 − ikq) d dq − kη )
f(q) = 0
ρ = ikq = ik(r − z)
(ρ d2
dρ 2 + (1 − ρ) d dρ + iη )
f(q) = 0
(15.59) a = − iη, b = 1 , q = 0
A c
(15.80) |r − z| → ∞
ψ c (r) = A c e ikz M(−iη, 1, ikq) (10.72)
)
M(−iη, 1, ikq) =
(1 (−ikq)iη + η2
+ eikq (ikq) −iη−1
Γ (1 + iη) ikq Γ (−iη)
= eπη/2
Γ (1 + iη) eiη log kq (
1 + η2
ikq
(1 +
)
(1 + iη)2
ikq
)
+ eπη/2 ikq−iη log kq
e
Γ (−iη) ikq
(−ikq) iη = e iη log(−ikq) = e iη(−iπ/2+log(kq))
|r − z| −2 A c = Γ (1 + iη) e −πη/2 q = r − z = 2r sin 2 (θ/2)
ψ c (r) −→ ψ in (r) + ψ out (r)
(
)(
)
ψ in (r) = exp ikz + iη log k(r − z) 1 + O(|r − z| −1 )
ψ out (r) = f c (θ)
f c (θ) = 1
2ik
= − η
2k
exp (ikr − iη log 2kr)
(
)
1 + O(|r − z| −1 )
r
Γ (1 + iη) exp ( − iη log sin 2 (θ/2) )
Γ (−iη) sin 2 (θ/2)
Γ (1 + iη) exp ( − iη log sin 2 (θ/2) )
Γ (1 − iη) sin 2 (θ/2)
(10.73)
(10.74)
(10.75)
, Γ (15.52) iΓ (−iη) = − Γ (1 − iη)/η
(Γ (1 + iη)) ∗ = Γ (1 − iη)
f c (θ) = −
Γ (1 + iη) = |Γ (1 + iη)| e iσ0
η
2k sin 2 (θ/2) eiδ = −
ZZ′ αc
4E sin 2 (θ/2) eiδ , δ = 2σ 0 − η log sin 2 (θ/2) (10.76)

10 201
, (10.73), (10.74) (10.2)
, ψ in exp(iη log k(r − z)) ψ in
ψ in J
J x = )
(ψ
m Im in
∗ ∂
∂x ψ in
J z = k m
= η
m
(
1 − η kr + O((r − z)−1 )
x
r(r − z)
)
(
)
1 + O((r − z) −1 )
z → −∞ J x , J y → 0 , J z → k/m , ψ in
ψ out e ikr /r , exp (−iη log 2kr) (10.3)
ψ out J
|f c | 2 d
(
)
J = e r
m r 2 kr − η log 2kr + Im (fc ∗ uf c ) k |f c | 2 ( )
dr
m r 3 = e r
m r 2 1 + O(r −1 )
, f c (θ) ,
(
dσ
ZZ
dΩ = |f c(θ)| 2 ′ ) 2
αc
=
4E sin 2 (10.77)
(θ/2)
( ? ),
, z = r (10.73), (10.74)
θ = 0
1 (10.75) η
, η 1 , , (10.76) δ = 0 f c (θ)
f born (θ) = −
, (10.16)
∫
V(q) = d 3 r e −iq·r V (r) = 2πZZ ′ αc
= 2πZZ′ αc
iq
ZZ′ αc
4E sin 2 (θ/2)
∫ ∞
0
∫ ∞
0
dr r 2 ∫ π
0
cos θ
e−iqr
dθ sin θ
r
dr
(e iqr − e −iqr)
(10.78)
µ e −µr ,
µ → + 0
V(q) = 2πZZ′ αc
iq
[ ] e
(−µ+iq)r
∞
−µ + iq + e(−µ−iq)r
µ + iq
0
= 4πZZ′ αc
q 2 + µ 2
µ→+0
−−−−→ 4πZZ′ αc
q 2
,
f born (θ) = −
m
2π 2 V(q) = − ZZ′ αc
4E sin 2 (θ/2)
(10.78) f born (10.76) ,
(10.77)
(10.35)
ψ c (r) = 1
kr
∞∑
χ l (r) P l (cos θ) (10.79)
l=0

10 202
( d
2
l(l + 1)
−
dρ2 ρ 2 − 2η )
ρ + 1 χ l (r) = 0 , ρ = kr (10.80)
χ l (ρ) = ρ l+1 e iρ v l (ρ)
(
z d2
dz 2 + (2l + 2 − z) d dz − (l + 1 + iη) )
v l = 0 , z = −2iρ (10.81)
(15.59) a = l + 1 + iη , b = 2l + 2 ,
B l
v l (ρ) = B l M(l + 1 + iη, 2l + 2, −2iρ)
(15.77), (15.78) W 1 , W 2
)
v l (ρ) = B l
(W 1 (l + 1 + iη, 2l + 2, −2iρ) + W 2 (l + 1 + iη, 2l + 2, −2iρ)
= B l e −iρ( )
e iρ W 1 (l + 1 + iη, 2l + 2, −2iρ) + e −iρ W 1 (l + 1 − iη, 2l + 2, 2iρ)
(
)
= 2B l e −iρ Re e iρ W 1 (l + 1 + iη, 2l + 2, −2iρ)
(15.77), (15.79)
W 1 (l + 1 + iη, 2l + 2, −2iρ) =
=
Γ (2l + 2)
Γ (l + 1 − iη) (2iρ)−l−1−iη g l (η, ρ)
(2l + 1)! eπη/2 e −iη log(2ρ)−i(l+1)π/2+iσ l
|Γ (l + 1 + iη)| (2ρ) l+1 g l (η, ρ) (10.82)
g l (η, ρ) = g(l + 1 + iη, −l + iη, 2iρ) = 1 + η + i ( l(l + 1) + η 2)
+ O(ρ −2 )
2ρ
, σ l Γ (l + 1 + iη)
Γ (l + 1 + iη) = |Γ (l + 1 + iη)| e iσ l
, e 2iσ l
=
Γ (l + 1 + iη)
Γ (l + 1 − iη)
(10.83)
B l = 2l |Γ (l + 1 + iη)|
e −πη/2
(2l + 1)!
χ l (ρ) = F l (η, ρ) ≡ B l ρ l+1 e iρ M(l + 1 + iη, 2l + 2, −2iρ)
(
)
= 2B l ρ l+1 Re e iρ W 1 (l + 1 + iη, 2l + 2, −2iρ)
[ (
)) ]
= Im g l (η, ρ) exp i
(ρ − η log(2ρ) − πl/2 + σ l
(10.84)
ρ → ∞
(
F l (η, ρ) −→ sin ρ − η log(2ρ) − πl )
2 + σ l , ρ = kr

10 203
(10.34) (10.83) σ l
F l (η, ρ) ρ → 0 F l (η, ρ) −→ B l ρ l+1
, η = 0 (15.85)
F l (0, ρ) = 2l |Γ (l + 1)|
(2l + 1)!! ρj l (ρ) = ρj l (ρ)
(2l + 1)!
(10.72) (10.79)
A c e ikz M(−iη, 1, ik(r − z)) = 1 ρ
∞∑
C l F l (η, ρ)P l (cos θ) ,
l=0
A c = Γ (1 + iη) e −πη/2
C l x = cos θ (15.27)
C l
F l (η, ρ)
ρ
= A 2l + 1
2 l+1 l!
∫ 1
−1
dx (1 − x 2 ) l dl
dx l eiρx M(−iη, 1, iρ(1 − x))
(15.74)
∫ (
)
e iρx M(−iη, 1, iρ(1 − x)) = v 0 (−iη, 1) dt exp iρx + iρ(1 − x)t t −iη−1 (1 − t) iη
= v 0 (−iη, 1)
C
∞∑ (iρ) n ∫
n=0
n!
C
(
) nt
dt t + (1 − t)x
−iη−1 (1 − t) iη
C l
F l (η, ρ)
ρ
= A c
2l + 1
2 l+1 l! v 0(−iη, 1)
∞∑ (iρ) n ∫
n=0
n!
C
dt t −iη−1 (1 − t) iη f n (t)
f n (t) =
∫ 1
−1
(
)
dx (1 − x 2 ) l dl
n
dx l t + (1 − t)x
ρ → 0 F l (η, ρ) → B l ρ l+1 , n = l (5.25)
f l (t) = (1 − t) l l!
∫ 1
−1
dx (1 − x 2 ) l = (1 − t) l l! 22l+1 (l!) 2
(2l + 1)!
(15.27)
C l = A c (2i) l ∫
l!
v 0 (−iη, 1) dt t −iη−1 (1 − t) l+iη
B l (2l)!
C
∫
M(−iη, l + 1, 0) = 1 = v 0 (−iη, l + 1) dt t −iη−1 (1 − t) l+iη
C
C l = A c (2i) l l! v 0 (−iη, 1)
B l (2l)! v 0 (−iη, l + 1) = A c (2i) l
B l (2l)!
Γ (l + 1 + iη)
Γ (1 + iη)
= (2l + 1)i l e iσ l
ψ c (r) = A c e ikz M(−iη, 1, ik(r − z)) = 1 ∞∑
(2l + 1) i l e iσ l
F l (η, ρ)P l (cos θ) (10.85)
ρ
l=0

10 204
η = 0
(10.31)
∞∑
e ikz = (2l + 1) i l j l (kr)P l (cos θ)
l=0
2 (10.80) , F l (η, ρ) G l (η, ρ) F l
W 1 + W 2 , W 1 − W 2
(
)
G l (η, ρ) = B l ρ l+1 e iρ i W 1 (l + 1 + iη, 2l + 2, −2iρ) − W 2 (l + 1 + iη, 2l + 2, −2iρ)
(
)
= − 2B l ρ l+1 Im e iρ W 1 (l + 1 + iη, 2l + 2, −2iρ)
[ (
)) ]
= Re g l (η, ρ) exp i
(ρ − η log(2ρ) − πl/2 + σ l
(10.82) ρ → ∞
G l (η, ρ) → cos
, (15.86) G l (0, ρ) = − ρ n l (ρ)
(
ρ − η log(2ρ) − πl )
2 + σ l
F l G l
(
)
u (+)
l
(η, ρ) = e −iσ l
G l (η, ρ) + i F l (η, ρ)
= 2B l ρ l+1 e iρ−iσl+iπ/2 W 1 (l + 1 + iη, 2l + 2, −2iρ)
( ) ∗
u (−)
l
(η, ρ) = u (+)
l
(η, ρ)
(10.86)
, ρ → ∞
u (±)
±i(ρ−η log(2ρ)−πl/2)
l
(η, ρ) −→ e
u (+) (η, ρ) , u (−) (η, ρ)
, (10.85)
ψ c (r) = 1
2ikr
F l (η, ρ) = e−iσ l (
2i
e 2iσ l
u (+)
l
∞∑ ( (2l + 1) i l e 2iσ l
u (+)
l=0
l
)
(η, ρ) − u (−)
l
(η, ρ)
)
(η, kr) − u (−)
l
(η, kr) P l (cos θ) (10.87)
10.4
x = cos θ
(15.27)
, (15.21)
l=0
f c (θ) = 1 Γ (1 + iη)
2ik Γ (−iη)
f c (θ) = 1
2ik
( 1 − x
2
) −1−iη
∞∑
(2l + 1)e 2iσ l
P l (x) (10.88)
l=0
∞∑
∞∑
)
(1 − x) (2l + 1)P l (x) =
((2l + 1)P l − (l + 1)P l+1 − lP l−1 = 0
l=0

10 205
{
∞∑
0 , x ≠ 1
(2l + 1)P l (x) =
∞ , x = 1
l=0
(15.28) x ′ = 1 , x ≠ 1 , ( θ ≠ 0 )
f c (θ) = 1
2ik
(10.39)
∞∑
(2l + 1) ( e 2iσ l
− 1 ) P l (cos θ)
l=0
10.5
Γ (15.52) σ l = σ l−1 + tan −1 (η/l)
1
Γ (z) = z eγz
∞
∏
n=1
[(1 + z/n) e −z/n]
γ = = lim
(1 + 1
n→∞ 2 + 1 3 + · · · + 1 )
n − log n = 0.57721 56649 01532 · · ·
∞∑ ( η
σ 0 = − γη +
n n)
− η tan−1
n=1
(10.89)
S l = e 2iσ l
E Γ (z) n ′ = 0, 1, 2, · · ·
z = −n ′ , (10.83) , E
l + 1 + iη = − n ′ , ik = ZZ′ αmc 2
c
1
l + 1 + n ′
E = 2 k 2
2m = − (ZZ′ α) 2 mc 2
2n 2 , n = l + 1 + n ′
(6.20) F l , χ l (r)
r → ∞
χ l (r) = D l
(
u (+)
l
(η, kr) − S −1
l
(
χ l (r) = D l e i(kr−η log(2kr)−πl/2) − S −1
2 1
l
)
u (−)
l
(η, kr)
i(kr−η log(2kr)−πl/2)
F l (η, ρ) = D l e
−i(kr−η
e
log(2kr)−πl/2))
r → ∞ χ l (r) → 0 ik < 0 ZZ ′ < 0 ,
, r → 0 u (+)
l
S −1
l
u (−)
l
u (−)
l
,

10 206
, ,
σ l ≠ 0 l , ,
,
V (r) = V 0 (r) + ZZ′ αc
r
V 0 (r) V (r) σ l + δ l , δ l l
0 , σ l l ,
f c (θ)
ψ(r) = 1
kr
∞∑
χ l (r)P l (cos θ)
l=0
( d
2
l(l + 1)
−
dr2 r 2 − 2m
2 V 0(r) − 2kη )
+ k 2 χ l (r) = 0 (10.90)
r
r V 0 (r) (10.80)
χ l (r) = a l
(
F l (η, kr) cos δ l + G l (η, kr) sin δ l
)
(
r→∞
−−−−→ a l sin kr − η log(2kr) − πl )
2 + σ l + δ l
σ l + δ l V (r) ,
G l , r G l
χ l (r) V 0 (r) ,
F l = i 2
(
e −iσ l
u (−) − e iσ l
u (+)) , G l = 1 (e −iσ l
u (−) + e iσ l
u (+))
2
(
χ l (r) = c l e 2iδ l+2iσ l
u (+)
l
)
(η, kr) − u (−) (η, kr)
, c l = e−iδ l−iσ l
a l
2i
, r → ∞ (10.2) ,
r → ∞ , (10.73), (10.74)
r → ∞
(
)
exp (ikr − iη log 2kr)
ψ(r) = exp ikz + iη log k(r − z) + f(θ)
r
1 U(r)
, 2 f c (θ) f(θ) V 0 (r)
f(θ)
f(θ) = f c (θ) + f 0 (θ)
ψ(r) = e ikz+iη log k(r−z) + f c (θ)
eikr−iη
log 2kr
r
+ f 0 (θ)
eikr−iη
log 2kr
r

10 207
1 2 ψ c (r)
= 1
2ikr
eikr−iη
log 2kr
ψ(r) = ψ c (r) + f u (θ)
r
∞∑ ( (2l + 1) i l e 2iσ l
u (+)
l=0
+ f 0 (θ)
eikr−iη
log 2kr
, χ l r → ∞
ψ ( r) = 1
kr
= 1
kr
r
∞∑ (
c l e 2iδ l+2iσ l
u (+)
l=0
∞∑ (
c l e 2iσ l
u (+)
l=0
+ 1
kr
∞∑
c l e 2iσ l
l=0
l
l
l
)
(η, kr) − u (−)
l
(η, kr) P l (cos θ) (10.91)
)
(η, kr) − u (−)
l
(η, kr) P l (cos θ)
(10.92)
)
(η, kr) − u (−)
l
(η, kr) P l (cos θ) (10.93)
(
e
2iδ l
− 1 ) u (+)
l
(η, kr)P l (cos θ) (10.94)
(10.91) (10.93) c l = (2l + 1)i l /(2i) u (+)
l
(η, kr)
(10.94) (10.92)
f 0 (θ) = 1
2ik
∞∑
(2l + 1) e 2iσ (
l e
2iδ l
− 1 ) P l (cos θ) (10.95)
l=0
dσ
dΩ = |f c(θ) + f 0 (θ)| 2 = |f c (θ)| 2 + |f 0 (θ)| 2 + 2Re(fc ∗ f 0 )
V 0 (r) δ l l 0 , (10.95)
10.8
(10.90) , δ l (10.90)
6.7 , r = 0 r ,
λ
q = r λ ,
ε = 2mλ2 E
2 , U 0 (q) = 2mλ2
2 V 0 (r)
( d
2
l(l + 1)
−
dq2 q 2 − U 0 (q) − 2√ )
ε η
+ ε χ l (q) = 0 (10.96)
q
η = ZZ′ αmcλ
√ ε
= ZZ′ αmc
k
q → 0 q 2 U 0 (q) → 0 χ l (q) (6.42) (10.96) q
χ max = χ l (q max ) ,
χ max−1 = χ l (q max − ∆q)

10 208
q = q max U 0 (q)
χ max = a l
(
F max cos δ l + G max sin δ l
)
, χ max−1 = a l
(
F max−1 cos δ l + G max−1 sin δ l
)
F max = F l
(
η,
√
εqmax
)
, Fmax−1 = F l
(
η,
√
ε(qmax − ∆q) )
G max = G l
(
η,
√
εqmax
)
, Gmax−1 = G l
(
η,
√
ε(qmax − ∆q) ) (10.97)
tan δ l = − F maxχ max−1 − F max−1 χ max
G max χ max−1 − G max−1 χ max
(10.98)
χ max χ max−1 , F l G l , δ l
δ l nπ , (10.95)
σ l , F l (η, ρ), G l (η, ρ) σ l
10.5
(10.89) (10.84), (10.86) F l (η, ρ) G l (η, ρ) (15.79)
g l (η, ρ) = g(l + 1 + iη, −l + iη, 2iρ) =
(a) k
D k+1 =
∞∑
D k ,
k=0
(l + k + 1 + iη)(k − l + iη)
D k
(k + 1) 2iρ
D k = (l + 1 + iη) k (−l + iη) k
k!
1
(2iρ) k
D 0 = 1 g l (η, ρ) ,
Re D k+1 =
Im D k+1 =
(2k + 1)η
2(k + 1)ρ Re D k − η2 + l(l + 1) − k(k + 1)
Im D k
2(k + 1)ρ
(2k + 1)η
2(k + 1)ρ Im D k + η2 + l(l + 1) − k(k + 1)
Re D k
2(k + 1)ρ
η = 0 D k = 0 , ( k > l ) g l (η, ρ) , η ≠ 0
, g l (η, ρ) , ρ
g l (η, ρ) ( 271 ), ρ , , |D k | < 10 −6
k ρ ,
,
{
− u0 , q < 1
U 0 (q) =
0 , q > 1
η = 0 , δ l (10.54)
,
u 0
ρ = √ ε , ρ 1 = √ ε + u 0
= 50 , ε = 20 , (10.54) (10.98)
δ l q min = 10 −3 , q max =
3 , ∆q = 0.01, 0.001
l
(10.98)
0.01 0.001
(10.54)
0 1.0295 1.0488 1.0514
1 0.3748 0.3797 0.3804
2 0.7494 0.7562 0.7571
3 −0.3499 −0.3219 −0.3178
4 −0.4628 −0.4626 −0.4626
5 −0.3969 −0.3885 −0.3873
6 0.0638 0.0694 0.0702
7 0.0027 0.0029 0.0029
∆q ≈ 0.1 ∼ 0.01 ,
0 , (10.98) δ l ,
∆q

11 209
11
11.1
3 α, −→ α , ,
,
| α 〉 , α
( ket vector )
, , | α 〉
1. 1
, 1
,
2. ,
| α 〉 ,
c 1 | α 〉 + c 2 | α 〉 = (c 1 + c 2 )| α 〉 , ( c 1 , c 2 )
,
,
( bra vector ) | α 〉 〈 α | 1
c c | α 〉 c 〈 α | c ∗ 〈 α |
⎛ ⎞
A ≡
⎜
⎝
a 1
a 2
a 3
.
⎟
⎠ ⇐⇒ A† = ( a ∗ 1 a ∗ 2 a ∗ 3 · · · )
, | α 〉 + 〈 β |
〈 α |, | β 〉 , 〈 α | β 〉
⎛
b 1
( a ∗ 1 a ∗ 2 a ∗ 3 · · · )
⎜
⎝
b 3
. b 2
⎞
⎟
⎠ = a∗ 1b 1 + a ∗ 2b 2 + a ∗ 3b 3 + · · ·
(
)
(
)
1. 〈 α | c | β 〉 + c ′ | β ′ 〉 = c 〈 α | β 〉 + c ′ 〈 α | β ′ 〉 , c 〈 β | + c ′ 〈 β ′ | | α 〉 = c 〈 β | α 〉 + c ′ 〈 β ′ | α 〉

11 210
2. 〈 α | α 〉
〈 α | α 〉 ≥ 0 ( | α 〉 )
√ 〈 α | α 〉 | α 〉
3. | β 〉 = c | α 〉 〈 β | = c ∗ 〈 α | 〈 β | α 〉 = c ∗ 〈 α | α 〉 〈 α | α 〉
〈 β | α 〉 ∗ = c 〈 α | α 〉 = 〈 α | β 〉
,
〈 α | β 〉 = 〈 β | α 〉 ∗ (11.1)
a·b = b·a
| α 〉 ,
| α ′ 〉 =
1
√
〈 α | α 〉
| α 〉
〈 α ′ | α ′ 〉 = 1 1 ( normalization )
,
〈 α | β 〉 = 0 , 2
11.2
, ,
( operator ) X ,
| α 〉 , X
, | α 〉X, X〈 α |
• , , c α c β
(
)
(
)
X c α | α 〉 + c β | β 〉 = c α X| α 〉 + c β X| β 〉 , c α 〈 α | + c β 〈 β | X = c α 〈 α |X + c β 〈 β |X
• XY = Y X , X Y
( commute )
• 〈 α | X| β 〉 , 〈 α |X
| β 〉 :
〈 α | (X| β 〉) = (〈 α |X) | β 〉
, 〈 α |X| β 〉
X ( Hermitian conjugate ) X †
| γ 〉 = X| β 〉
〈 β |X † | α 〉 = 〈 α |X| β 〉 ∗ (11.2)
〈 β |X † | α 〉 = 〈 α | γ 〉 ∗ = 〈 γ | α 〉 , 〈 γ | = 〈 β |X †
X| β 〉 〈 β |X † 〈 β |X X = X † , X
( Hermitian )

11 211
1.
〈 α |XY | β 〉 = 〈 β |(XY ) † | α 〉 ∗
| γ 〉 = Y | β 〉 〈 γ | = 〈 β |Y †
〈 α |XY | β 〉 = 〈 α |X| γ 〉 = 〈 γ |X † | α 〉 ∗ = 〈 β |Y † X † | α 〉 ∗
(XY ) † = Y † X † (11.3)
n
2. (11.2)
〈 β | ( X †) †
| α 〉 = 〈 α |X † | β 〉 ∗ = 〈 β |X| α 〉
( X †) †
= X
3. | α 〉〈 β |
( )
( )
| α 〉〈 β | | γ 〉 = | α 〉〈 β | γ 〉 , 〈 γ | | α 〉〈 β | = 〈 γ | α 〉〈 β |
| α 〉〈 β | | γ 〉 | α 〉 〈 β | γ 〉
, | α 〉〈 β | X = | α 〉〈 β |
〈 α ′ |X| β ′ 〉 ∗ = 〈 β ′ |X † | α ′ 〉
( )
〈 α ′ |X| β ′ 〉 ∗ = (〈 α ′ | α 〉〈 β | β ′ 〉) ∗ = 〈 α | α ′ 〉〈 β ′ | β 〉 = 〈 β ′ | | β 〉〈 α | | α ′ 〉
X † = | β 〉〈 α |,
(
| α 〉〈 β |) †
= | β 〉〈 α |
11.3
A :
A| a 〉 = a| a 〉 (11.4)
a | α 〉 A| α 〉 | α 〉
(11.4) | a 〉 a A
( eigenvalue ) , | a 〉 , | a 〉 a
(11.4) a | a 〉
, , ( A † = A ) ,
2

11 212
1.
(11.4) 〈 a | 〈 a |A| a 〉 = a〈 a | a 〉 ,
(11.1) 〈 a | a 〉
a ∗ 〈 a | a 〉 = 〈 a |A| a 〉 ∗ = 〈 a |A † | a 〉 = 〈 a |A| a 〉 = a〈 a | a 〉
, a = a ∗ a
2.
A 2 a, a ′ :
A † = A a ′ , 2
A| a 〉 = a| a 〉 (11.5)
A| a ′ 〉 = a ′ | a ′ 〉 (11.6)
〈 a ′ |A = a ′ 〈 a ′ | (11.7)
(11.5) 〈 a ′ | 〈 a ′ |A| a 〉 = a〈 a ′ | a 〉 , (11.7) | a 〉 〈 a ′ |A| a 〉 =
a ′ 〈 a ′ | a 〉 2
(a − a ′ )〈 a ′ | a 〉 = 0
, a ≠ a ′ 〈 a ′ | a 〉 = 0 , | a 〉 | a ′ 〉
⎧
⎨ 1 , a ′ = a
δ a′ a =
⎩
0 , a ′ ≠ a
,
〈 a ′ | a 〉 = δ a ′ a〈 a | a 〉
〈 a | a 〉 = 1
〈 a ′ | a 〉 = δ a′ a (11.8)
,
, ,
:
1. A , A 1
, A
, a
|〈 a | α 〉| 2
, | α 〉 , | a 〉, | α 〉
2. , ,
,

11 213
3. ( complete system ) , , | α 〉
| α 〉 = ∑ a
c a | a 〉 (11.9)
(11.8)
〈 a | α 〉 = ∑ a ′ c a ′〈 a | a ′ 〉 = ∑ a ′ c a ′δ a a ′ = c a (11.10)
, a |c a | 2
, (11.9) ,
2. 3. ( observable )
(11.10) (11.9)
| α 〉 = ∑ a
| a 〉〈 a | α 〉 =
( ∑
a
| a 〉〈 a |
)
| α 〉
| α 〉 ∑ a
| a 〉〈 a |
∑
| a 〉〈 a | = 1 (11.11)
a
( completeness ) , A
,
∑
|〈 a | α 〉| 2 = ∑ ( )
∑
〈 α | a 〉〈 a | α 〉 = 〈 α | | a 〉〈 a | | α 〉 = 〈 α | α 〉 = 1
a
a
a
, | α 〉 A () 〈A〉 α
〈A〉 α = ∑ a
a|〈 a | α 〉| 2 = ∑ a
a〈 α | a 〉〈 a | α 〉 = ∑ a
〈 α |A| a 〉〈 a | α 〉 = 〈 α |A| α 〉 (11.12)
(11.8) (11.11)
, 3 3 e i
(11.8) , e i e i·e j = δ ij
A i A i A i = e i·A
A =
3∑
i=1
A i e i = ∑ i
e i (e i·A)
| α 〉 = ∑ a
| a 〉〈 a | α 〉
,

11 214
11.1
2 | 1 〉 , | 2 〉 E 0 , λ
(
)
H = E 0 | 1 〉〈 1 | − E 0 | 2 〉〈 2 | + λ | 1 〉〈 2 | + | 2 〉〈 1 |
1. H
2. | 1 〉 , | 2 〉 , H c 1 , c 2
c 1 | 1 〉 + c 2 | 2 〉
H
11.4
, 1,2,· · ·
α i = 〈 i | α 〉 , X ij = 〈 i |X| j 〉 , i, j = 1, 2, 3, · · ·
| i 〉
⎛ ⎞
α ≡
⎜
⎝
α 1
α 2
.
⎟
⎠ , α† = (α1 ∗ α2 ∗ · · · ) , X ≡ ⎜
⎝
⎛
⎞
X 11 X 12 · · ·
X 21 X 22 · · · ⎟
⎠
.
. . ..
,
〈 α |X| β 〉 = ∑ ij
〈 α | i 〉〈 i |X| j 〉〈 j | β 〉 = α † Xβ
, | β 〉 = X| α 〉 〈 i | β 〉 = 〈 i |X| α 〉 = ∑ j
〈 i |X| j 〉〈 j | α 〉
β i = ∑ j
X ij α j = (Xα) i β = Xα
→ , → , →
, | i 〉
,
⎛ ⎞
1
0
| 1 〉 ⇒
⎜
⎝
0 ⎟
⎠ ,
.
A | a 〉 a
⎛ ⎞
0
| 2 〉 ⇒ 1
⎜
⎝
0 ⎟
⎠ , · · ·
.
A| a 〉 = a| a 〉

11 215
| a 〉 a
A
Aa = a a , A ij = 〈 i|A| j〉 (11.13)
A ∗ ij = 〈 i |A| j 〉 ∗ = 〈 j |A † | i 〉 = 〈 j |A| i 〉 = A ji
, A (11.13)
, a
det (A ij − a δ ij ) = 0
11.2 11.1
( ) ( )
1
0
| 1 〉 ⇒ , | 2 〉 ⇒
0
1
, H , H
11.5
2 A, B AB − BA = 0
〈 a ′ | (AB − BA) | a 〉 = (a ′ − a)〈 a ′ |B| a 〉 = 0
,
〈 a ′ |B| a 〉 = δ a′ a〈 a |B| a 〉
| a ′ 〉 a ′
∑
a ′ | a ′ 〉〈 a ′ |B| a 〉 = ∑ a ′ | a ′ 〉δ a′ a〈 a |B| a 〉 = | a 〉〈 a |B| a 〉
(11.11)
B| a 〉 = | a 〉〈 a |B| a 〉
, | a 〉 A , b = 〈 a |B| a 〉 B
A B | a, b 〉
A|a, b〉 = a| a, b 〉 , B| a, b 〉 = b| a, b 〉
a 1 , | a 〉
, | a, b 〉 b , 1 2 a
( ( degeneracy ) ), b , b
a b | a, b 〉 ,
∑
〈 a, b | a ′ , b ′ 〉 = δ a a ′δ b b ′ , | a, b 〉〈 a, b | = 1
a,b

11 216
, 3
A, B, C,· · ·
, 1 a, b, c, · · · , A, B, C,· · · | a, b, c, · · · 〉
1 ,
∑
〈 a, b, c, · · · | a ′ , b ′ , c ′ , · · · 〉 = δ a a ′δ b b ′δ c c ′ · · · , | a, b, c, · · · 〉〈a, b, c, · · · | = 1
a,b,c,···
11.6
, ,
,
Q, q (11.4)
Q| q 〉 = q| q 〉
f(q) g(q)
∫
∫
| α 〉 = dq f(q)| q 〉 , | β 〉 =
dq g(q)| q 〉 (11.14)
q
∫ ∫
∫
∫
〈 α | β 〉 = dq f ∗ (q) dq ′ g(q ′ )〈 q | q ′ 〉 = dq f ∗ (q)G(q) , G(q) =
dq ′ g(q ′ )〈 q | q ′ 〉
, q ′ ≠ q 〈 q | q ′ 〉 = 0 , 〈 q | q 〉
G(q) = 0 , 〈 α | β 〉 = 0 〈 α | β 〉
, 〈 q | q 〉 G(q)
, , (11.8)
〈 q | q ′ 〉 = δ(q − q ′ ) (11.15)
G(q) = g(q)
∫
〈 α | β 〉 =
dq f ∗ (q) g(q)
,
| q 〉 , | α 〉 (11.14)
∫
∫
〈q| α 〉 = dq ′ f(q ′ )〈 q | q ′ 〉 = dq ′ f(q ′ )δ(q − q ′ ) = f(q)
∫
| α 〉 =
∫
dq | q 〉f(q) =
dq | q 〉〈 q | α 〉
∫
dq | q 〉〈 q | = 1 (11.16)

11 217
| α 〉 , A , a
|〈 a | α 〉| 2 , Q q q + dq
|〈q| α 〉| 2 dq
(11.16)
∫
∫
(∫
dq |〈 q | α 〉| 2 = dq 〈 α | q 〉〈 q | α 〉 = 〈 α |
)
dq | q 〉〈 q | | α 〉 = 〈 α | α 〉 = 1
| α 〉 Q 〈Q〉 α
∫
∫
〈Q〉 α = dq q|〈 q | α 〉| 2 = dq q〈 α | q 〉〈 q | α 〉
∫
(∫ )
= dq 〈 α |Q| q 〉〈 q | α 〉 = 〈 α |Q dq | q 〉〈 q | | α 〉
= 〈 α |Q| α 〉
, (11.12)
11.7 ,
1 ˆx , ˆp
, ˆˆx ˆp
[ ˆx , ˆp ] ≡ ˆxˆp − ˆpˆx = i (11.17)
ˆx | x 〉
ˆx| x 〉 = x| x 〉 , 〈 x |x ′ 〉 = δ(x − x ′ )
ˆx , x
(11.17)
[ ˆx , ˆp 2 ] = ˆp [ ˆx , ˆp ] + [ ˆx , ˆp ]ˆp = 2i ˆp , [ ˆx , ˆp 3 ] = ˆp [ ˆx , ˆp 2 ] + [ ˆx , ˆp ]ˆp 2 = 3i ˆp 2
[ ˆx , ˆp n ] = n i ˆp n−1 ,
[ ˆx , F (ˆp) ] = iF ′ (ˆp) (11.18)
F ′ (x) F (x) , S(x) = e −iαx/
[ ˆx , S(ˆp) ] = αS(ˆp)
(
)
ˆxS(ˆp) − S(ˆp)ˆx | x 〉 = ˆxS(ˆp)| x 〉 − xS(ˆp)| x 〉 = αS(ˆp)| x 〉
ˆxS(ˆp)| x 〉 = (x + α) S(ˆp)| x 〉
S(ˆp)| x 〉 ˆx x + α
S(ˆp)| x 〉 = | x + α 〉

11 218
α → 0 S(ˆp) = 1 − iαˆp/
| x + α 〉 − | x 〉
ˆp | x 〉 = i lim
= i ∂
α→0 α
∂x | x 〉
〈 x | ˆp | x ′ 〉 = i ∂
∂x ′ 〈 x | x′ 〉 = i ∂
∂x ′ δ(x − x′ ) = − i ∂
∂x δ(x − x′ ) (11.19)
∂ x = ∂
∂x
∫
∫ ( )
〈 x | ˆp 2 | x ′ 〉 = dy 〈 x | ˆp | y 〉〈 y | ˆp | x ′ 〉 = (−i) 2 dy ∂ x δ(x − y) ∂ y δ(y − x ′ )
∫
= (−i) 2 ∂ x dy δ(x − y)∂ y δ(y − x ′ )
= (−i∂ x ) 2 δ(x − x ′ )
〈 x | ˆp n | x ′ 〉 = (−i∂ x ) n δ(x − x ′ )
F (ˆp)
〈 x |F (ˆp)| x ′ 〉 = F (−i∂ x ) δ(x − x ′ ) (11.20)
〈 x | x ′ 〉 = δ(x − x ′ )
〈 x |F (ˆp)| x ′ 〉 = F (−i∂ x ) 〈 x | x ′ 〉
〈 x ′ | x ′ (11.20)
〈 x |F (ˆp) = F (−i∂ x ) 〈 x | (11.21)
, | α 〉
〈 x |F (ˆp)| α 〉 = F (−i∂ x ) 〈 x | α 〉 (11.22)
, ˆx F (ˆx) 〈 x |F (ˆx) = 〈 x |F (x)
〈 x |F (ˆx)| x ′ 〉 = F (x)〈 x | x ′ 〉 = F (x)δ(x − x ′ ) , 〈 x |F (ˆx)| α 〉 = F (x)〈 x | α 〉 (11.23)
F (ˆx) , F (x)
, | x 〉 1
, ˆp −i∂ x
1. ψ α (x) = 〈 x | α 〉
∫
| α 〉 = dx | x 〉 ψ α (x)
x x + dx
|〈 x | α 〉| 2 dx = |ψ α (x)| 2 dx
∫
〈 α | α 〉 =
∫
dx 〈 α | x 〉〈 x | α 〉 =
dx |ψ α (x)| 2 = 1
ψ α (x) | α 〉
| x 〉

11 219
2. 〈 β |A| α 〉
∫
〈 β |A| α 〉 =
∫
∫ ∫
dx dx ′ 〈 β | x 〉〈 x |A| x ′ 〉〈 x ′ | α 〉 = dx dx ′ ψβ(x)〈 ∗ x |A| x ′ 〉ψ α (x ′ )
〈 β |A| α 〉 , 2 x x ′
〈 x |A| x ′ 〉 , A ˆx ˆp , (11.22) (11.23)
∫
∫
〈 β |F (ˆx)| α 〉 = dx〈 β | x 〉〈 x |F (ˆx)| α 〉 = dx ψβ(x)F ∗ (x)ψ α (x)
∫
∫
〈 β |F (ˆp)| α 〉 = dx〈 β | x 〉〈 x |F (ˆp)| α 〉 = dx ψβ(x)F ∗ (−i∂ x ) ψ α (x) (11.24)
3. t | t 〉 , | t 〉
( )
ˆp
2
2m + V (ˆx) | t 〉 = i d dt | t 〉 (11.25)
, m , V 〈 x |
, (11.22) (11.23)
(− 2
∂ 2 )
2m ∂x 2 + V (x) ψ(x, t) = i ∂ ψ(x, t) , ψ(x, t) = 〈 x | t 〉 (11.26)
∂t
(11.24), (11.26)
, , ˆp −i∂ x
4. ˆp | p 〉
ˆp | p 〉 = p | p 〉 (11.27)
u p (x) = 〈 x | p 〉 (11.22) 〈 x | ˆp | p 〉 = −i ∂
∂x 〈 x | p 〉 ,
u p (x)
−i ∂
∂x u p(x) = p u p (x)
u p (x) = N p exp (ipx/)
N p 〈 p | p ′ 〉 = δ(p − p ′ )
∫
∫
〈 p | p ′ 〉 = dx 〈 p | x 〉〈 x | p ′ 〉 = Np ∗ N p ′ dx exp (i(p ′ − p)x/) = |N p | 2 2π δ(p − p ′ )
, N p = 1/ √ 2π
u p (x) = 1 √
2π
exp (ipx/) (11.28)
| α 〉 | p 〉 ψ α (p) = 〈p| α 〉
∫
∫
ψ α (p) = dx 〈 p | x 〉〈 x | α 〉 = dx u p (x) ∗ ψ α (x) = √ 1 ∫
dx exp (−ipx/) ψ α (x)
2π
∫
ψ α (x) =
dp 〈 x | p 〉〈 p | α 〉 = √ 1 ∫
dp exp (ipx/) ψ α (p)
2π

11 220
5. A | a 〉
| α 〉 = ∑ a
| a 〉〈 a | α 〉
ψ α (x) = 〈 x | α 〉 = ∑ a
〈 x | a 〉〈 a | α 〉 = ∑ a
c a u a (x)
c a = 〈 a | α 〉 , u a (x) = 〈 x | a 〉
u a (x) A | a 〉 , ( eigenfunction )
∑
| a 〉〈 a | = 1
a
∑
〈 x | a 〉〈 a | x ′ 〉 = 〈 x | x ′ 〉 = δ(x − x ′ )
a
∑
u a (x) u ∗ a(x ′ ) = δ(x − x ′ )
a
3 ˆx | x 〉
ˆx| x 〉 = x| x 〉 , ˆx = (ˆx, ŷ, ẑ) , x = (x, y, z)
〈 x | x ′ 〉 = δ(x − x ′ ) ,
∫
d 3 x | x 〉〈 x | = 1
ˆp | x 〉 − i∇
11.3
ˆp | p 〉 | α 〉
〈 p | ˆx | α 〉 = i ∂〈 p | α 〉
∂p
(11.28) , (11.25)
( p
2
(i
2m + V ∂ ))
ψ(p, t) = i ∂ ψ(p, t) , ψ(p, t) = 〈 p | t 〉
∂p
∂t

12 221
12
12.1
c 1 , c 2
T
| a 〉 = c 1 |a 1 〉 + c 2 |a 2 〉
(
) ( ) ( )
T c 1 |a 1 〉 + c 2 |a 2 〉 = c ∗ 1 T |a 1 〉 + c ∗ 2 T |a 2 〉
[ ( )] ∗ [ ( )] ∗ [ ( )] ∗
〈 b | T | a 〉 = c1 〈 b | T | a 1 〉 + c2 〈 b | T | a 2 〉
[ ( ∗
〈 b | T | a 〉)]
| a 〉
[ ( ] ∗
〈 b | T | a 〉)
= 〈 b ′ | a 〉
〈 b ′ |
〈 b ′ | = 〈 b |T
(
F 〈 b |
( )
〈 b |
T | a 〉
≠
(
〈 b |T
[ ( ) ] ∗ ( )
〈 b | T | a 〉 = 〈 b |T | a 〉 (12.1)
) ( )
F | a 〉 = 〈 b |F | a 〉 ,
)
| a 〉 , T
[(
) ] [(
)( )] ∗
c 1 〈 a 1 | + c 2 〈 a 2 | T | b 〉 = c 1 〈 a 1 | + c 2 〈 a 2 | T | b 〉
= c ∗ 1
= c ∗ 1
[ ( )] ∗ [ ( )] ∗
〈 a 1 | T | b 〉 + c
∗
2 〈 a 2 | T | b 〉
( ) ( )
〈 a 1 |T | b 〉 + c ∗ 2 〈 a 2 |T | b 〉
(
) ( ) ( )
c 1 〈 a 1 | + c 2 〈 a 2 | T = c ∗ 1 〈 a 1 |T + c ∗ 2 〈 a 2 |T
, T
2 T 1 , T 2 T 1 T 2
| b 2 〉 = T 2 | b 〉
〈 a |(T 1 T 2 )| b 〉 = 〈 a | (T 1 | b 2 〉) =
〈 a |(T 1 T 2 )| b 〉 = 〈 a | (T 1 T 2 | b 〉) = (〈 a |T 1 T 2 ) | b 〉
〈 a |(T 1 T 2 )| b 〉
[
] ∗ [
∗
(〈 a |T 1 ) | b 2 〉 = (〈a|T 1 )(T 2 | b 〉)]
≠ (〈a|T1 )(T 2 | b 〉)
| b 〉 = T | a 〉
〈 b | = 〈 a |T †

12 222
T † T †
( )
(
〈 c | T | a 〉 = 〈 c | b 〉 = 〈 b | c 〉 ∗ = 〈 a |T †) ( )
| c 〉 ∗ = 〈 a | T † | c 〉
(12.2)
F 〈 c |F | a 〉 = 〈 a |F † | c 〉 ∗
T
T T † = T † T = 1
, T T , 1
| a 〉 = T |a〉 , 〈 a | = 〈a|T † , F = T F T †
, F
T 〈 a 1 | F | a 2 〉 = 〈 a 1 |F | a 2 〉 , ,
T F T †
〈 a 1 | F | a 2 〉 =
(
〈 a 1 |T †)( T F T †)( ) (
T | a 2 〉 = 〈 a 1 |T †)( ) (
T F T † T | a 2 〉 = 〈 a 1 |T †)( )
T F | a 2 〉
(12.1)
〈 a 1 | F | a 2 〉 =
[( )(
∗
〈 a 1 |T † T F | a 2 〉)]
= 〈 a1 |F | a 2 〉 ∗ (12.3)
[ x , p ]
[ x , p ] = T xT † T pT † − T pT † T xT † = T (xp − px) T † = T i T † = − i T T † = − i
c T c T † = c ∗ ,
12.2
{ | a 〉 } K A
K A | a 〉 = | a 〉 (12.4)
K A | α 〉
∑
K A | α 〉 = K A | a 〉〈 a | α 〉 = ∑ 〈 a | α 〉 ∗ K A | a 〉 = ∑ 〈 a | α 〉 ∗ | a 〉 (12.5)
a
a
a
〈 a | α 〉 K A | α 〉 = | α 〉 , 〈 a | α 〉
| α 〉 K A | α 〉 = | α 〉 K A ≠ 1 , K A
(12.4) K A 1
KA 2
∑
KA| 2 α 〉 = K A 〈 a | α 〉 ∗ | a 〉 = ∑ 〈 a | α 〉 | a 〉 = | α 〉
a
a

12 223
K † A | α 〉 = ∑ a
)
| a 〉〈 a |
(K † A | α 〉 = ∑ a
K 2 A = 1 (12.6)
[( ) ] ∗ ∑
| a 〉 〈 a |K † A
| α 〉 = | a 〉〈 a | α 〉 ∗ = K A | α 〉
a
K † A = K A (12.7)
(12.6) (12.7) K † A K A = K A K † A = K2 A = 1, K A
{| a 〉} | α 〉 ψ α (a) = 〈 a | α 〉 , | α 〉 = K A | α 〉
(12.5)
( )
ψ α (a) ≡ 〈 a | K A | α 〉 = 〈 a | α 〉 ∗ = ψα(a)
∗
K A {| a 〉}
, F = K A F K † A | a 〉 = K A| a 〉 = | a 〉 (12.3)
〈 a | F | a ′ 〉 = 〈 a | F | a ′ 〉 = 〈 a | F | a ′ 〉 ∗
a, a ′ 〈 a | F | a ′ 〉 , F
K A F K † A = ⎧
⎨
⎩
F ,
− F ,
〈 a | F | a ′ 〉
〈 a | F | a ′ 〉
{ | a 〉 } { | b 〉 } K A K B
K B | b 〉 = | b 〉
(12.8)
|α〉
∑
K B | α 〉 = K B | b 〉〈 b | α 〉 = ∑ | b 〉〈 b | α 〉 ∗ = ∑
b
b
ba
| b 〉〈 b | a 〉 ∗ 〈 a | α 〉 ∗
〈 a | b 〉 a, b
K B | α 〉 = ∑ ba
| b 〉〈 b | a 〉〈 a | α 〉 ∗ = ∑ a
| a 〉〈 a | α 〉 ∗ = K A | α 〉
K B = K A , 〈 a | b 〉 K B ≠ K A
K B | a 〉 = | a 〉 K
T
T = T K 2 = F K ,
F = T K
T K F ,
K

12 224
12.3 : 0
dp(t)
dt
= F (r(t)) , p(t) = m dr(t)
dt
r(t) = r(−t) (12.9)
( s = −t )
p(t) = m dr(t)
dt
dp(t)
dt
= − dp(s)
dt
= m ds dr(s)
= −p(s) = −p(−t) (12.10)
dt ds
= dp(s)
ds
= F (r(s)) = F (r(t))
, , r(t)
∂ψ(r, t)
i =
∂t
(− 2
2m ∇2 + V (r)
)
ψ(r, t)
ψ(r, t) ≡ ψ ∗ (r, −t)
∂ψ(r, t)
i
∂t
= i ds ∂ψ ∗ [
(r, s)
= −i ∂ψ∗ (r, s)
= i
dt ∂s
∂s
] ∗
∂ψ(r, s)
∂s
)
∗ )
∂ψ(r, t)
i =
[(− 2
∂t 2m ∇2 + V (r) ψ(r, s)]
=
(− 2
2m ∇2 + V (r) ψ(r, t)
ψ(r, t) = ψ ∗ (r, −t) t − t ψ(r, −t)
∫
∫
r av (t) = d 3 r rψ ∗ (r, t)ψ(r, t) , r av (t) = d 3 r rψ ∗ (r, t)ψ(r, t)
∫
p av (t) = −i d 3 r ψ ∗ (r, t)∇ψ(r, t) ,
∫
p av (t) = −i d 3 r ψ ∗ (r, t)∇ψ(r, t)
ψ(r, t)
r av (t) = r av (−t) (12.11)
,
∫
p av (t) = i d 3 r ψ ∗ (r, −t)∇ψ(r, −t) = − p av (−t) (12.12)
(12.11) (12.12) (12.9) (12.10)
, ψ(r, t) ψ ∗ (r, −t)
ˆx ˆp , T
T ˆxT † = ˆx , T ˆpT † = − ˆp , T T † = T † T = 1 (12.13)

12 225
T
ˆx | r 〉
ˆx| r 〉 = r| r 〉
K| r 〉 = | r 〉 , KK † = K † K = 1 (12.14)
ˆx, ˆp
〈 r | ˆx | r ′ 〉 = rδ(r − r ′ ) = , 〈 r | ˆp | r ′ 〉 = − i ∂δ(r − r′ )
∂r
=
, (12.8)
K ˆxK † = ˆx ,
K ˆpK † = − ˆp
T = K
H
, (12.13)
H = ˆp2
2m + V (ˆx)
T HT † = H , [ H , T ] = 0
H
i ∂ ∂t | t 〉 = H| t 〉
T , T i = −iT , T H = HT
−i ∂ ∂t T | t 〉 = HT | t 〉
i ∂ ∂t T | − t 〉 = HT | − t 〉
| t 〉 | t 〉 ≡ T | − t 〉
( ) [( ) ∗
ψ(r, t) ≡ 〈 r | T | − t 〉 = 〈 r |T | − t 〉]
= 〈 r | − t 〉 ∗ = ψ ∗ (r, −t)
ˆx, ˆp (12.13)
〈 t | ˆx | t 〉 =
[ (
∗
〈− t | T † ˆxT | − t 〉)]
= 〈− t | ˆx | − t 〉 ∗ = 〈− t | ˆx | − t 〉
〈 t | ˆp | t 〉 = − 〈− t | ˆp | − t 〉 (12.11), (12.12)
| α 〉 | α 〉 = T | α 〉 ,
, ψ α (r) = ψ ∗ α(r) , | p 〉
ˆpT = − T ˆp
ˆp | p 〉 = − p | p 〉 , | p 〉 ≡ T | p 〉
, | p 〉 = | − p 〉 | p 〉
ψ p (r)
ψ p (r) =
1
exp (ip·r/)
(2π)
3/2

12 226
ψ p (r) = ψ ∗ p(r) =
1
(2π) 3/2 exp (−ip·r/) = ψ −p(r)
L = ˆx× ˆp T LT † = − L , L 2 T L z
, L 2 , L z | l, m 〉
L 2 | l, m 〉 = 2 l(l + 1)| l, m 〉 , L z | l, m 〉 = m| l, m 〉
| l, m 〉 , | l, −m 〉 | l, m 〉
Y lm (θ, φ) | l, m 〉 Ylm ∗ (θ, φ)
Y ∗
lm(θ, φ) = (−1) m Y lm (θ, φ)
, | l, m 〉 = (−1) m | l, −m 〉 , L z ≠ 0
, z , L z
| α 〉 H :
H| α 〉 = E α | α 〉
T , E α
T H| α 〉 = E α T | α 〉
H , H T
HT | α 〉 = E α T | α 〉 (12.15)
, T | α 〉 H | α 〉 E α
T | α 〉 = e iθ | α 〉
ψ α (r) = ψ ∗ α(r) = e iθ ψ α (r)
e iθ/2 ψ α (r) ψ α (r) ψα(r) ∗ = ψ α (r) ,
,
12.1 | α 〉 H T
F 〈 α | F | α 〉 0
12.2 | p 〉
K p | p 〉 = | p 〉
T T = P K p P
P ˆxP † = − ˆx , P ˆpP † = − ˆp , P † = P
ψ α (p) = 〈 p | α 〉
ψ α (p) = ψα(− ∗ p)
| p 〉 ,

12 227
12.4 : 0
S ˆx× ˆp
T ˆx× ˆpT † = − ˆx× ˆp
T ˆxT † = ˆx , T ˆpT † = − ˆp , T ST † = − S (12.16)
T
ˆx, S 2 , S z | r, m 〉 :
ˆx| r, m 〉 = r| r, m 〉 , S 2 | r, m 〉 = s(s + 1)| r, m 〉 , S z | r, m 〉 = m| r, m 〉
(12.14) K
K| r, m 〉 = | r, m 〉 (12.17)
S ± ≡ S x ± iS y
〈m|S + |m ′ 〉 = δ m m′ +1√
s(s + 1) − m′ (m ′ + 1) , 〈m|S − |m ′ 〉 = δ m m′ −1√
s(s + 1) − m′ (m ′ − 1)
, 〈 m |S| m ′ 〉 S z S x = (S + + S − )/2 , S y = (S + − S − )/(2i)
(12.8)
K ˆxK † = ˆx , K ˆpK † = − ˆp
KS x K † = S x , KS y K † = − S y , KS z K † = S z
K (12.16)
, , T F T = F K , (12.16)
F K 2 = 1 F = T K, F † = K † T † , K
(12.16)
F ˆxF † = ˆx , F ˆpF † = ˆp
F S x F † = − S x , F S y F † = S y , F S z F † = − S z
F S x S z , y π
F = η e −iπSy/ , |η| 2 = 1
η ,
T
T = η e −iπSy/ K (12.18)
, K (12.17) Kis y K = is y K
K 2 = 1 KiS y = iS y K K iS y
T = η e −iπSy/ K = Kη ∗ e −iπSy/ ,
T 2 = e −2πiSy/ = (−1) 2s
T 2 = (−1) 2s , S 2 , S y

12 228
1/2
e −iπS y
= e −iπσ y/2 = cos(π/2) − iσ y sin(π/2) = −iσ y
T = − iσ y K = −Kiσ y , T 2 = cos π − iσ y sin π = − 1
T † T = T T † = 1 T † = − T , (12.2)
, | α 〉
( ) ( ) ( )
〈 α | T | α 〉 = 〈 α | T † | α 〉 = − 〈 α | T | α 〉 = 0
| α 〉 T | α 〉
| α 〉 H , H , (12.15)
T | α 〉 H 1/2 | α 〉 T | α 〉 ,
H 2 ,
n 1/2 , k S (k)
, (12.18) S y y
T 2 =
n∏
k=1
n∑
k=1
S (k)
y
e −2πiS(k) y / = (−1) n
, 1/2
( ) [( ) ] ∗
〈 r, m | α 〉 = − 〈 r, m | Kiσ y | α 〉 = − 〈 r, m |K iσ y | α 〉
= i 〈 r, m | σ y | α 〉 ∗ = i ∑ m ′ 〈 m | σ y | m ′ 〉 ∗ 〈 r, m ′ | α 〉 ∗
ψ α (r) =
(
0 −1
1 0
)
ψ ∗ α(r)
ψ 2
12.3
ψ α (5.53) n·σ ± 1
ψ α n·σ ∓ 1

13 229
13
13.1
B ,
( B = ∇×A = 0 ) ,
A ≠ 0
AB q
( ρ , θ , z ) B
{
B ez , ρ ≤ a
B =
0 , ρ > a
B = 0 D
y
a
B ≠0 a
Dρ 1 ρ 2
x
D = {( ρ , θ , z ) | ρ 1 ≤ ρ ≤ ρ 2 , − d ≤ z ≤ d } ,
ρ 1 > a
x = ρ cos θ , y = ρ sin θ
∂
∇ = e ρ
∂ρ + e θ
ρ
∂
∂θ + e ∂
z
∂z , e ρ = e x cos θ + e y sin θ , e θ = − e x sin θ + e y cos θ (13.1)
, A = A(ρ) e θ , xy ρ C
∮
dr·A(r) = 2πρA(ρ)
C
∮
∫
∫ {
πρ 2 B , ρ ≤ a
dr·A(r) = dS·(∇×A) = dS·B =
C
S
S
πa 2 B , ρ > a
ρ > a D
A =
⎧
⎪⎨
A(ρ) =
⎪⎩
Bρ
2 , ρ ≤ a
πa 2 B
2πρ ,
ρ > a
Φ
2πρ e θ , Φ = πa 2 B = (13.2)
de ρ /dθ = e θ , de θ /dθ = − e ρ
H = −
(∇ 2
− i q ) 2
2m c A 2 [
∂
= − e ρ
2m ∂ρ + e θ
ρ
[
= − 2 ∂ 2
2m ∂ρ 2 + 1 ∂
ρ
( ) ] 2
∂
∂θ − iα ∂
+ e z
∂z
∂ρ + 1 ( ) ]
2 ∂
ρ 2 ∂θ − iα + ∂2
∂z 2
α =
q Φ
2πc = Φ Φ 0
,
Φ 0 = 2πc
q

13 230
ψ(r) = ϕ(ρ, θ) Z(z) Hψ = Eψ
[
− 2 ∂ 2
2m ∂ρ 2 + 1 ∂
ρ ∂ρ + 1 ( ) ] 2 ∂
ρ 2 ∂θ − iα ϕ(ρ, θ) = E ′ ϕ(ρ, θ) ,
dZ
2m dz 2 = E zZ(z)
− 2
E = E ′ + E z
k z = √ 2mE z / 2 d 2 Z/dz 2 = − k 2 zZ Z(−d) = 0
Z = C z sin k z (z + d) ,
C z =
Z(d) = 0 sin 2dk z = 0
k z = nπ
(
2d , E z = k2 z
2m = 2 nπ
) 2
, n = 1, 2, 3, · · ·
2m 2d
Z
ϕ
e iαθ ∂ ∂θ e−iαθ ϕ(ρ, θ) =
( ∂
∂θ − iα )
ϕ(ρ, θ)
iαθ ∂2
e
∂θ 2 e−iαθ ϕ(ρ, θ) = e iαθ ∂ ∂θ e−iαθ e iαθ ∂ ∂θ e−iαθ ϕ(ρ, θ) = e iαθ ∂ ( ) ∂
∂θ e−iαθ ∂θ − iα ϕ(ρ, θ)
=
( ∂
∂θ − iα ) 2
ϕ(ρ, θ)
ϕ α (ρ, θ) = e −iαθ ϕ(ρ, θ)
ϕ(ρ, θ)
[
− 2 ∂
2
2m ∂ρ 2 + 1 ∂
ρ ∂ρ + 1 ∂ 2 ]
ρ 2 ∂θ 2 ϕ α (ρ, θ) = E ′ ϕ α (ρ, θ)
[
− 2 d
2
2m dρ 2 + 1 ρ
ϕ α (ρ, θ) = e iνθ R(ρ)
]
d
dρ − ν2
ρ 2 R(ρ) = E ′ R(ρ) (13.3)
,
ψ
ψ(ρ, θ, z) = e iαθ ϕ α (ρ, θ)Z(z) = e i(α+ν)θ R(ρ)Z(z)
r 1 ψ(ρ, θ, z) = ψ(ρ, θ + 2π, z)
ν = m z − α = m z − Φ Φ 0
,
m z =
E ′ R(ρ) ν , Φ ,
, Φ/Φ 0 =
E ′ R(ρ) B = 0 , Φ ,
Φ 0
ψ(ρ, θ, z) = e im zθ R(ρ)Z(z) ,
L z = − i ∂ ∂θ
ψ L z m z

13 231
(13.3) J ν , N ν
, , ρ 2 → ρ 1
ρ 1 , R(ρ) =
(13.3)
E ′ = 2 ν 2
2mρ 2 =
(m 2
1 2mρ 2 z − Φ ) 2
1 Φ 0
E ′ Φ/Φ 0
E ′ Φ 0
ρ = ρ 1 > a ,
ρ ≤ a
2mρ 2 1 E ′ / 2
4
1
0
0 1 2 3
Φ/Φ 0
Φ 0 D
, B ≠ 0 B = 0 D D B = ∇× A = 0
A = ∇χ D P C
C B ≠ 0
∮ ∮
∫
dr·A = dr·∇χ = dχ = χ(P ′ ) − χ(P)
C
C
χ(P ′ ) 1 P χ
∮ ∫
∫
dr·A = dS·(∇×A) = dS·B = Φ
C
S
S
χ(P ′ ) − χ(P) = Φ
Φ ≠ 0 χ(r) r 1
(
ψ χ (r) = exp − i qχ(r) )
ψ(r)
c
∇ψ χ (r) = exp
(
− i qχ(r) ) (
∇ − i q )
(
c
c ∇χ ψ(r) = exp − i qχ(r) ) (
∇ − i q )
c
c A ψ(r)
[
−
(∇ 2
− i q ) ]
2
2m c A(r) + V (r) ψ(r) = E ψ(r)
(
)
− 2
2m ∇2 + V (r) ψ χ (r) = E ψ χ (r) (13.4)
, A , ψ(r) r 1
ψ(P ′ ) = ψ(P) , ψ χ (r)
( )
(
ψ χ (P ′ ) = exp − i qχ(P′ )
ψ(P ′ ) = exp − i qΦ
c
c
) (
exp − i qχ(P)
c
)
= exp
(− 2πi Φ/Φ 0 ψ χ (P) ,
)
ψ(P)
Φ 0 = 2πc
q
(13.5)

13 232
(13.4) (13.5) , E Φ
AB (13.5) Φ Φ 0
, E Φ 0 Φ
χ(θ) = Φ 2π θ
, (13.1)
(13.2)
∇χ = e θ
ρ
dχ
dθ =
Φ
2πρ e θ
∇×∇ χ = Φ 2π
χ(P ′ ) − χ(P) = χ(θ + 2π) − χ(θ) = Φ
( dρ
−1
dρ e ρ×e θ + 1 ρ 2 e θ × de θ
dθ
)
= Φ 2π
(
− e ρ×e θ
+ e ρ×e θ
)
ρ 2 ρ 2
ρ ≠ 0 ∇×∇χ = 0 ρ = 0 ∇χ ∇×∇χ
∮
∫
dr·∇χ = dS·(∇×∇ χ)
C
S z ( ρ = 0 ) 0 , 0
S
13.2
ˆρ(r) ĵ(r)
ˆρ(r) = δ(r − ˆx) , ĵ(r) = 1 (
)
ˆp δ(r − ˆx) + δ(r − ˆx) ˆp
2m
r 3 x, y, z , ˆx , ˆp
,
[ ˆx i , ˆp j ] = i δ ij , i, j = x, y, z
r 3 ˆp
ˆx | r 〉
ˆx| r 〉 = r| r 〉
ˆp (11.21)
〈 r | ˆp = −i ∂〈 r |
(
∂r , ∂
∂
∂r ≡ ∇ = ∂x , ∂ ∂y , ∂ )
∂z
r
| α 〉 ˆρ
∫
〈 α |ˆρ(r)| α 〉 = d 3 x 〈 α | x 〉〈 x |ˆρ(r)| α 〉
δ(r − ˆx) 〈 x | , δ(r − x)
〈 x |δ(r − ˆx) = δ(r − x)〈 x |

13 233
∫
〈 α |ˆρ(r)| α 〉 = d 3 x 〈 α | x 〉δ(r − x)〈 x | α 〉 = 〈 α | r 〉〈 r | α 〉 = |ψ α (r)| 2
ψ α (r) ≡ 〈r|α〉 ,
〈 α |ĵ(r)| α 〉 = 1 ∫
2m
= 1
2m
(
)
d 3 x 〈 α | ˆp | x 〉〈 x |δ(r − ˆx)| α 〉 + 〈 α |δ(r − ˆx)| x 〉〈 x | ˆp | α 〉
(
)
〈 α | ˆp | r 〉〈 r | α 〉 + 〈 α | r 〉〈 r | ˆp | α 〉
〈 r | ˆp = − i ∂〈 r |
∂r
〈 r | ˆp | α 〉 = − i ∂〈 r | α 〉
∂r
i
〈 α |ĵ(r)| α 〉 =
2m
= − i ∂ψ α(r)
∂r
, 〈 α | ˆp | r 〉 = 〈 r | ˆp | α 〉 ∗ = i ∂ψ∗ α(r)
∂r
(
ψ α (r) ∂ψ∗ α(r)
− ψ ∗
∂r
α(r) ∂ψ )
α(r)
= [
∂r m Im ψα(r) ∗ ∂ψ ]
α(r)
∂r
(1.4) φ(r) ψ α (r) = e iθ φ(r)
, 0
(11.18)
∂F (ˆx)
[ F (ˆx) , ˆp ] = i
∂ ˆx
(13.6)
, ∂F (ˆx)/∂ˆx “F (r) x ∂F (r)/∂x r ˆx
” F (ˆx) = δ(r − ˆx)
∂δ(r − ˆx) ∂δ(r − ˆx)
[ δ(r − ˆx) , ˆp ] = i = − i
∂ ˆx
∂r
(13.7)
ˆρ ĵ
[ ˆρ(x) , ĵ(y) ] = 1 (
)
[ δ(x − ˆx) , ˆp ]δ(y − ˆx) + δ(y − ˆx) [ δ(x − ˆx) , ˆp ]
2m
( ∂δ(x − ˆx)
= − i
2m
∂x
δ(y − ˆx) + δ(y − ˆx)
)
∂δ(x − ˆx)
∂x
= − i ∂δ(x − ˆx)
δ(y − ˆx) (13.8)
m ∂x
= − i ∂δ(x − y)
δ(y − ˆx) (13.9)
m ∂x
= − i ∂δ(x − ˆx)
δ(y − x) (13.10)
m ∂x
(13.8) , ˆx

13 234
(13.7)
[ ˆρ(r) , ˆp 2 ] = ˆp · [ ˆρ(r) , ˆp ] + [ ˆρ(r) , ˆp ] · ˆp
, H
, V (ˆx) ˆρ
∂δ(r − ˆx) ∂δ(r − ˆx)
= − i ˆp · − i · ˆp
∂r
∂r
= − i ∂ (
)
∂r · ˆp δ(r − ˆx) + δ(r − ˆx) ˆp = − i 2m∇·ĵ(r)
H = ˆp2
2m + V (ˆx)
1
[ ˆρ(r) , H ] = − ∇·ĵ(r) (13.11)
i
t | t 〉
i ∂ ∂t | t 〉 = H| t 〉 , −i ∂ ∂t 〈 t | = 〈 t |H
t ρ(r, t) = 〈 t |ˆρ(r)| t 〉
∂ρ(r, t)
∂t
= 1 〈 t |[ˆρ(r) , H ]| t 〉 = − ∇·〈 t |ĵ(r)| t 〉 = − ∇·j(r, t) , j(r, t) = 〈 t |ĵ(r)| t 〉
i
F (ˆx)
∫
∫
F (ˆx) = d 3 x F (x) δ(x − ˆx) = d 3 x F (x) ˆρ(x)
F (ˆx) ĵ(y) , F (x)
ĵ(y) , (13.9)
∫
[ F (ˆx) , ĵ(y) ] = d 3 x F (x) [ ˆρ(x) , ĵ(y) ] = − i ∫
m δ(y − ˆx) d 3 ∂δ(x − y)
x F (x)
∂x
[ F (ˆx) , ĵ(y) ] = i m δ(y − ˆx) ∫
d 3 ∂F (x)
x δ(x − y)
∂x
= i (y)
δ(y − ˆx)∂F
m ∂y
(13.12)
(13.12) y (13.6)
ˆρ(r) ĵ(r)
ˆρ(r) = ∑ i
δ(r − r i ) , ĵ(r) = 1
2m
∑
)
(p i δ(r − r i ) + δ(r − r i )p i
i
(13.13)
, r i i , p i ˆ
[ ˆρ(x) , ĵ(y) ] = 1
2m
= − i m
∑(
)
[ δ(x − r i ) , p i ]δ(y − r i ) + δ(y − r i ) [ δ(x − r i ) , p i ]
i
∑
i
∂δ(x − y)
∂x
δ(y − r i ) = − i ∂δ(x − y)
ˆρ(y) (13.14)
m ∂x

13 235
ˆF = ∑ i
F (r i )
ˆF = ∑ i
∫
∫
d 3 x F (x) δ(x − r i ) = d 3 x F (x) ˆρ(x) (13.15)
, (13.12)
∫
[ ˆF , ĵ(y) ] = d 3 x F (x) [ ˆρ(x) , ĵ(y) ] = − i ∫
d 3 ∂δ(x − y)
x F (x) ˆρ(y)
m
∂x
= i m
ˆρ(y)∂F
(y)
∂y
[ ˆF † , ĵ(y) ] = i m ˆρ(y)∂F ∗ (y)
∂y
|α〉 :
(13.16)
(13.17)
T |α〉 = e iθ |α〉
T O
(
〈α|O ĵ(r)|α〉 = 〈α|T †) ) [
)] ∗
O
(T ĵ(r) |α〉 = 〈α|
(T † O ĵ(r)T |α〉
T † r i T = r i , T † p i T = − p i , (13.13)
T † ĵ(r)T = − ĵ(r)
,
(
T † OT ) †
= O (13.18)
O
T † Oĵ(r)T = (T † OT ) (T † ĵ(r)T ) = − (T † OT ) ĵ(r) = − O† ĵ(r)
〈 α |Oĵ(r)| α 〉 = − 〈 α |O† ĵ(r)| α 〉 ∗ = − 〈α| ĵ(r)O|α〉 = 1 〈 α |[ O , ĵ(r) ]| α 〉 (13.19)
2
O = 1 (13.18)
〈 α | ĵ(r)| α 〉 = 1 〈 α |[ 1 , ĵ(r) ]| α 〉 = 0
2
0 1 K|α〉 = e iθ |α〉
φ ψ(r) = e −iθ/2 φ(r)
0
Energy–weighted sum rule
(
T † F (r)ˆρ(r)T ) †
= (F ∗ (r)ˆρ(r)) † = F (r)ˆρ(r)
(13.15) ˆF ˆF † (13.18) , (13.17)
〈 α | ˆF † ĵ(r)| α 〉 = 1 2 〈 α |[ ˆF † ,
i
ĵ(r) ]| α 〉 =
2m ρ α(r) ∂F ∗ (r)
∂r

13 236
ρ α (r) ≡ 〈 α |ˆρ(r)| α 〉
H , 1
1
[ H , ˆρ(r) ] = ∇·ĵ(r)
i
〈 α | ˆF † [ H , ˆρ(r) ]| α 〉 = i 〈 α | ˆF † ∂
∂
· ĵ(r)| α 〉 = i
∂r ∂r · 〈 α | ˆF † ĵ(r)| α 〉
〈 α | ˆF † [ H , ˆρ(r) ]| α 〉 = − 2
2m
∂
∂r ·ρ α(r) ∂F ∗ (r)
∂r
∫
〈 α | ˆF † [ H , ˆF ]| α 〉 = d 3 r F (r) 〈 α | ˆF † [ H , ˆρ(r) ]| α 〉
∫
= − 2
2m
∫
= 2
2m
d 3 r F (r) ∂
∂r ·ρ α(r) ∂F ∗ (r)
∂r
d 3 r ρ α (r) ∂F ∗ (r) ∂F (r)
·
∂r ∂r
r → ∞ ρ α (r) → 0
| α 〉 H :
H| α 〉 = E α | α 〉
(13.20)
(13.20)
〈 α | ˆF † [ H , ˆF ]| α 〉 =
∑
α ′ 〈 α | ˆF † | α ′ 〉〈 α ′ |[ H , ˆF ]| α 〉
〈 α ′ |[ H , ˆF ]| α 〉 = 〈 α ′ |(H ˆF − ˆF H)| α 〉 = (E α ′ − E α )〈 α ′ | ˆF | α 〉
〈 α | ˆF
∑
† [ H , ˆF ]| α 〉 = (E α ′ − E α )〈 α | ˆF † | α ′ 〉〈 α ′ | ˆF | α 〉 = ∑ ∣
(E α ′ − E α ) ∣〈 α ′ | ˆF ∣
| α 〉
α ′ α ′
∑
∣
(E α ′ − E α ) ∣〈 α ′ | ˆF ∫
| α 〉 ∣ 2 = 2
2m
α ′
d 3 r ρ α (r) ∂F ∗ (r)
∂r
·
∂F (r)
∂r
Energy–weighted sum rule H
,
ρ α (r) | α 〉
( - )
, F (r) = z
∂F ∗ (r)
∂r
N
·
∂F (r)
∂r
= (0, 0, 1) · (0, 0, 1) = 1
∑
∣
N∑
∫
(E α ′ − E α ) ∣〈 α ′ | z i | α 〉 ∣ 2 = 2
2m
α ′ i=1
d 3 r ρ α (r) = 2
2m N
∣ 2

14 237
14
14.1 ( Levi–Civita )
i, j, k 1, 2, 3 i, j, k 3 ijk 2
, ijk i ′ j ′ k ′
ijk i ′ j ′ k ′ , ,
123 213 123 → 213 ,
123 → 321 → 231 → 213 ,
, () ijk → i ′ j ′ k ′
() 3 ε ijk
⎧
⎪⎨ 0 2
ε ijk = +1 ijk 123
⎪⎩
−1 ijk 123
ε 123 = ε 231 = ε 312 = 1 , ε 132 = ε 213 = ε 321 = −1 , ε ijk = 0
ε ijk ( Levi–Civita )
(14.1)
• ikj ijk 1 , ijk () ikj
() ε ijk = − ε ikj ε ijk = ε kij ijk
•
∑
ε kij ε kmn = δ im δ jn − δ in δ jm (14.2)
k
, i = 1, j = 2 ε kij ≠ 0 k k = 3
∑
ε k12 ε kmn = ε 312 ε 3mn
k
ε 3mn ≠ 0 m = 1, n = 2 m = 2, n = 1
⎧
∑
⎪⎨
(ε 312 ) 2 = 1 , m = 1, n = 2
ε k12 ε kmn = ε 312 ε 321 = −1 , m = 2, n = 1
k
⎪⎩
0 ,
∑
ε k12 ε kmn = δ m1 δ n2 − δ n1 δ m2
k
, (14.2) i = 1, j = 2
• A x, y, z A 1 , A 2 , A 3 A×B k
(A×B) k
= ∑ ij
ε kij A i B j (14.3)
(A×B) 1
= ε 123 A 2 B 3 + ε 132 A 3 B 2 = ε 123 A 2 B 3 − ε 123 A 3 B 2 = A 2 B 3 − A 3 B 2

14 238
14.1 (14.2), (14.3)
A×(B×C) = B(C ·A) − C(A·B) ,
(A×B)·(C ×D) = (A·C)(B·D) − (B·C)(A·D)
(
)
A×(B×C) = B iA·C − C i A·B
i
14.2
“” δ(x) :
⎧
⎨ ∞ , x = 0
∫ ∞
δ(x) =
, δ(x) dx = 1
⎩ 0 , x ≠ 0
−∞
⎧
⎨
δ(x) = lim δ ε (x) , δ ε (x) =
ε→0 ⎩
f(x)
∫ ∞
−∞
dx f(x) δ(x − a) = f(a)
∫ ∞
−∞
1/(2ε) ,
|x| ≤ ε
0 , |x| > ε
dx δ(x − a) = f(a)
x ≠ a δ(x − a) = 0 , x ≠ a f(a)
, − ∞ ∞ ,
,
δ(x) = δ(−x) (14.4)
x δ(x) = 0 (14.5)
x δ ′ (x) = − δ(x) (14.6)
δ(cx) = 1 δ(x) ,
|c|
( c ≠ 0 ) (14.7)
δ(x 2 − c 2 ) = 1 (
)
δ(x − c) + δ(x + c) ,
2|c|
( c ≠ 0 ) (14.8)
δ(g(x)) = ∑ n
1
|g ′ (x n )| δ(x − x n) , ( g(x n ) = 0 , g ′ (x n ) ≠ 0 ) (14.9)
,
, δ ′ (x) δ(x) 1 , (14.9) , g(x)
(14.6)
∫ ∞
−∞
dx f(x) x δ ′ (x) =
= −
= −
[ ] ∞
f(x)xδ(x) −
−∞
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
( ′δ(x)
dx f(x)x)
(
)
dx f ′ (x)x + f(x) δ(x)
dx f(x) δ(x)

14 239
ε > 0
δ ε (x) = 1 ∫ (
∞
[exp(ikx
dk exp(ikx − ε|k| ) = 1
− εk)
2π
2π ix − ε
−∞
= 1 π
ε
x 2 + ε 2
] ∞
0
[ exp(ikx + εk)
+
ix + ε
] 0
−∞)
, x ≠ 0 δ ε (x)
∫ b
−adx δ ε (x) = 1 π
ε→0
−−−−→ 0 , a, b
[
tan −1 x ] b
ε→+0
−−−−→ 1 (
tan −1 (∞) − tan −1 (−∞) ) = 1
ε −a π
, ε → +0 δ ε (x) δ(x) :
1
δ(x) = lim
ε→+0 2π
∫ ∞
−∞
dk exp(ikx − ε|k| ) = 1
2π
∫ ∞
−∞
dk e ikx (14.10)
1 , 3
, r = ( x, y, z ) ,
δ(r) ≡ δ(x) δ(y) δ(z)
( d 3 r = dx dy dz )
∫
∫ ∫ ∫
d 3 r δ(r) = dx δ(x) dy δ(y) dz δ(z) = 1
F (x, y, z) F (r)
∫
d 3 r F (r) δ(r − r 0 ) = F (r 0 )
(14.10)
δ(r) = 1 ∫ ∞
(2π) 3 d 3 k exp (ik·r) (14.11)
−∞
k·r = k x x + k y y + k z z , d 3 k = dk x dk y dk z
14.3
F (x)
F (x) = 1
2π
∫ ∞
F (x) =
∫ ∞
δ(x − y) = 1
2π
−∞
−∞
dy δ(x − y) F (y)
∫ ∞
−∞
dk exp (ik(x − y))
∫ ∞
dk e ikx dy e −iky F (y) = √ 1 ∫ ∞
dk e ikx ˜F (k) (14.12)
2π
−∞
−∞

14 240
˜F (k) = √ 1 ∫ ∞
dx e −ikx F (x) (14.13)
2π
−∞
˜F (k) F (x) , F (x) ˜F (k) , F (x) ˜F (k)
∫ ∞
˜F (k) = dx e −ikx F (x) , F (x) = 1 dk e ikx ˜F (k) (14.14)
−∞
2π −∞
3
∫
1 ∞
∫
˜F (k) =
d 3 1 ∞
r exp(−ik·r) F (r) , F (r) =
d 3 k exp(ik·r)
(2π) 3/2 (2π) ˜F (k)
3/2
−∞
, exp(ik·r)
∫ ∞
−∞
(
∇ 2 − m 2) F (r) = − g δ(r) (14.15)
, m, g ( m > 0 )
∇ 2 = ∇·∇ = ∂2
∂x 2 + ∂2
∂y 2 + ∂2
∂z 2
F (r)
∫ d 3 k
F (r) =
(2π) 3 exp(ik·r) ˜F (k) (14.16)
x exp(ik x x)
∂ 2 ∫ d 3 ∫
∂x 2 F (r) = k
(2π) ˜F 3 (k) ∂2
d 3
∂x 2 exp(ik·r) = − k
(2π) ˜F 3 (k) kx 2 exp(ik·r)
y, z
∫ d
∇ 2 3 k
F (r) = −
(2π) ˜F 3 (k) k 2 exp(ik·r) , k 2 = k·k = kx 2 + ky 2 + kz
2
(14.11) , (14.15)
∫ d 3 k
(2π) ˜F 3 (k) ( k 2 + m 2) ∫ d 3 k
exp(ik·r) = g
(2π) 3 exp(ik·r)
(k 2 + m 2 ) ˜F (k) = g ,
,
F (r) =
g ∫
(2π) 3 d 3 k exp(ik·r)
k 2 + m 2
3 r k θ k ( k, θ, φ )
, k·r = k r cos θ
F (r) =
g ∫ ∞ ∫ π
(2π) 3 dk k 2 dθ sin θ
g
=
(2π) 2 ir
= g
4iπ 2 r
0
∫ ∞
0
∫ ∞
−∞
dk k
0
dk exp(ikr)
∫ 2π
0
dφ
exp(ikr) − exp(−ikr)
k 2 + m 2
exp(ikr cos θ)
k 2 + m 2
k
k 2 + m 2 = g e −mr
4π r
(14.17)

14 241
(14.17) (14.15)
(14.17) (14.15)
(
∇ 2 − m 2) e −mr
r
= − 4π δ(r) , m = 0 ∇ 2 1 r
= − 4π δ(r)
14.4
z f(z)
f(z) − f(z 0 )
lim
z→z 0 z − z 0
, f(z) z = z 0
z 0 , ,
f(z) = u(x, y) + iv(x, y) ,
z = x + iy
f ′ f(z + ∆z) − f(z)
(z) = lim
∆z→0 ∆z
= lim
∆z→0
( u(x + ∆x, y + ∆y) − u(x, y)
+ i
∆x + i∆y
)
v(x + ∆x, y + ∆y) − v(x, y)
∆x + i∆y
∆y = 0 f ′ (z) = ∂u
∂x + i ∂v
∂x ,
∆x = 0 f ′ (z) = ∂v
∂y − i ∂u
∂y
∂u
∂x = ∂v
∂y ,
∂v
∂x = − ∂u
∂y
, u, v
f(z)
D , f(z) D f(z)
z = z 0 , f(z)
() 2 z a , z b 2 C z ,
t z = z(t) , ( z a z b ) C
f(z)
∫
C
dz f(z) ≡
∫ tb
t a
dt dz
dt f(z(t)) , z a = z(t a ) , z b = z(t b )
, C R z =
R e iθ , ( 0 ≤ θ ≤ π )
∫
∫ π
dz f(z) = i R dθ e iθ f(Re iθ )
C
0
O

14 242
✓
✏
D f(z) , C
D
✒
C
C
∫
C
dz f(z) = 0
∫ ∫ (
)
dz f(z) = u(x, y) + iv(x, y) (dx + idy)
∫ (
)
= u(x, y)dx − v(x, y)dy
C
∫ (
)
+ i v(x, y)dx + u(x, y)dy
C
C R , ,
∫ (
) ∫ ( ∂Q
P (x, y)dx + Q(x, y)dy =
∂x − ∂P )
dxdy
∂y
∫
C
C
∫
dz f(z) = −
C
R
( ∂v
∂x + ∂u ) ∫ ( ∂u
dxdy + i
∂y
C ∂x − ∂v )
dxdy
∂y
u, v , 0 ,
(14.17)
f(z) =
z
z 2 + m 2 eirz
f(z) z = ±im ,
f(z) , C
C [−R, R ],
R C R , z = im
ε C ε , C 1 ,
−R
O
iR
C 2
C 1
R
− im
C 2 C 1 C 2 , ,
C 1 C 2 , 2 f(z) 2
C , f(z) C 1 0
∫ ∫ R ∫
∫
dz f(z) = dx f(x) + dz f(z) + dz f(z) = 0 (14.18)
C
−R
C R C ε
C R z = Re iθ = R cos θ + iR sin θ
|e irz | = | exp(−rR sin θ + irR cos θ)| = exp(−rR sin θ)
✑
|f(z)| =
sin θ
sin θ
Re−rR Re−rR
|R 2 e i2θ + m 2 ≤
| R 2 − m 2
∣∫
∫ ∣∣∣ π
dz f(z)
∣ ≤ R dθ ∣ ∣e iRθ f(Re iθ ) ∣ R 2 ∫ π
−rR sin θ
≤
C R 0
R 2 − m 2 dθ e
0

14 243
0 ≤ θ ≤ π/2 sin θ ≥ 2θ/π
∫ π
0
dθ e −rR sin θ = 2
∫ π/2
0
dθ e −rR sin θ ≤ 2
∫ π/2
0
dθ e −2rR θ/π = π
rR
∣∫
∣∣∣ dz f(z)
∣ ≤ π R 2 (
1 − e
−rR )
C R
rR R 2 − m 2 −→ 0 ( R −→ ∞ )
, (14.18) R −→ ∞
∫ ∞
∫
dx f(x) = − dz f(z) = i
−∞
C ε
∫ 2π
0
dθ α f(im + α) =
α = εe iθ ε −→ + 0
(14.17)
∫ ∞
−∞
∫ 2π
dx f(x) = i ∫ 2π
2 e−mr dθ = iπe −mr
0
0
(
1 − e
−rR )
dθ im + α
2m − iα e−mr+irα
z = α f(z) , f(z)
f(z) =
∞∑
k=−∞
a k (z − α) k
z = α ε C , C
z = α + ε e iθ ,
0 ≤ θ ≤ 2π
∫
(z − α) k dz =
∫ 2π
(
ε e
iθ ) ∫ {
2π
k
iε e iθ dθ = i ε k+1 e i(k+1)θ dθ =
C
0
0
0 , k ≠ −1
2πi ,
k = −1
∫
f(z) dz = 2πi a −1 (14.19)
C
a −1 f(z) α
a −1 = Res(α)
(14.19) z = α D ,
, D α 1 , α 2 , · · · , α m
∫
m∑
f(z) dz = 2πi Res(α m ) (14.20)
D
k=1
f(z) l
∞∑
f(z) = a k (z − α) k , a −l ≠ 0
k=−l
(z − α) l f(z) = a −l + a −l+1 (z − α) + a −l+2 (z − α) 2 + · · · + a −1 (z − α) l−1 + · · ·

14 244
d l−1
dz l−1 (z − α)l f(z) = (l − 1)! a −1 + l! a 0 (z − α) + · · ·
Res(α) = a −1 =
1
(l − 1)! lim d l−1
z→α dz l−1 (z − α)l f(z)
f(z) = e irz /(z 2 + m 2 ) , z = im l = 1
r
∫ ∞
−∞
(14.17)
e−mr
Res(im) = lim (z − im)f(z) =
z→im 2im
dx f(x) = − dz f(z) = 2πi
∫C e−mr
ε
2im = πe−mr
m
F (r) =
∫ ∞
∫ ∞
−∞
i
4π 2 r
−∞
dx
dx
∫ ∞
e irx
x 2 + m 2 = πe−mr
m
x
x 2 + m 2 eirx = iπe −mr
−∞
dk
k
k 2 + m 2 eikr = − 1 e −mr
4π r
a
∫ ∞
−∞
√ π
dx e −ax2 = , Re a ≥ 0 (14.21)
a
a a = re iφ Re a ≥ 0
|φ| ≤ π/2 z = e iφ/2 x
∫ ∞
∫
I = dx e −ax2 = e −iφ/2 dz e −rz2
−∞
C 1
C 1 C 2
−R φ/2
C 4
C 3
R
C 1 tan(φ/2) C 1 + C 2 +
C 3 + C 4 ( R → ∞ ), e −rz2
∫
∫
dz e −rz2 + dz e −rz2 +
C 1 C 2
∫ −R
R
∫
dz e −rz2 + dz e −rz2 = 0
C 4
C 2 z = Re iθ/2 , θ φ 0
∫
J 2 = dz e −rz2 = iR
C 2
2
∫ 0
φ
dθ exp ( −rR 2 e iθ)
|J 2 | ≤ R 2
∫ |φ|
0
dθ ∣ ∣exp ( −rR 2 e iθ)∣ ∣ = R 2
∫ |φ|
0
dθ exp ( −rR 2 cos θ )

14 245
cos θ 1 − 2θ/π 0 < θ < |φ| ≤ π/2 cos θ > 1 − 2θ/π
|J 2 | ≤ R 2
∫ |φ|
0
dθ exp
( (−rR 2 1 − 2θ ))
= π [ ( (
exp − 1 − 2|φ| ) )
rR 2 − exp ( −rR 2)]
π 4R
π
1 − 2|φ|/π ≥ 0 R → ∞ J 2 → 0
∫
C 4
dz e −rz2 → 0
∫
dz e −rz2 = −
C 1
∫ −R
R
∫ ∞
−∞
dz e −rz2 =
∫ R
−R
√ √ π π
dx e −ax2 =
r e−iφ/2 =
a
dz e −rz2 →
√ π
r , R → ∞
Re a = 0 a = ik k > 0 φ = π/2
∫ ∞
−∞
∫ ∞
−∞
, a, b
∫ ∞
dx e −ikx2 =
√ π
k e−iπ/4 =
√ π
2k (1 − i )
dx cos ( kx 2) =
∫ ∞
−∞
dx sin ( √
kx 2) π
=
2k
C 4
dx e −a(x+b)2
−∞
−R
, ( R → ∞ )
∫ ∞
∫
∫
∫
∫
dx e −a(x+b)2 = dz e −az2 = − dz e −az2 − dz e −az2 − dz e −az2
−∞
C 1 C 2 C 3 C 4
C 2 z = R + iy y b i = Im b 0
I 2 =
∫
dz e −az2 = i
C 2
∫ 0
dy e −a(R+iy)2
b i
C 1 b
C 2
C 3
a(R + iy) 2 (
= a r R 2 − y 2) ( (
− 2a i Ry + i a i R 2 − y 2) )
+ 2a r Ry , a r = Re a , a i = Im a i
a r > 0 R → ∞ I 2 → 0 C 4
∫ ∞
∫
∫ ∞
√ π
dx e −a(x+b)2 = − dz e −az2 = dx e −ax2 = , Re a > 0 (14.22)
−∞
C 2 −∞
a
a r = 0 b i = 0
∫ ∞
−∞
√ π
dx e −a(x+b)2 =
a
, b i ≠ 0 e −a(x+b)2 x → ∞ x → − ∞
R

14 246
14.5
f(z) = u(z) + iv(z) z 0 f(z)
f(z) = f(z 0 ) + f ′ (z 0 )(z − z 0 ) + · · ·
, f ′ (z 0 ) = |f ′ (z 0 )|e iα , z − z 0 = re iθ
f(z) = f(z 0 ) + |f ′ (z 0 )|re i(α+θ) + · · ·
u(z) = u(z 0 ) + |f ′ (z 0 )|r cos(θ + α) + · · · , v(z) = v(z 0 ) + |f ′ (z 0 )|r sin(θ + α) + · · ·
, v sin(θ + α) = 0 z u
v(z) = u
f(z) z = z 0 = x 0 + iy 0 df/dz = 0
f(z) = f(z 0 ) + f ′′ (z 0 )
2
(z − z 0 ) 2 + · · · = f(z 0 ) + |f ′′ (z 0 )|
2
, f ′′ (z 0 ) = |f ′′ (z 0 )|e iβ , z − z 0 = re iθ
u(z) = u(z 0 ) + |f ′′ (z 0 )|
2
r 2 cos (2θ + β) + · · · , v(z) = v(z 0 ) + |f ′′ (z 0 )|
2
θ cos (2θ + β) > 0 u(z 0 )
, cos (2θ + β) < 0 ,
z = z 0 u(z) = = u(z 0 ) 2 ,
u = u(z) u(z)
z = z 0
u = u(z) , z = z 0
z 0 ( saddle point )
v(z) = u(z)
, z = z 0
2 z = z 0 ,
A
r 2 e i(2θ+β) + · · ·
r 2 sin (2θ + β) + · · ·
2θ + β = π , 2θ + β = 0 u(z)
z 0
B
,
∫
F (t) = dz e tf(z) g(z)
C 0
|t| → ∞ tf(z) = u(z) + iv(z)
e tf(z) = e u(z)( )
cos v(z) + i sin v(z)
C 0 u(z) C |t| → ∞
e u(z) , z = z 0
, |t| → ∞ cos v(z) sin v(z)

14 247
, v(z) ,
z = z 0
tf(z) = tf(z 0 ) + tf ′′ (z 0 )
2
(z − z 0 ) 2 + · · ·
tf ′′ (z 0 ) = |tf ′′ (z 0 )|e iβ ,
z − z 0 = re iθ
, θ z = z 0 C , r > 0
θ , A → B B → A θ π
C e i(2θ+β) = − 1
tf(z) = tf(z 0 ) + |tf ′′ (z 0 )|
2
r 2 e i(2θ+β) + · · · = tf(z 0 ) − |tf ′′ (z 0 )|
2
r 2 + · · ·
|t| → ∞ r − ε ≤ r ≤ ε tf(z)
2 g(z) ≈ g(z 0 )
∫
F (t) = dz e tf(z) g(z) ≈ e tf(z0) g(z 0 )
x = √ |t| r
C
∫ ε
−ε
dr e iθ e −|tf ′′ (z 0 )|r 2 /2
∫
F (t) ≈ e tf(z0)+iθ 1 X
g(z 0 ) √ dx e −|f ′′ (z 0 )|x 2 /2 , X = √ |t| ε
|t|
−X
|t| → ∞ X → ∞
∫
√
F (t) = dz e tf(z) g(z) ≈ e tf(z0)+iθ 2π
g(z 0 )
C 0
|tf ′′ (z 0 )|
(14.23)
2θ + β = π + 2nπ
√
F (t) ≈ ± i e tf(z0) 2π
g(z 0 )
tf ′′ (z 0 )
C 0 C

15 248
15
15.1
2 59
dn
H n (q) = (−1) n e q2
dq n = e−q2
[n/2]
∑
k=0
(−1) k n!
k! (n − 2k)! (2q)n−2k (15.1)
d n
dq n qf(q) = q dn f
dq n + f
ndn−1
dq n−1
(−1) n+1 e q2
d n+1
)
e−q2 = − 2 dn
dqn+1 dq n = − 2
(q dn
qe−q2 dq n + n dn−1
e−q2 dq n−1 e−q2
H n
H n+1 (q) = 2qH n (q) − 2nH n−1 (q) (15.2)
dH n
dq = d (
(−1)n
dn
eq2
dq dq n = (−1) n 2q e e−q2 q2
)
dn
dq n + e e−q2 q2 dn+1
dq n+1 e−q2
= 2qH n (q) − H n+1 (q) = 2nH n−1 (q) (15.3)
d 2 H n
dq 2
(15.3)
= d dq
(2qH n − H n+1
)
= 2H n + 2q dH n
dq − dH n+1
= 2H n + 2q dH n
dq
dq
− 2(n + 1)H n
( d
2
dq 2 − 2q d dq + 2n )
H n = 0 (15.4)
( d
dq e−q2 d dq + 2ne−q2 )
H n = 0 (15.5)
f(q) = e −q2
H n (q) = (−1) n e q2
dn
dq n e−q2
= (−1) n e q2 f (n) (q)
,
∞∑
n=0
H n (q)
n!
t n = e q2
f(q + t) =
∞ ∑
n=0
∞∑
n=0
f (n) (q)
n!
t n
f (n) (q)
(−t) n = e q2 f(q − t) = e −t2 +2tq
n!
(15.6)

15 249
(4.22)
∞∑
n=0
√
ϕ n (q) α
√ t n /2
=
n! π 1/2 e−q2
∞∑
n=0
H n (q)
n!
( ) n √ t α +t
√ =
2 )/2+ √ 2 tq
2 π 1/2 e−(q2
(15.7)
I(n, n ′ ) =
∫ ∞
−∞
dq e −q2 H n (q)H n ′(q)
n ≥ n ′ H n
∫ ∞
I(n, n ′ ) = (−1) n dq dn e
[d −q2
dq n H n ′(q) n−1 ∫
e −q2
∞
=
dq n−1 H n ′ − (−1) n dq dn−1 e −q2 dH n ′
dq n−1 dq
−∞
,
] ∞
−∞
∫ ∞
= (−1) n+1 dq dn−1 e −q2 dH n ′
dq n−1 dq
−∞
d k e −q2
dq k = q × e −q2 → 0 , q → ± ∞
H n ′ (15.1)
I(n, n ′ ) =
∫ ∞
−∞
dq e −q2 dn H n ′
dq n =
∫ ∞
−∞
∫ ∞
−∞
(4.22)
−∞
(
)
dq e −q2 dn
dq n (2q) n′ − n ′ (n ′ − 1)(2q) n′ −2 + · · ·
⎧
⎪⎨ 0 , n > n ′
=
∫ ∞
⎪⎩ 2 n n! dq e −q2 = 2 n n! √ π , n ′ = n
−∞
dq e −q2 H n (q) H n ′(q) = √ π n! 2 n δ nn ′ (15.8)
∫ ∞
−∞
, (15.6)
∞∑
∞∑
n=0 m=0
e −q2
∞∑
∞∑
n=0 m=0
t n s m
n! m!
∫ ∞
−∞
e 2ts
∞∑
dq e −q2 H n (q)H m (q) =
∞∑
n=0 m=0
t n s m
n! m!
dx ϕ n (x)ϕ n ′(x) = δ nn ′
H n (q) H m (q)
t n s m = e −t2 +2tq−s 2 +2sq
n! m!
∫ ∞
−∞
∫ ∞
t n s m (15.8)
−∞
dq exp ( 2ts − (q − s − t) 2) = √ π e 2ts (15.9)
dq e −q2 H n (q)H m (q) = √ π
∞∑ 2 n t n s n
n=0
n!

15 250
(4.22) (15.2) H n (q)
H 0 (q) = 1 , H 1 (q) = 2q , H 2 (q) = 4q 2 − 2 , H 3 (q) = 8q 3 − 12q
15.1 (15.5) H n ′(x) n n ′ n ≠ n ′
I(n, n ′ ) = 0
15.2
(15.6)
∫ ∞
−∞
dq q 2 e −q2 H n (q) H n ′(q)
〈 n ′ | x 2 | n 〉 (4.23)
15.3
1. − d dq e−ikq = ik e −ikq H n (15.1)
∫
1 ∞
√ dk (ik) n e −ikq−k2 /4 = e −q2 H n (q)
4π
−∞
e −q2 H n (q)
∞∑
2. (4.22) ϕ n (x)ϕ n (x ′ ) = δ(x − x ′ )
3. ( Mehler )
∞∑
n=0
n=0
z n
(
n! 2 n H n(q)H n (q ′ 1 2qq ′
) = √ exp z − (q 2 + q ′2 )z 2 )
1 − z
2 1 − z 2
(15.10)
z → 1 ϕ n (x)
15.2
( Legendre )
P m l
Pl m (x) = 1
2 l l! (1 − x2 m/2 dl+m
)
dx l+m (x2 − 1) l , − l ≤ m ≤ l , |x| ≤ 1 (15.11)
(x) (1 − x2 ) m/2 l − m m = 0 P l (x) P l (x)
l m ≥ 0
P m l
(x) = (1 − x 2 m/2 dm
)
dx m P l(x)
P m l (−x) = (−1) l+m P m l (x) (15.12)
P l l (x) = (2l)!
2 l l! (1 − x2 ) l/2 = (2l − 1)!! (1 − x 2 ) l/2 , P l l+1(x) = (2l + 1) x P l l (x) (15.13)

15 251
P −m
l
(x)
P 0 (x) = 1 , P 1 (x) = x , P 2 (x) = 3x2 − 1
2
, P 3 (x) = 5x3 − 3x
2
P −m
l
(x) = (−1)m
2 l l!
(1 − x 2 ) m/2 F (x) , F (x) = (x 2 −m dl−m
− 1)
dx l−m (x2 − 1) l
u = x − 1 , v = x + 1 ,
l−m
F (x) = u −m −m dl−m
∑
v
dx l−m ul v l = u −m v −m (l − m)! d k u l d l−m−k v l
k! (l − m − k)! dx k dx l−m−k
= ∑ k
k=0
(l − m)! l! l!
k! (l − m − k)! (l − k)! (k + m)! ul−k−m v k
k n = k + m
F (x) = ∑ n
(l − m)! l! l!
(n − m)! (l − n)! (l + m − n)! n! ul−n v n−m
,
d l+m
dx l+m ul v l = ∑ k
(l + m)! l! l!
k! (l + m − k)! (l − k)! (k − m)! ul−k v k−m (l + m)!
=
(l − m)! F (x)
P −m
m (l − m)!
l
(x) = (−1)
(l + m)!
1
2 l l! (1 − x2 )
m/2 dl+m
dx l+m ul v l
m (l − m)!
= (−1)
(l + m)! P l m (x) (15.14)
( x 2 − 1 ) l
d l+m
1
2 l l! dx l+m (x2 − 1) l = 1
2 l l!
= ∑ k
l∑ (−1) k l!
k! (l − k)!
k=0
d l+m
x2l−2k
dxl+m (−1) k (2l − 2k)!
2 l k! (l − k)! (l − m − 2k)! xl−m−2k
k m = 0
P l (x) =
[l/2]
∑
k=0
(−1) k (2l − 2k)!
2 l k! (l − k)! (l − 2k)! xl−2k (15.15)
x = 0
⎧
⎪⎨
Pl m (0) =
⎪⎩
(−1) (l−m)/2 (l + m)!
2 l( (l − m)/2 ) ! ( (l + m)/2 ) !
l − m
0 l − m

15 252
x = 1 − 2u
1 d l+m
2 l l! dx l+m (x2 − 1) l = (−1)l+m d l+m
2 m l! du l+m ul (u − 1) l = 1 l∑
2 m
m ≥ 0 k ≥ m , m ≤ 0 k ≥ 0
⎧
⎪⎨ u m/2 (1 − u) m/2 (l + m)!
Pl m m! (l − m)!
(x) =
⎪⎩ u −m/2 m/2
(−1)m
(1 − u)
(−m)!
x = 1 ( u = 0 )
k=0
(−1) k+m
k! (l − k)!
d l+m
ul+k
dul+m = 1 ∑ (−1) k+m (l + k)!
2 m k! (k − m)! (l − k)! uk−m (15.16)
k
(
(l + 1 + m)(l − m)
1 − u + · · ·
m + 1
)
, m ≥ 0
(
)
l(l + 1)
1 +
m − 1 u + · · · , m ≤ 0
P l (1) = 1 , P l (−1) = (−1) l , P m l (± 1) = 0 ( m ≠ 0 ) , P ′ l(1) =
l(l + 1)
2
(15.17)
1/ √ 1 − x
(
2xt − t
2 ) n
1
√ = ∑
∞ (2n)! (
2xt − t
2 ) n
1 − 2xt + t
2 (2 n n!) 2
n=0
1
√ = ∑
∞
1 − 2xt + t
2
=
n=0
∞∑
n=0 k=0
l = n + k l k
[l/2]
1
√ = ∑
∞ ∑
1 − 2xt + t
2
l=0 k=0
x m
(2n)! ∑
n
(2 n n!) 2
n∑
(2m − 1)!! (1 − x 2 ) m/2
(1 − 2xt + t 2 ) m+1/2 =
k=0
n!
k! (n − k)! (2xt)n−k (−t 2 ) k
(−1) k (2n)!
2 n+k n! k! (n − k)! xn−k t n+k
(−1) k (2l − 2k)!
2 l (l − k)! k! (l − 2k)! xl−2k t l =
∞∑
P l (x) t l (15.18)
l=0
∞∑
t l−m Pl m (x) , m ≥ 0 (15.19)
l=m
(15.18), (15.19) , x, t | t | < min ∣ ∣x ± √ x 2 − 1 ∣ ∣
x |x| ≤ 1 | t | < 1
(15.19) t
(2m − 1)!! (1 − x 2 ) m/2 (2m + 1)(x − t)
(1 − 2xt + t 2 ) = ∑
∞ (l − m)t l−m−1 P m m+3/2 l (x) (15.20)
1 − 2xt + t 2 (15.19)
∞∑
l=m
l=m
(
)
(2m + 1)(x − t)t l−m Pl m (x) − (l − m)t l−m−1 (1 − 2xt + t 2 )Pl m (x) = 0

15 253
∞∑
l=m
(
)
(l + m)Pl−1 m − (2l + 1)xPl m + (l + 1 − m)Pl+1
m t l−m = 0
(l + 1 − m)P m l+1 − (2l + 1)xP m l + (l + m)P m l−1 = 0 (15.21)
(15.14) m < 0 , (15.13) Pm−1 m = 0 l = m
m ≥ 0 Pm−1 m = 0 (15.13) , l ≥ m Pl
m(x)
(15.19) x
∞∑
(1 − x 2 ) dP l
m
dx tl−m = (2m − 1)!!(1 − x2 ) m/2 (
− mx + t(2m + 1)(1 − )
x2 )
(1 − 2xt + t 2 ) m+1/2 1 − 2xt + t 2
l=m
(15.22)
∞∑
(1 − x 2 ) dP l
m
dx tl−m =
l=m
1 − x 2
(x − t)2
= 1 −
1 − 2xt + t2 1 − 2xt + t 2
(
) (2m − 1)!!(1 − x 2 ) m/2
(2m + 1)t − mx
(1 − 2xt + t 2 ) m+1/2
− t(x − t) (2m − 1)!!(1 − x2 ) m/2 (2m + 1)(x − t)
(1 − 2xt + t 2 ) m+3/2
(15.19), (15.20)
∞∑
(1 − x 2 ) dP l
m
dx tl−m =
l=m
=
∞∑
l=m
∞∑
l=m
((
)
(2m + 1)t − mx t l−m − (x − t)(l − m)t l−m) Pl
m
(
(l + m)P m l−1 − lxP m l
)
t l−m
(1 − x 2 ) dP m l
dx = (l + m)P m l−1 − lxP m l = (l + 1)xP m l − (l + 1 − m)P m l+1 (15.23)
(15.21) m < 0
(15.22)
∞∑
(x − t) dP l
m
dx tl−m = (2m − 1)!!(1 − x2 ) m/2 (
−
(1 − 2xt + t 2 ) m+1/2
l=m
(15.19), (15.20)
∞∑
(x − t) dP l
m
dx tl−m = −
l=m
mx(x − t)
1 − x 2 ∞ ∑
l=m
t l−m P m l (x) +
)
mx(x − t) (2m + 1)(x − t)
1 − x 2 + t
1 − 2xt + t 2
∞∑
(l − m)t l−m Pl m (x)
l=m
t
x dP m l
dx
− dP l−1
m (
dx = l −
m )
1 − x 2 Pl m + mx
1 − x 2 P l−1
m

15 254
(15.23) P m l−1
(l + m) dP m l−1
dx
= lxdP m l
dx + ( m
2
1 − x 2 − l2 )
P m l
(15.23) x
( d
dx (1 − x2 ) d
)
dx −
m2
1 − x 2 + l(l + 1) Pl m (x) = 0 (15.24)
x = cos θ
( 1
sin θ
d
dθ sin θ d dθ −
)
m2
sin 2 θ + l(l + 1) Pl m (cos θ) = 0 (15.25)
P m l
(15.24)
d
dx
(
(1 − x 2 ) dP m k
dx
)
− P m k
(
d
(1 − x 2 ) dP l
m
dx
dx
) (
)
+ k(k + 1) − l(l + 1) Pl m Pk m = 0
=
=
(
) ∫ 1
l(l + 1) − k(k + 1) dx Pl
m
∫ 1
−1
[
P m l
(
dx
P m l
−1
(
d
(1 − x 2 ) dP k
m
dx
dx
(x) (1 − x 2 ) dP m k
dx
P m k
)
− P m k
− P m
k (x) (1 − x 2 ) dP m l
dx
(
d
(1 − x 2 ) dP l
m
dx
dx
] 1
−1
= 0
))
l ≠ k
F (x) = x 2 − 1
∫ 1
−1
dx P m l (x) P m k (x) = 0
∫ 1
dx (P m l (x)) 2 =
∫ 1
−1
−1
dx dl+m F l
H(x) ,
dxl+m (−1)m
H(x) ≡
(2 l l!) 2 F m dl+m F l
dx l+m
∫ 1
−1
[
dx (Pl m (x)) 2 d l+m−1 F l ] 1
=
dx l+m−1 H −
−1
∫ 1
−1
dx dl+m−1 F l
dx l+m−1
[· · · ] 2l + 1 1 0
∫ 1
−1
∫ 1
−1
dx (P m l
(x)) 2 = −
∫ 1
−1
dx dl+m−1 F l
dx l+m−1
∫ 1
∫
dx (Pl m (x)) 2 = (−1) l+m dx F l dl+m H
1
(
−1 dx l+m = (−1)l
(2 l l!) 2 dx F l dl+m
−1 dx l+m F m dl+m F l )
dx l+m
F l = x 2l − lx 2l−2 + · · ·
d l+m F l
dx l+m = 2l(2l − 1) · · · (l − m + 1)xl−m + · · · = (2l)!
(l − m)! xl−m + · · ·
dH
dx
dH
dx

15 255
d l+m (
dx l+m F m dl+m F l ) (
)
dx l+m = dl+m (2l)!
dx l+m (l − m)! xl+m + · · ·
(l + m)!
= (2l)!
(l − m)!
∫ 1
−1
dx (Pl m (x)) 2 = (2l)! (l + m)!
(2 l l!) 2 (l − m)!
(5.25)
15.4
∫ 1
−1
∫ 1
−1
dx F (x)P l (x) = 1
2 l l!
∫ 1
−1
dx (1 − x 2 ) l = 2 (l + m)!
2l + 1 (l − m)!
dx Pl m (x) Pk m (x) = 2 (l + m)!
2l + 1 (l − m)! δ lk (15.26)
∫ 1
−1
l > n 0
15.5
∫ 1
−1
( 2
dx P l(x)) ′ = l(l + 1)
dx dl F
dx l (1 − x2 ) l , F (x) n
15.6
|x| ≤ 1 F (x)
∞∑
l=0
2l + 1
2
a l = 2l + 1
2
∫ 1
−1
P l (x)P l (x ′ ) = δ(x − x ′ ) ,
F (x) =
∞∑
a l P l (x) (15.27)
l=0
dx F (x)P l (x) = 2l + 1
2 l+1 l!
∞∑
n=0
∫ 1
−1
dx dl F
dx l (1 − x2 ) l
(−1) n (4n + 1)(2n)!
2 2n+1 (n!) 2 P 2n (x) = δ(x) (15.28)
15.3
Y lm (θ, φ)
Y lm (θ, φ) = (−1) m √
2l + 1 (l − m)!
4π (l + m)! eimφ Pl m (cos θ) , − l ≤ m ≤ l (15.29)
(15.13) (15.17)
Y ll (θ, φ) = (−1)l
2 l l!
√
2l + 1
4π (2l)! sinl θ e ilφ , Y lm (0, φ) =
√
2l + 1
4π δ m0 (15.30)
C lm ≡ (−1) m √
2l + 1 (l − m)!
4π (l + m)!

15 256
(15.12), (15.14)
Y lm (π − θ, φ + π) = C lm (−1) m e imφ P m l (− cos θ) = (−1) l Y lm (θ, φ) (15.31)
Ylm(θ, ∗ φ) = C lm e −imφ m (l + m)!
(−1)
(l − m)! P −m
l
(cos θ) = (−1) m Y l −m (θ, φ) (15.32)
(15.25)
L 2 Y lm = l(l + 1)Y lm , L 2 = (−ir×∇) 2 = − 1
sin θ
x = cos θ
P m l
(x) (15.11)
(1 − x 2 ) dP m l
dx = 1
2 l l!
(
L ± = e ±iφ ± ∂ ∂θ + i cot θ ∂ )
∂φ
L ± Y lm (θ, φ) = C lm
e i(m±1)φ
√
1 − x
2
(
∓ (1 − x 2 ) d
dx − mx )
P m l
∂
∂θ sin θ ∂ ∂θ − 1
sin 2 θ
(x)
∂ 2
∂φ 2
)
(− mx(1 − x 2 m/2 dl+m
)
dx l+m + (1 − x2 m/2+1 dl+m+1
)
dx l+m+1 (x 2 − 1) l
= − mxP m l + √ 1 − x 2 P m+1
l
L + Y lm (θ, φ) = − C lm e i(m+1)φ P m+1
l
= √ (l − m)(l + m + 1) Y l m+1
L − Y lm (θ, φ) = C lm
e i(m−1)φ
√
1 − x
2
(15.24)
(
(1 − x 2 ) d
dx − mx )
F (x) =
( F (x)
dF
√ = C lme i(m−1)φ
1 − x
2 dx − m − 1 )
1 − x 2 xF
(
(1 − x 2 ) d
)
dx + (m − 1)x P m−1
l
( )
dF (m − 1)
2
dx = 1 − x 2 − l(l + 1) + m − 1 P m−1
m−1
dPl
l
+ (m − 1)x
dx
L − Y lm (θ, φ) = − C lm e i(m−1)φ( )
l(l + 1) − m(m − 1)
P m−1
l
= √ (l + m)(l − m + 1) Y l m−1
L ± Y lm (θ, φ) , Y lm
(15.26)
∫
∫
dΩ Y lm (θ, φ) Yl ∗ ′ m ′(θ, φ) = δ ll ′ δ mm ′ , dΩ · · · =
θ , φ F (θ, φ) Y lm (θ, φ)
F (θ, φ) =
∞∑
l=0 m=−l
l∑
a lm Y lm (θ, φ)
∫ π
0
dθ sin θ
∫ 2π
0
dφ · · · (15.33)

15 257
(15.33)
∫
dΩ F (θ, φ) Y ∗
lm(θ, φ) =
∞∑
l ′ =0
∑l ′ ∫
a l′ m ′
m ′ =−l ′
dΩ Y ∗
lmY ∗
l ′ m ′ = a lm
F (θ, φ) =
=
∞∑
l∑
l=0 m=−l
∫ π
0
∫
Y lm (θ, φ) dΩ ′ F (θ ′ , φ ′ ) Ylm(θ ∗ ′ , φ ′ )
∫ 2π
dθ ′ dφ ′ F (θ ′ , φ ′ ) sin θ ′
0
∞ ∑
l∑
l=0 m=−l
Y lm (θ, φ)Y ∗
lm(θ ′ , φ ′ )
sin θ ′
∞ ∑
l∑
l=0 m=−l
Y lm (θ, φ)Y ∗
lm(θ ′ , φ ′ ) = δ(θ − θ ′ ) δ(φ − φ ′ )
∞∑
l∑
l=0 m=−l
Y lm (θ, φ)Y ∗
lm(θ ′ , φ ′ ) = δ(Ω − Ω ′ ) , δ(Ω − Ω ′ ) ≡ δ(θ − θ′ ) δ(φ − φ ′ )
sin θ
, δ(Ω − Ω ′ )
∫
dΩ F (θ, φ) δ(Ω − Ω 0 ) =
∫ π
0
dθ
∫ 2π
0
dφ F (θ, φ) δ(θ − θ 0 ) δ(φ − φ 0 ) = F (θ 0 , φ 0 )
2 r 1 r 2
r 1 = r 1 ( sin θ 1 cos φ 1 , sin θ 1 sin φ 1 , cos θ 1 ) , r 2 = r 2 ( sin θ 2 cos φ 2 , sin θ 2 sin φ 2 , cos θ 2 )
r 1 r 2 α
cos α = r 1·r 2
r 1 r 2
= sin θ 1 sin θ 2 cos(φ 1 − φ 2 ) + cos θ 1 cos θ 2
c 1 = cos θ 1 , s 1 = sin θ 1 , u = cos α F (u)
1
s 1
(
∂
∂θ 1
s 1
∂
∂u dF
( ) dF
F (u) = s 1
∂θ 1 ∂θ 1 du = c 1 u − c 2
du
) ( ) (
∂
u − c1 c 2 dF (u −
F (u) =
∂θ 1 s 2 − 2u
1 du + c1 c 2 ) 2 ) d
s 2 + c 2 2 − u 2 2 F
1
du 2
s 1
∂F
= − s 1 s 2 sin(φ 1 − φ 2 ) dF
∂φ 1 du ,
∂ 2 F
∂φ 2 1
= − (u − c 1 c 2 ) dF (
du + s 2 1s 2 2 − (u − c 1 c 2 ) 2) d 2 F
du 2
L 1 = − i r 1 ×∇ 1
( 1
L 2 ∂ ∂
1F (u) = − s 1 + 1 s 1 ∂θ 1 ∂θ 1 s 2 1
∂ 2
∂φ 2 1
)
F (u) = ( u 2 − 1 ) d 2 F
du 2 + 2udF du
(15.34)

15 258
L 2 2F (u) , L 2 = − i r 2 ×∇ 2
(
(L 1 ) ± F (u) = e ±iφ 1
± ∂
)
∂
(
)
+ i cot θ 1 F = ± c 1 s 2 e ±iφ 2
− s 1 c 2 e ±iφ 1 dF
∂θ 1 ∂φ 1 du
( ∂
(L 1 + L 2 ) ± F (u) = 0 , (L 1 + L 2 ) z F (u) = − i +
∂ )
F (u) = 0
∂φ 1 ∂φ 2
F (u) = P l (u) , (15.24) L 2 1P l (u) = l(l + 1)P l (u)
P l (u) =
l∑
m=−l
a m (θ 2 , φ 2 ) Y lm (θ 1 , φ 1 )
L 2 2P l (u) = l(l + 1)P l (u) (L 2 ) z P l (u) = − (L 1 ) z P l (u)
L 2 2a m (θ 2 , φ 2 ) = l(l + 1) a m (θ 2 , φ 2 ) , (L 2 ) z a m (θ 2 , φ 2 ) = − m a m (θ 2 , φ 2 )
a m (θ 2 , φ 2 ) = c m Y l −m (θ 2 , φ 2 )
(L 1 ) + P l (u) = ∑ m
c m
√
(l − m)(l + m + 1) Yl m+1 (θ 1 , φ 1 )Y l −m (θ 2 , φ 2 )
(L 2 ) + P l (u) = ∑ m
= ∑ m
c m
√
(l + m)(l − m + 1) Ylm (θ 1 , φ 1 )Y l −m+1 (θ 2 , φ 2 )
c m+1
√
(l + m + 1)(l − m) Yl m+1 (θ 1 , φ 1 )Y l −m (θ 2 , φ 2 )
(L 1 + L 2 ) + P l (u) = 0 c m + c m+1 = 0 c m = (−1) m c 0
∑
∑
P l (u) = c 0 (−1) m Y lm (θ 1 , φ 1 )Y l −m (θ 2 , φ 2 ) = c 0 Y lm (θ 1 , φ 1 )Ylm(θ ∗
2 , φ 2 )
m
θ 2 = 0 (15.30)
√
∑
2l + 1
P l (cos θ 1 ) = c 0 Y lm (θ 1 , φ 1 )
4π δ 2l + 1
m0 = c 0
4π P l(cos θ 1 )
,
m
P l (cos α) =
4π
2l + 1
l∑
m=−l
θ 1 = θ 2 = θ, φ 1 = φ 2 = φ P l (1) = 1
15.7
1. (15.18)
l∑
m=−l
1
∞
|r 1 − r 2 | = ∑ r<
l
r>
l+1 P l (cos α) =
l=0
m
Y lm (θ 1 , φ 1 )Y ∗
lm(θ 2 , φ 2 ) (15.35)
|Y lm (θ, φ)| 2 = 2l + 1
4π
∞∑
l∑
l=0 m=−l
4π
2l + 1
r<
l
r>
l+1
Y lm (ˆr 1 )Y ∗
lm(ˆr 2 )
, |r 1 |, |r 2 | r > , r < , α r 1 r 2

15 259
2. 1. F (r 1 , r 2 )
F (r 1 , r 2 ) =
∞∑
l∑
l=0 m=−l
4π
2l + 1 F l(r 1 , r 2 ) Y lm (ˆr 1 )Y ∗
lm(ˆr 2 )
F l (r 1 , r 2 ) = θ(r 1 − r 2 ) rl 2
r l+1
1
+ θ(r 2 − r 1 ) rl 1
r2
l+1 , θ(x) =
{
1 , x > 0
0 , x < 0
∇ 2 1F (r 1 , r 2 ) = ∇ 2 2F (r 1 , r 2 ) = − 4πδ(r 1 − r 2 )
∇ 2 1 r
= − 4πδ(r)
15.4
( 1 d 2
) (
l(l + 1)
d
2
ρ + 1 −
ρ dρ2 ρ 2 R l (ρ) =
dρ 2 + 2 d
ρ dρ
)
l(l + 1)
+ 1 −
ρ 2 R l (ρ) = 0 (15.36)
15.10 ,
R l (ρ) = ρ l χ l (ρ)
1 d 2
ρ dρ 2 ρR l = 1 d 2
( d
ρ dρ 2 ρl+1 χ l = ρ l 2
2(l + 1) d
+
dρ2 ρ dρ
)
l(l + 1)
+
ρ 2 χ l
( )
d
2
2(l + 1) d
+
dρ2 ρ dρ + 1 χ l = 0 (15.37)
( ) d
2
2(l + 2) d χ
′
+
dρ2 ρ dρ + 1 l
ρ = 1 (
)
χ ′′′ 2(l + 1)
l + χ ′′ 2(l + 1)
l −
ρ
ρ
ρ 2 χ ′ l + χ ′ l
= 1 ( )
d d
2
2(l + 1) d
+
ρ dρ dρ2 ρ dρ + 1 χ l = 0
χ ′ l /ρ (15.37) l l + 1 C l
χ l+1 (ρ) = C l
ρ
( 2 ( ) l+1
dχ l
1
dρ = C d
1 d
l C l−1 χ l−1 = · · · = N l+1 χ 0
ρ dρ)
ρ dρ
N l+1 N l = (−1) l
(
R l (ρ) = ρ l − 1 ) l
d
R 0
ρ dρ
(15.36) l = 0 (ρR 0 ) ′′ + ρR 0 = 0 ρR 0 sin ρ cos ρ
, ρR 0 = sin ρ ρR 0 = − cos ρ
(
j l (ρ) = ρ l − 1 ) l
d sin ρ
ρ dρ ρ , n l(ρ) = − ρ
(− l 1 ) l
d cos ρ
ρ dρ ρ
(15.38)

15 260
j l , n l
l
(ρ) = j l (ρ) + in l (ρ) = − i ρ
(− l 1 ) l
d e iρ
ρ dρ ρ
l
(ρ) = j l (ρ) − in l (ρ) = i ρ
(− l 1 ) l
d e −iρ
ρ dρ ρ
h (1)
h (2)
(15.39)
(15.40)
, h (1)
l
(ρ) 1 , h (2)
l
(ρ) 2
(15.36) j l (ρ), n l (ρ), h (1)
l
(ρ), h (2)
l
(ρ) 2
( ∇ 2 + k 2) ψ(r) = 0 , Y lml
(6.1)
ψ(r) =
( 1
r
∞∑
l∑
l=0 m l =−l
F lml (r) Y lml (θ, φ)
d 2
)
l(l + 1)
r −
dr2 r 2 + k 2 F lml (r) = 0
ρ = kr (15.36) A lml , B lml
F lml (r) = A lml j l (kr) + B lml n l (kr)
h (1)
l
(kr), h (2)
l
(kr) ,
ψ(r) =
∞∑
l∑
l=0 m l =−l
(
)
A lml j l (kr) + B lml n l (kr) Y lml (θ, φ) (15.41)
, j l , n l , h (1)
l
, h (2)
l
f l ( ∇ 2 + k 2) f l (kr) Y lml = 0 ,
f l (kr) Y lml k → 0 (15.46), (15.47) ,
(15.41) ∇ 2 ψ = 0 , e ik·r ( ∇ 2 + k 2) ψ(r) = 0
(15.41) ( (15.50) )
l (15.38)
j 0 (ρ) = sin ρ
ρ , j sin ρ − ρ cos ρ
1(ρ) =
ρ 2 , j 2 (ρ) = (3 − ρ2 ) sin ρ − 3ρ cos ρ
ρ 3
n 0 (ρ) = − cos ρ
ρ
h (1)
0 (ρ) = − ieiρ
ρ ,
, n 1 (ρ) = −
cos ρ + ρ sin ρ
ρ 2 , n 2 (ρ) = − (3 − ρ2 ) cos ρ + 3ρ sin ρ
ρ 3
(ρ + h(1)
i)eiρ
1 (ρ) = −
ρ 2 , h (1)
2 (ρ) = − (3ρ + i(3 − ρ2 ))e iρ
ρ 3 (15.42)
j l , n l , h (1)
l
, h (2)
l
f l
(
f l (ρ) = ρ l − 1 ρ
) l
d
f 0 (ρ)
dρ
df l
(−
dρ = 1 l
d
lρl−1 f 0 + ρ
ρ dρ) l d (
− 1 ) l
d
f 0 = l dρ ρ dρ ρ f l − f l+1

15 261
df l+1
dρ
= d dρ
(15.36) d 2 f l /dρ 2
df l+1
dρ = l + 2
ρ
( l
ρ f l − df )
l
= − l
dρ ρ 2 f l + l df l
ρ dρ − d2 f l
dρ 2
(
)
df l
dρ + l(l + 2)
1 −
ρ 2 f l = l + 2
ρ
= f l − l + 2
ρ
( ) (
)
l
ρ f l(l + 2)
l − f l+1 + 1 −
ρ 2 f l
f l+1
df l
dρ = f l−1 − l + 1 f l = l ρ ρ f l − f l+1 , f l−1 + f l+1 = 2l + 1 f l (15.43)
ρ
, j −1 = − n 0 , n −1 = j 0 l = 0
1 d
ρ dρ
sin ρ
ρ
=
∞∑
n=1
sin ρ
ρ
(−1) n
(2n + 1)! 2n ρ2(n−1) ,
=
∞∑
n=0
( 1
ρ
(−1) n
(2n + 1)! ρ2n
) 2
d sin ρ
dρ ρ
=
∞∑
n=2
(−1) n
(2n + 1)! 22 n(n − 1) ρ 2(n−2)
(
− 1 ) l
d sin ρ ∑
∞
= (−1) l (−1) n
ρ dρ ρ
(2n + 1)! 2l n(n − 1) · · · (n − l + 1) ρ 2(n−l)
=
∞∑
k=0
n=l
(−1) k 2 l (k + l)!
(2k + 2l + 1)! k! ρ2k =
∞∑
k=0
(−1) k
2 k (2k + 2l + 1)!! k! ρ2k
1 d
ρ dρ
cos ρ
ρ
=
l
∞∑
n=0
(2n + 1)!! = (2n + 1)(2n − 1) · · · 3 · 1 =
j l (ρ) = ρ l
cos ρ
ρ
(−1) n
(2n)! (2n − 1) ρ2n−3 ,
∞ ∑
k=0
=
(2n + 1)!
2 n n!
(−1) k
2 k (2k + 2l + 1)!! k! ρ2k (15.44)
∞∑
n=0
( 1
ρ
n l (ρ) = − 1
ρ l+1
(−1) n
(2n)! ρ2n−1
) 2
d cos ρ
dρ ρ
∞ ∑
n=0
=
(−1) n+l C nl
(2n)!
∞∑
n=0
(−1) n
(2n)!
(2n − 1)(2n − 3) ρ2n−5
ρ 2n (15.45)
, C n0 = 1, l ≥ 1 C nl = (2n − 1)(2n − 3) · · · (2n + 1 − 2l) ,
⎧
⎪⎨
1
C nl = (2n − 1)!! × (2n − 2l − 1)!! , n ≥ l
⎪⎩ (−1) l−n (2l − 2n − 1)!! , n ≤ l

15 262
(−1)!! = 1 n = 0 n = l
ρ → 0 (15.44), (15.45)
j l (ρ) =
n l (ρ) = −
ρ l (
1 −
(2l + 1)!!
ρ 2 )
2(2l + 3) + · · ·
(
(2l − 1)!! ρ 2 )
ρ l+1 1 +
2(2l − 1) + · · ·
(15.46)
(15.47)
n l (ρ) ρ → ∞
( 1
ρ
) l
d f(ρ)
dρ ρ
= 1 ( d l f l(l + 1)
ρ l+1 −
dρl 2ρ
d l−1 f
dρ l−1 + O( ρ −2))
( l = 1, 2, 3 ), (15.39), (15.40)
(
h (1)
l(l + 1)
l
(ρ) = − i 1 + i
2ρ
(
h (2)
l(l + 1)
l
(ρ) = i 1 − i + · · ·
2ρ
) e
i(ρ−lπ/2)
+ · · ·
ρ
) e
−i(ρ−lπ/2)
ρ
(15.48)
j l (ρ) = 1 ρ
n l (ρ) = 1 ρ
(
l(l + 1)
sin(ρ − lπ/2) +
2ρ
(15.87), (15.88)
)
cos(ρ − lπ/2) + · · ·
(
l(l + 1)
− cos(ρ − lπ/2) + sin(ρ − lπ/2) + · · ·
2ρ
) (15.49)
(15.43)
dn l
j l
dρ − n dj l
l
dρ = j l n l−1 − n l j l−1 = j l+1 n l − n l+1 j l
l
(
)
ρ 2 dn l
j l
dρ − n dj l
l = ρ 2( )
j 1 n 0 − n 1 j 0 = 1
dρ
15.8 (15.43) d dρ ρl+1 f l (ρ) = ρ l+1 f l−1 (ρ) ,
d
dρ ρ−l f l (ρ) = − ρ −l f l+1 (ρ)
15.9
(15.36)
(
)
d
ρj l
dρ ρn d
l − ρn l
dρ ρj l = ρ 2 dn l
j l
dρ − n dj l
l
dρ
15.10
(15.36) R l (ρ)
R l (ρ) = ρ α
∞ ∑
n=0
a n ρ n , a 0 ≠ 0

15 263
1. (15.36) α = l, − l − 1 , a 1 = 0
(
)
(α + 2 + n) (α + 3 + n) − l(l + 1) a n+2 + a n = 0
2. n a n = 0 , n n = 2k b k = a 2k
b k = −
α = l ,
b k−1
(α + 2k) (α + 1 + 2k) − l(l + 1)
b k = (−1)k (2l + 1)!!
2 k k! (2l + 2k + 1)!! b 0
b 0 = 1/(2l + 1)!! (15.44) α = − l − 1
b 0 = − (2l − 1)!! (15.45)
15.11
|x| ≤ 1 , 15.6
e iqx =
∞∑
l=0
a l (q)P l (x) , a l (q) = 2l + 1
2 l+1 l!
∫ 1
−1
dx dl e iqx
dx l (1 − x 2 ) l
1. a l (q) = i l (2l + 1)I l (q)
2. I l (q) = j l (q)
3. (15.35)
I l (q) = 1
2 l l! ql (
1 + d2
dq 2 ) l
j 0 (q)
exp(ik·r) = 4π
∞∑
l∑
l=0 m=−l
â a
i l j l (kr)Y lm (ˆr)Y ∗
lm(ˆk) (15.50)
15.5
Γ (x) =
∫ ∞
0
dt e −t t x−1 (15.51)
, x − 1 > −1
Γ (x + 1) = − [ e −t t x ] ∫ ∞
t=∞
t=0 + x dt e −t t x−1 = xΓ (x) (15.52)
x < 0
0
Γ (x) =
Γ (x + 1)
x
=
Γ (x + 2)
x(x + 1) = · · · = Γ (x + n)
x(x + 1) · · · (x + n − 1)
(15.53)
x = 1
Γ (n + 1) = n! Γ (1) = n! (15.54)

15 264
x = 1/2
u = √ t
Γ (n + 1/2) = 1 2 · 3
2 · · · 2n − 1 Γ (1/2) =
2
(2n − 1)!!
2 n Γ (1/2)
(2n − 1)!! = 1·3 · · · (2n − 3)·(2n − 1) = (2n)!
2 n n!
Γ (1/2) =
∫ ∞
0
Γ (n + 1/2) =
dt e −t t −1/2 = 2
∫ ∞
0
du e −u2 = √ π
(2n − 1)!! √ (2n)! √ π = π (15.55)
2 n 2 2n n!
k x = − k + δ , (15.53) n = k + 1
Γ (−k + δ) =
Γ (1 + δ)
(−k + δ)(−k + 1 + δ) · · · (−1 + δ) δ
δ→0
−−−−→
, Γ (x) x 0
(15.53) y = 1 − x − n
Γ (1 − x) = Γ (y + n) = y(y + 1) · · · (y + n − 2)(y + n − 1)Γ (y)
Γ (1)
(−k)(−k + 1) · · · (−1) δ = (−1)k 1
k! δ
= (−1) n (x + n − 1)(x + n − 2) · · · (x + 1)x Γ (1 − x − n)
n Γ (x + n)
= (−1) Γ (1 − x − n)
Γ (x)
Γ (x)Γ (1 − x) = (−1) n Γ (x + n)Γ (1 − x − n) (15.56)
, Γ (x)Γ (1 − x) = π/ sin(πx)
B(p, q) =
∫ 1
2 t = x/(1 − x)
0
dx x p−1 (1 − x) q−1 =
∫ ∞
Γ t = (1 + u)s , ( 1 + u > 0 ) x = p + q
∫ ∞
u p−1 u
∫ ∞
0
t = us
0
ds e −(1+u)s s p+q−1 =
∫ ∞
du u p−1 ds e −(1+u)s s p+q−1 = Γ (p + q)
0
=
=
∫ ∞
0
∫ ∞
0
∫ ∞
0
0
dt
Γ (p + q)
(1 + u) p+q
du
∫ ∞
ds e −s s p+q−1 du u p−1 e −us
0
t p−1
(1 + t) p+q (15.57)
u p−1
= Γ (p + q) B(p, q)
(1 + u)
p+q
ds e −s s p+q−1 1 ∫ ∞
s p dt t p−1 e −t = Γ (q)Γ (p)
B(p, q) =
0
Γ (p)Γ (q)
Γ (p + q)
(15.58)

15 265
15.6
a , b 0 ,
)
(x d2
d
+ (b − x)
dx2 dx − a w(x) = 0 (15.59)
w(x)
(15.59)
k=0
w(x) = x α
∞ ∑
k=0
c k x k , c 0 ≠ 0
∞∑
∞∑
c k (α + k) (α + k − 1 + b) x α+k−1 − c k (α + k + a) x α+k = 0
1 2 1 k ′ = k − 1 k ′ k
∞∑
c k (α + k) (α + k − 1 + b) x α+k−1 =
k=0
k = −1 k ≥ 0
c 0 α (α − 1 + b) x α−1 +
∞∑
k=−1
k=0
c k+1 (α + k + 1) (α + k + b) x α+k
∞∑ (
)
c k+1 (α + k + 1) (α + k + b) − c k (α + k + a) x α+k = 0
k=0
x
c 0 α (α − 1 + b) = 0 , c k+1 (α + k + 1) (α + k + b) − c k (α + k + a) = 0 (15.60)
c 0 ≠ 0 1 α = 0 α = 1 − b
α = 0 (15.60) 2
c k+1 = 1 a + k
k + 1 b + k c k (15.61)
c 1 = a b c 0 , c 2 = 1 a + 1
2 b + 1 c 1 = 1 a(a + 1)
2 b(b + 1) c 0 , c 3 = 1 a + 2
3 b + 2 c 2 = 1 a(a + 1)(a + 2)
3! b(b + 1)(b + 2) c 0
c k = 1 (a) k
c 0 ,
k! (b) k
(a) k ≡
{
1 , k = 0
a(a + 1) · · · (a + k − 1) , k > 0
(15.62)
w(x) = c 0 M(a, b, x) , M(a, b, x) ≡
∞∑
k=0
(a) k x k
(b) k k!
(15.63)
M(a, b, x) (15.62) (15.53) n = 0, 1, 2, · · ·
⎧
⎪⎨ (−1) k n!
a = − n (−n) k = (n − k)! , k ≤ n
Γ (a + k)
, a ≠ − n (a) k = (15.64)
⎪⎩
Γ (a)
0 , k > n

15 266
α = 1 − b , (15.60) 2
c k+1 = 1 a − b + 1 + k
c k
k + 1 2 − b + k
(15.61) a , b a − b + 1 , 2 − b
w(x) = c 0 x 1−b M(a − b + 1, 2 − b, x)
, C , D
w(x) = C M(a, b, x) + D x 1−b M(a − b + 1, 2 − b, x) (15.65)
b M(a − b + 1, 2 − b, x) ,
w(x) = C M(a, b, x)
a = − n, ( n = 0, 1, 2, · · · ) (15.64) k > n (a) k = 0 M(−n, b, x)
x n
M(0, b, x) = 1 , M(−1, b, x) = 1 − x b , M(−2, b, x) = 1 − 2 b x + x2
b(b + 1)
(15.66)
, a ≠ −n M(a, b, x) (15.59) x → ∞
(x d2
dx 2 − x d )
dx − a M(a, b, x) = 0
|x dM/dx| ≫ |aM|
( d
2
x
dx 2 − d )
M(a, b, x) = 0 ,
dx
∴ M(a, b, x) = Ce x + D
M(a, b, x) Re x → ∞ e x (15.80)
b ≠ − n , ( n = 0, 1, 2, · · · ) , b = − n k ≥ n + 1 (b) k = 0
M(a, b, x) a = − m, ( m = 0, 1, 2, · · · )
m < n M(−m, −n, x) =
m∑
k=0
(−m) k x k
(−n) k k! = m!
n!
m∑
k=0
(n − k)!
(m − k)! k! xk (15.67)
, b = − n M(−m, −n, x)
n M(−n, b, x)
M(−n, b, x) =
d n
dx n xb+n−1 e −x =
n∑
k=0
n∑
k=0
(−1) k n!
k! (n − k)! (b) k
x k (15.68)
n! d k e −x d n−k x b+n−1 ∑
n
k! (n − k)! dx k dx n−k = e −x (−1) k n! d n−k x b+n−1
k! (n − k)! dx n−k
k=0
d n−k x b+n−1
dx n−k
= (b + n − 1)(b + n − 2) · · · (b + n − 1 − (n − k) + 1)x b−1+k
=
b(b + 1) · · · (b + n − 1)
b(b + 1) · · · (b + k − 1) zb−1+k = (b) n
(b) k
z b−1+k

15 267
d n
dx n xb+n−1 e −x = e −x x b−1
ν + b > 0 ,
I ν (n, b) =
∫ ∞
0
n ∑
k=0
M(−n, b, x) = ex x −b+1
(−1) k n! (b) n
x k = e −x x b−1 (b) n M(− n, b, x)
k! (n − k)! (b) k
(b) n
d n
dx n xb+n−1 e −x (15.69)
dx e −x x b+ν−1 M 2 (−n, b, x) = 1 ∫ ∞
dx x ν M(−n, b, x) dn f
(b) n dx n
f(x) = x b+n−1 e −x
I ν (n, b) = 1
(b) n
[
x ν M(−n, b, x) dn−1 f
dx n−1 ] ∞
0
0
− 1 ∫ ∞ (
) ′ d
dx x ν n−1 f
M(−n, b, x)
(b) n dx n−1
x → ∞ [· · · ] 0 x → 0 x ν+b ν + b > 0 0
,
I ν (n, b) = (−1)n
(b) n
∫ ∞
0
dx x b+n−1 e
0
−x dn
dx n xν M(−n, b, x) (15.70)
ν I ν ν = 0, ± 1 (15.68)
(
)
x ν M(−n, b, x) = (−1)n x n+ν − n(b + n − 1)x n+ν−1 + · · ·
(b) n
d n
(−1)n
M(−n, b, x) = n! ,
dxn (b) n
d n
(
)
(−1)n
xM(−n, b, x) = n! n + 1)x − n(b + n − 1)
dxn (b) n
I 0 (n, b) = n! ∫ ∞
[(b) n ] 2 dx x b+n−1 e −x = n!
n! Γ (b)
Γ (b + n) =
0
[(b) n ]
2
(b) n
I 1 (n, b) = n! ∫ ∞
[(b) n ] 2 dx x b+n e −x( )
n! Γ (b)
( )
(n + 1)x − n(b + n − 1) = b + 2n
(b) n
0
ν = −1
d n
( )
dx n x−1 M(−n, b, x) = dn 1
dx n x + (n − 1) = (−1)n n!
x n+1
I −1 (n, b) = n! ∫ ∞
dx x b−2 e −x =
(b) n
ν + b > 0 ν = 0, ± 1
∫ ∞
0
dx e −x x b+ν−1 M 2 (−n, b, x) =
0
n! Γ (b − 1)
(b) n
(
n! Γ (b + ν)
1 +
(b) n
)
ν(ν + 1)n
b
(15.71)

15 268
15.12
(4.7) ρ = q 2 , h
h = CM(−n/2, 1/2, ρ) + Dρ 1/2 M(−n/2 + 1/2, 3/2, ρ)
, n = 2k h = CM(−k, 1/2, q 2 ),
n = 2k + 1 h = CqM(−k, 3/2, q 2 ) H n (x)
H 2n (x) = (−1) n (2n)!
n!
M(−n, 1/2, x 2 n (2n + 1)!
) , H 2n+1 (x) = (−1) 2x M(−n, 3/2, x 2 )
n!
(15.59) M(a, b, x)
∫
M(a, b, x) = dz e xz v(z) (15.72)
C z ,
)
∫
)
(x d2
d
+ (b − x)
dx2 dx − a M(a, b, x) = dz v(z)
(x d2
d
+ (b − x)
C dx2 dx − a
∫ (
)
= dz v(z) xz 2 + (b − x)z − a e xz
∫
=
C
C
C
e xz
( (z
dz v(z)
2 − z ) )
d
dz + bz − a e xz
1
) (x d2
d
+ (b − x)
dx2 dx − a M(a, b, x) = [( z 2 − z ) e xz v(z) ] C
∫ ( ( )
+ dz e xz (b − 2)z − a + 1 v(z) − ( z 2 − z ) )
dv
dz
C
, [ f(z) ] C f(z) ,
[ (
z 2 − z ) e xz v(z) ] (
)
C = 0 , (b − 2)z − a + 1 v(z) − ( z 2 − z ) dv
dz = 0
(15.72) (15.59) v 0 v(z) = v 0 z a−1 (1−z) b−a−1
M(a, b, x) = v 0
∫
C
dz e xz z a−1 (1 − z) b−a−1 ,
[
e xz z a (1 − z) b−a ] C = 0 (15.73)
, a, b z z a , (1 − z) b−a
Re(b) > Re(a) > 0 , C z = 0 z = 1 ,
z = 0, 1 e xz z a (1 − z) b−a = 0 (15.73)
M(a, b, 0) = 1 (15.57)
1 = v 0
∫ 1
0
∫ 1
M(a, b, x) = v 0 dz e xz z a−1 (1 − z) b−a−1
0
dz z a−1 (1 − z) b−a−1 = v 0 B(a, b − a) = v 0
Γ (a)Γ (b − a)
Γ (b)

15 269
M(a, b, x) =
Γ (b)
Γ (a)Γ (b − a)
∫ 1
0
dz e xz z a−1 (1 − z) b−a−1 , Re(b) > Re(a) > 0
(15.57), (15.58)
Γ (b)
∞∑ x k ∫ 1
M(a, b, x) =
dz z a+k−1 (1 − z) b−a−1 =
Γ (a)Γ (b − a) k!
k=0
M(a, b, x) (15.63)
, a b
b z =
0 z = 1 C
C 0
z = re iθ0 ,
1 − z = r ′ e iθ1
0
∞∑
k=0
(a) k x k
(b) k k!
C
C 0
0 1
z
, z 0 1 θ 0 =
θ 1 = 0 z = 1 θ 0 θ 1 2π (1−z) b−a
e 2πi(b−a) = e −2πia , z = 0 θ 0 2π z a e 2πia
e −2πia × e 2πia = 1 C 0 e xz z a (1 − z) b−a
(15.73)
M(a, b, x) = v 0
∫
C 0
dz e xz z a−1 (1 − z) b−a−1
ε , z = 0 z = εe iθ , ( θ : −2π → 0 )
iε a ∫ 0
−2π
dθ e xz e iaθ (1 − z) b−a−1 = iε a ∫ 0
z = 1 z = 1 − εe iθ , ( θ : 0 → 2π )
0
0
−2π
dθ e iaθ = ε a 1 − e−2πia
a
∫ 2π
∫ 2π
− iε b−a dθ e xz z a−1 e i(b−a)θ = − iε b−a e x dθ e i(b−a)θ = ε b−a e x 1 − e−2πia
b − a
, z = re −2πi 1 − z = r ′ e 2πi ,
(∫ 1−ε ∫ ε
)
M(a, b, x) = v 0 + e −2πia dz e xz z a−1 (1 − z) b−a−1
ε
1−ε
(
+ v 0 1 − e
−2πia ) ( ε a a + εb−a e x )
b − a
(
= v 0 1 − e
−2πia ) (∫ 1−ε
dz e xz z a−1 (1 − z) b−a−1 + εa a + εb−a e x )
v 0 b > Re(a) > 0 ε → 0
x = 0
ε
(
M(a, b, x) = v 0 1 − e
−2πia ) ∫ 1
dz e xz z a−1 (1 − z) b−a−1
1 = v 0
(
1 − e
−2πia ) ∫ 1
∫
M(a, b, x) = v 0 (a, b)
C
0
0
b − a
dz z a−1 (1 − z) b−a−1 = v 0
(
1 − e
−2πia ) Γ (a)Γ (b − a)
Γ (b)
dz e xz z a−1 (1 − z) b−a−1 , v 0 (a, b) =
1 Γ (b)
1 − e −2πia Γ (a)Γ (b − a)
(15.74)

15 270
C C
, , (15.74)
a
C , x = |x|e iφ
Z R = − Re −iφ , ( R > 0 ) z
y > 0 z = (−R+iy)e −iφ e xz = e −|x|(R+iy)
, R → ∞ ,
M(a, b, x) = W 1 (a, b, x) + W 2 (a, b, x) ,
W k (a, b, x) = v 0
∫
Z R C 1
C 2
z
0 1
C k
dz e xz z a−1 (1 − z) b−a−1
2
) (x d2
d
+ (b − x)
dx2 dx − a [
W k (a, b, x) = v 0 e xz z a (1 − z) b−a ] C k
, C 1 e xz = e −|x|R
W 2 (15.59) 2
C 0 C 1
Z R z = − |z|e −iφ = − r x , r : 0 → ∞
( Re(a) > 0 )
W 1 = − v 0
x
∫ ∞
0
dr e −r ( − r x
R→∞
−−−−→ 0 C 2 , W 1
z = − |z|e −iφ−2πi = − re−2πi
x
) a−1 (
1 + r ) b−a−1 v 0 −
x x
= Γ (b)
Γ (b − a) (−x)−a g(a, a + 1 − b, − x)
g(a, b, x) = 1
Γ (a)
∫ ∞
0
∫ 0
∞
, r : ∞ → 0
( ) a−1 (
dr e −r − re−2πi
1 + r ) b−a−1
x
x
dr e −r r a−1 ( 1 − r x) − b
(15.75)
C 2 1 z = 1−r/x , ( r : ∞ → 0 ), z = 1−re 2πi /x ,
( r : 0 → ∞ ) ( b > Re(a) )
W 2 = − v 0
x
∫ 0
∞dr e x−r ( 1 − r x) a−1 ( r
x
= Γ (b)
Γ (a) ex x a−b g(b − a, 1 − a, x)
) b−a−1
−
v 0
x
∫ ∞
0
dr e x−r ( 1 − r x
) a−1
( re
2πi
x
) b−a−1
M(a, b, x) = W 1 (a, b, x) + W 2 (a, b, x) (15.76)
W 1 (a, b, x) =
Γ (b)
Γ (b − a) (−x)−a g(a, a + 1 − b, − x) (15.77)
W 2 (a, b, x) = Γ (b)
Γ (a) ex x a−b g(b − a, 1 − a, x) = e x W 1 (b − a, b, −x) (15.78)

15 271
b > Re(a) > 0 , a W 1 W 2
2
f(x) = (1 − x) −b
(1 − x) −b =
∞∑
k=0
f (k) (0)
k!
g(a, b, x) = 1
Γ (a)
=
∞∑
k=0
∞∑
k=0
(b) k
k!
x k =
(b) k
k!
∞∑
k=0
Γ (a + k)
Γ (a)
b(b + 1) · · · (b + k − 1)
k!
∫
1 ∞
x k dr e −r r a+k−1
0
1
∞
x k = ∑
k=0
(a) k (b) k
k!
x k =
∞∑
k=0
(b) k
k! xk
1
x k = 1 + ab
x + · · · (15.79)
, |x| → ∞
M(a, b, x) =
Γ (b) (
)
a(a + 1 − b)
Γ (b − a) (−x)−a 1 − + · · ·
x
+ Γ (b) (
)
Γ (a) ex x a−b (1 − a)(b − a)
1 + + · · ·
x
(15.80)
(15.79) , k = ∞ , x
, g(a, b, x)
b , z a (1 − z) b−a 269
e 2πi(b−a) e 2πia = e 2πib ≠ 1 ,
(15.73) , C
z = 1
e 2πi(b−a) e −2πi(b−a) = 1 z = 0
z a (1 − z) b−a b
0 1
, (15.76) ∼ (15.78)
15.13 C
∫
1
Γ (b)
M(a, b, x) =
(1 − e −2πia )(1 − e 2πi(b−a) ) Γ (a)Γ (b − a)
C
dz e xz z a−1 (1 − z) b−a−1
, z = 0 z = 1 ,
269 (15.76)
f n (x) =
n∑
k=0
c
)
k
, lim
xk xn( f(x) − f n (x) = 0 (15.81)
|x|→∞
f n (x) f(x) lim
n→∞ f n(x)
f(x) = f n (x) + O(|x| −n ) , f n (x) |x|
g(a, b, x)
z
∞∑
g(a, b, x) = D k ,
k=0
D k = (a) k(b) k
k!
1 (a + k − 1)(b + k − 1)
= D
xk k−1
kx

15 272
a , b ,
, k |D k | ≈ |kD k−1 /x| > |D k−1 | k = n
g n (a, b, x)
x n( )
g(a, b, x) − g n (a, b, x) =
∞∑
k=n+1
(a) k (b) k
k!
1
x k−n
x→∞
−−−−→ 0
g n (a, b, x)
lim
n→∞ f n(x) , x n ,
, , |x| ,
g(a, b, x) (15.75)
g( 1, b, −x) =
∫ ∞
0
( ) b ∫ x
∞
dr e −r = x b e x dr e−r
r + x
r b
b = 1 g n ( 1, b, −x) |1 − g n /g| x n
n ≈ x ,
n = 1 , n = 15 g n (1, b, −x) ,
x
1.0
10 −1
10 −2
x = 5
0.9
10 −3
10 −4
10 −5
10 −6
10 0 x = 10
x = 15
0 5 10 15 20
n
0.8
0.7
0.6
n = 1
n = 15
0 5 10 15
x
(15.36)
R l (ρ) = ρ l e iρ S(z) ,
z = −2iρ
z d2 S
dS
+ (2l + 2 − z) − (l + 1)S = 0 (15.82)
dz2 dz
H (1)
l
(ρ) = 2l+1 l!
(2l + 1)! ρl e iρ W 1 (l + 1, 2l + 2, −2iρ) = (−i)
H (2)
l
(ρ) = 2l+1 l!
(2l + 1)! ρl e iρ W 2 (l + 1, 2l + 2, −2iρ) = i
(15.36) 2 (15.64), (15.79)
H (1)
l+1 eiρ
l
(ρ) = (−i)
ρ
∞∑
k=0
(l + 1) k (−l) k
k!
l+1 eiρ
l+1 e−iρ
ρ
1
eiρ
= (−i)l+1
(2iρ)
k
ρ
ρ
g(l + 1, −l, 2iρ)
g(l + 1, −l, −2iρ)
l∑
k=0
(l + k)!
( ) k i
(l − k)! k! 2ρ

15 273
H (1)
l+1 + H(1)
eiρ
l−1
= (−i)l+2
ρ
∑l−1
F =
k=0
(
F +
(
= 2l + 1 l+1 eiρ
(−i)
ρ ρ
(2l + 1)!
l!
( ( (l + 1 + k)! (l − 1 + 1 i
−
(l + 1 − k)! (l − 1 − k)!)
k! 2ρ
H (1)
l+1 + H(1) l−1 = 2l + 1 l+1 eiρ
(−i)
ρ
ρ
( ) l ( ) ) l+1 i (2l + 2)! i
+
2ρ (l + 1)! 2ρ
ρ (2l − 1)!
F +
(2l + 1) i (l − 1)!
) k ∑l−1
= 2(2l + 1)
= i 2l + 1
ρ
l∑ (l + k)! 1
(l − k)! k!
k=0
( ) l−1 i
+ (2l)! ( ) ) l i
2ρ l! 2ρ
k=1
∑l−2
k=0
(l − 1 + k)! 1
(l + 1 − k)! (k − 1)!
(l + k)! 1
(l − k)! k!
( ) k i
2ρ
( ) k i
= 2l + 1 H (1)
l
2ρ ρ
(15.43) , l = 0, 1 (15.42)
(
H (1)
0 (ρ) = − ieiρ
ρ = h(1) 0 (ρ) , H(1) 1 (ρ) = − 1 + i ) e
iρ
ρ ρ = h(1) 1 (ρ)
( ) k i
2ρ
, l h (1)
l
(ρ) = H (1)
l
(ρ) h (1) (ρ)
h (1)
l
(ρ) = 2l+1 l!
(2l + 1)! ρl e iρ l+1 eiρ
l∑
( ) k
(l + k)! i
W 1 (l + 1, 2l + 2, −2iρ) = (−i) (15.83)
ρ (l − k)! k! 2ρ
h (2)
l
(ρ) = 2l+1 l!
(2l + 1)! ρl e iρ l+1 e−iρ
W 2 (l + 1, 2l + 2, −2iρ) = i
ρ
j l (ρ) , n l (ρ)
j l (ρ) = h(1) l
(ρ) + h (2)
l
(ρ)
2
k=0
l∑
k=0
(
(l + k)!
− i ) k
(15.84)
(l − k)! k! 2ρ
= 2l l!
(2l + 1)! ρl e iρ( W 1 (l + 1, 2l + 2, −2iρ) + W 2 (l + 1, 2l + 2, −2iρ)
= 2l l!
(2l + 1)! ρl e iρ M(l + 1, 2l + 2, −2iρ) (15.85)
)
n l (ρ) = h(1) l
(ρ) − h (2)
l
(ρ)
2i
2 l l!
= − i
(2l + 1)! ρl e iρ( )
W 1 (l + 1, 2l + 2, −2iρ) − W 2 (l + 1, 2l + 2, −2iρ)
(15.83), (15.84)
(15.86)
j l (ρ) =
sin(ρ − lπ/2)
ρ
+
cos(ρ − lπ/2)
ρ 2
[l/2]
∑
k=0
(l + 2k)! (−1) k
(l − 2k)! (2k)! (2ρ) 2k
[(l−1)/2]
∑
k=0
(l + 2k + 1)! (−1) k
(l − 2k − 1)! (2k + 1)! (2ρ) 2k (15.87)

15 274
n l (ρ) = −
+
cos(ρ − lπ/2)
ρ
sin(ρ − lπ/2)
ρ 2
[l/2]
∑
k=0
[(l−1)/2]
∑
k=0
(l + 2k)! (−1) k
(l − 2k)! (2k)! (2ρ) 2k
(l + 2k + 1)! (−1) k
(l − 2k − 1)! (2k + 1)! (2ρ) 2k (15.88)
15.14
(15.82) 2 M(a, b, z) z 1−b M(a − b + 1, 2 − b, z)
a = l + 1 , b = 2l + 2 = 2a M(a, b, z) j l (ρ) z 1−b M(a − b + 1, 2 − b, z)
h (1)
l
15.7
L α n(x) ≡ ex x −α d n
n! dx n e−x x n+α = (α + 1) n
M(−n, α + 1, x) (15.89)
n!
( Laguerre ) , α = 0 , L n (x)
III ,
(L k n) Schiff = n! (−1) k L k n−k , (L k n) Messiah = (n + k)! L k n
(15.89) , (15.68)
L α n(x) =
n∑
m=0
(−1) m C nm (α)
x m (15.90)
m! (n − m)!
C nm (α) = (α + 1) n
(α + 1) m
=
{
1 , m = n
(α + n)(α + n − 1) · · · (α + m + 1) , 0 ≤ m ≤ n − 1
k = 0, 1, 2, · · · C nm (k) = (n + k)!/(m + k)!
L k n(x) =
n∑
m=0
m − k m
(−1) m n+k
(n + k)!
∑ (−1) m (n + k)!
m! (n − m)! (m + k)! xm , L n+k (x) =
m! (n + k − m)! m! xm
d k
n+k
dx k L ∑
n+k(x) =
m=k
d k
dx k L n+k(x) = (−1) k
n ∑
m=0
m=0
(−1) m (n + k)! m!
m! (n + k − m)! m! (m − k)! xm−k
(−1) m (n + k)!
m! (n − m)! (m + k)! xm = (−1) k L k n(x) (15.91)

15 275
| t | < 1
f(x, t) =
1
(
(1 − t) α+1 exp −
xt )
=
1 − t
∞∑ (−x) m
t m (1 − t) −m−α−1
m!
m=0
(1 − t) −m−α−1
(1 − t) −m−α−1 = 1 +
f(x, t) =
∞∑
∞∑
m=0 k=0
∞∑
k=1
1
k! (α + m + 1)(α + m + 2) · · · (α + m + k) tk =
(−x) m C m+k,m (α)
m! k!
k = n − m
L α n+1
(n + 1)L α n+1 = ex x −α
t m+k =
1
(
(1 − t) α+1 exp −
xt )
=
1 − t
n!
∞∑
n∑
n=0 m=0
(−x) m C nm (α)
m! (n − m)!
∞∑
k=0
t n =
C m+k,m (α)
k!
∞∑
L α n(x) t n
n=0
∞∑
L α n(x) t n , | t | < 1 (15.92)
n=0
d n+1
dx n+1 e−x x n+1+α = ex x −α d n (
n! dx n (n + 1 + α)e −x x n+α − e −x x n+1+α)
f(x) = e −x x n+α
n − 1
= (n + 1 + α)L α n − ex x −α
d n
dx n e−x x n+1+α = dn xf(x)
dx n = x dn f(x)
dx n + n dn−1 f(x)
dx n−1
(n + 1)L α n+1 = (n + 1 + α − x)L α n − ex x −α
n!
d n−1
d n
dx n e−x x n+1+α
(n − 1)! dx n−1 e−x x n+α
d
dx e−x x n+α = (n + α)x n+α−1 e −x − e −x x n+α
d n−1
dx n−1 e−x x n+α = (n + α) dn−1
dx n−1 e−x x n+α−1 − dn
dx n e−x x n+α
t k
e x x −α d n−1
(n − 1)! dx n−1 e−x x n+α = (n + α)L α n−1 − nL α n (15.93)
(n + 1)L α n+1 = (2n + α + 1 − x)L α n − (n + α)L α n−1 (15.94)
,
xL α+1
n
= ex x −α
n!
d n
dx n e−x x n+α+1 = ex x −α
n!
(x dn
dx n e−x x n+α + n dn−1
dx n−1 e−x x n+α )

15 276
(15.93)
xL α+1
n = (x − n)L α n + (n + α)L α n−1 = (n + α + 1)L α n − (n + 1)L α n+1 (15.95)
L α n
dL α n
dx = Lα n − α x Lα n + ex x −α d n+1
n! dx n+1 e−x x n+α
d n+1
dx n+1 e−x x n+1+α = x dn+1
dx n+1 e−x x n+α + (n + 1) dn
dx n e−x x n+α
L α n+1 = ex x −α
(n + 1)! x dn+1
dx n+1 e−x x n+α + L α n
x dLα n
dx = (x − α)Lα n + (n + 1)(L α n+1 − L α n) (15.96)
= n L α n − (n + α)L α n−1 (15.97)
(15.97)
x(L α n) ′′ + (L α n) ′ = n(L α n) ′ − (n + α)(L α n−1) ′
(15.96)
x(L α n−1) ′ = nL α n + (x − α − n)L α n−1 = nL α n + x − α − n (nL α n − x(L α
n + α
n) ′)
(
)
x d2
d
+ (α + 1 − x)
dx2 dx + n L α n(x) = 0 (15.98)
( d
dx e−x x α+1 d
dx + n e−x x α )
L α n(x) = 0 (15.99)
(15.98) a = −n, b = α + 1
α + 1 > 0
I nm ≡
∫ ∞
0
dx e −x x α L α n(x)L α m(x) = 1 n!
I nm = (−1)n
n!
∫ ∞
0
dx e −x x
∫ ∞
0
n+α dn
dx L α m(x) dn
dx n e−x x n+α
dx n Lα m(x)
n ≥ m L α m(x) m , n > m n
0 (15.90)
L α n(x) = (−1)n x n + (n − 1) , ∴ I nn = 1 n!
n!
(15.99) L α m
I nm = 0
∫ ∞
0
dx e −x x α L α n(x)L α m(x) =
∫ ∞
0
dx e −x x n+α =
Γ (n + α + 1)
n!
δ nm
Γ (n + α + 1)
n!
n, m n ≠ m

15 277
15.8
w(q)
d 2 w
dq 2
d 2 w
− qw(q) = 0 (15.100)
dq2 w(q) =
∫ ∞
∫ ∞
= − dk e ikq k 2 ϕ(k) , qw(q) =
−∞
−∞
dk e ikq ϕ(k)
∫ ∞
−∞
dk ϕ(k)(− i) ∂
∂k eikq = i
k → ± ∞ ϕ(k) → 0 (15.100)
∫ ∞
(
dk e ikq k 2 + i d )
dϕ
ϕ(k) = 0 , ∴
dk
dk = ik2 ϕ
−∞
∫ ∞
−∞
ikq dϕ(k)
dk e
dk
ϕ(k) ∝ e ik3 /3 C
w(q) = CAi(q) , Ai(q) ≡ 1
2π
∫ ∞
−∞
( ) ik
3
dk exp
3 + ikq
Ai(q) (Airy) z = aq
( )
( d
2
d
dq 2 − q Ai(aq) = a 2 2
dz 2 − z )
a 3 Ai(z)
(15.101)
a = e ±2πi/3 a 3 = 1 Ai(e ±2πi/3 q) (15.100)
(
)
Bi(q) = e iπ/6 Ai(e 2πi/3 q) + e −iπ/6 Ai(e −2πi/3 q) = 2Re e iπ/6 Ai(e 2πi/3 q)
, (15.100) C, D w(q) = CAi(q) + DBi(q)
Ai(q) Bi(q) q → ∞ Ai(q) , Bi(q)
, q → − ∞ |q| −1/4 a n Ai(a n ) = 0
(15.102) (15.103) , |q| 2
, q → − ∞ (15.102) , a n
2
3 |a n| 3/2 + π [ 3π
4 ≈ nπ , ∴ a n ≈ −
2
(
n − 1 4)] 2/3
, n = 1, 2, 3, · · ·
1.0
Bi(q)
0.5
a 5
a 4
a 3
−5
a 2
a 1
−1
O
Ai(q)
1 2
q
−0.5

15 278
(15.101) k = √ |q| z
Ai(q) =
√
|q|
2π
∫ ∞
−∞
dz e λf ±(z) , λ = |q| 3/2 , f ± (z) = iz3
3 ± iz
q > 0 f + (z), q < 0 f − (z) z
z = Re iθ
Re f ± (z) = −
(sin R3
3θ ± 3 )
3
R 2 sin θ
sin 3θ > 0,
0 < θ < π 3 , 2π
3 < θ < π , 4π
3 < θ < 5π 3
R → ∞ e λf±(z) → 0 0 < θ < π/3
2π/3 < θ < π 2 ,
C C D 1 , D 2
( ∫ R ∫ ∫ )
− + + dz e
−R C D 1
∫D λf ±(z) = 0
2
−R
D 2
C
D 1
R
R → ∞ D 1 D 2 0
∫ ∞
∫
dz e λf ±(z) = dz e λf ±(z)
−∞
C
, (15.101) f ± (z)
, (14.23) |q| → ∞ Ai(q)
q > 0
f +(z ′ 0 ) = i ( z0 2 + 1 ) = 0
z 0 z 0 = ± i z = x + iy
i
C 1
z 0 = i
( )
x
2
Imf + (z) = x
3 − y2 + 1 = = Imf + (z 0 ) = 0
C −
C ′
−1
1
C +
z 0 = i
√
x
2
x = 0 y =
3 + 1
f + (z) z = z 0 ( z = z 0 + re iθ )
f + (z) = f + (z 0 ) + f +(z ′′ 0 )
(z − z 0 ) 2 + · · · = f + (z 0 ) + iz 0 (z − z 0 ) 2 + · · · = − 2 2
3 − r2 e 2iθ + · · ·
e 2iθ = 1 θ = 0, π x = 0 θ = ± π/2 x = 0
, y = √ x 2 /3 + 1 z = i x θ = 0, π
C 1 C 1
Ai(q) =
√ q
2π
∫
C 1
dz e λf+(z)

15 279
λ → ∞ r = 0 − ε ≤ r ≤ ε
√ ∫ q ε
Ai(q) ≈ dr dz √ ∫ q ε
(
2π dr eλf +(z) = dr exp − 2 )
2π
3 λ − λr2 + · · ·
−ε
, θ = 0 dz/dr = e iθ = 1
√ ∫ q
ε
Ai(q) ≈
2π e−2λ/3 dr e −λr2 =
λ → ∞ ε √ λ → ∞
Ai(q) ≈
−ε
−ε
√ q
2π e−2λ/3 1
(
1
2 √ π q exp − 2 )
1/4 3 q3/2
√
∫ ε λ
√ ds e −s2
λ
−ε √ λ
q < 0 f −(z ′ 0 ) = 0 z 0 z 0 = ± 1
( )
x
2
z 0 = ± 1 x
3 − y2 − 1 = = Im f − (z 0 ) = ∓ 2 3
C ± C + + C −
√ ∫ |q|
Ai(q) = I + + I − , I ± = dz e λf−(z)
2π C ±
f − (z) z 0 = ± 1 ( z − z 0 = re iθ± )
f − (z) = f − (z 0 ) + f −(z ′′ 0 )
(z − z 0 ) 2 + · · · = ∓ 2i
2
3 − r2 e 2iθ±∓iπ/2 + · · ·
e 2iθ±∓iπ/2 = 1 , θ ± = ± π/4 dz = dre iθ±
q > 0
( (
1
2
I ± ≈
2 √ π |q| exp ∓ i
1/4 3 |q|3/2 − π ))
4
Ai(q) ≈
(
1 2
√ cos π |q|
1/4 3 |q|3/2 − π )
(
1 2
= √
4
sin π |q|
1/4 3 |q|3/2 + π )
4
⎧
⎪⎨
Ai(q) →
⎪⎩
(
1
2 √ π q exp − 2 )
1/4 3 q3/2 , q → ∞
1
√ π |q|
1/4 sin ( 2
3 |q|3/2 + π 4
)
, q → − ∞
(15.102)
Bi(q)
Ai(e 2πi/3 q) = 1
2π
∫ ∞
( ) ik
3
∫ ( )
dk exp
3 + iqe2πi/3 k = e−2πi/3
ik
3
dk exp
2π C 3 + iqk ′
√ ∫ |q|
= e −2πi/3 dz e λf±(z)
2π C ′
−∞
, k ′ = e 2πi/3 k k ′ k 2π/3
C ′ q < 0 , C −
Ai(e 2πi/3 q) = − e −2πi/3 I − ≈
(i e−iπ/6
2 √ π |q| exp 2 1/4 3 |q|3/2 + iπ )
4

15 280
q > 0 , C ′ C 1 −π/3 < θ < 0 0
q > 0 z 0 = − i
√
x
2
x = 0 y = −
3 + 1
z = z 0
f + (z) = f + (z 0 ) + f +(z ′′ 0 )
(z − z 0 ) 2 + · · · = 2 2
3 + r2 e 2iθ + · · ·
x = 0 θ = ± π/2 f + (z) = 2/3 − r 2 + · · · x = 0, ,
C ′ Re z < 0 C ′ Im z < 0
Ai(e 2πi/3 q) ≈ e −2πi/3 √ q
2π
∫ ε
−ε
( )
dr e iπ/2 e 2λ/3−λr2 ≈ e−πi/6 2
2 √ π q exp 1/4 3 q3/2
(
)
Bi(q) = 2Re e iπ/6 Ai(e 2πi/3 q) →
⎧
⎪⎨
⎪⎩
( )
1 2
√ exp π q
1/4 3 q3/2 , q → ∞
(
1 2
√ cos π |q|
1/4 3 |q|3/2 + π )
, q → − ∞
4
(15.103)
15.9
z
∞∑
k=0
H k (q)
k!
∞∑
k=0
C
H k (q)
z k = e −z2 +2qz
k!
∫
∫
dz z k−n−1 = dz e −z2 +2qz 1
C 1
∫
{
2πi , n = k
dz z k−n−1 =
C
0 , n ≠ k
H n (q) = n!
2πi
∫
C
∫
dz e −z2 +2qz 1 n!
=
zn+1 2πi
n → ∞ H n (q)
C
C
z n+1
(
dz exp −z 2 + 2qz − (n + 1) log z
ω(n + 1/2) , q 2 =
2n + 1 q = √ 2n + 1 q ′ , z = √ 2n + 1 z ′
H n ( √ 2n + 1 q ) = 1 n!
dz exp
((2n + 1)f(z) −
2πi (2n + 1)
∫C
1 )
n/2 2 log z
)
f(z) = −z 2 + 2qz − 1 2 log z

15 281
, q ′ , z ′ q, z n → ∞
z = Re iθ
Re f(z) = −R 2 cos 2θ + 2qR cos θ − 1 2 log r
cos 2θ > 0 R → ∞ e (2n+1)f(z)
→ 0 ,
− π/4 < θ < π/4 , 3π/4 < θ < 5π/4
f(z) z f ′ (z) = −2z + 2q − 1/(2z) = 0
z 2 ± = qz ± − 1/4
4z + z − = 1
z = z ± = q ± √ q 2 − 1
2
f(z ± ) = 1 4 + qz ± − 1 2 log z ±
f ′′ (z ± ) = 1 − 4z2 ±
2z 2 ±
= 2 z ∓ − z ±
z ±
= 8z ∓ (z ∓ − z ± )
|q| < 1
q = sin φ ,
√
1 − q2 = cos φ , |φ| < π 2
z ± = 1 2
(
)
sin φ ± i cos φ = 1 2 e∓i(π/2−φ) , f ′′ (z ± ) = − 4 cos φ e ±iφ
, z = z ± z − z ± = re iθ ±
f(z) = f ± + f ′′ (z ± )
(z − z ± ) 2 = f ± − 2r 2 cos φ e i(2θ±±φ) , f ± = f(z ± )
2
z = z ± r 2 e i(2θ ±±φ) = 1 ,
θ ± = ∓ φ 2 , π ∓ φ 2
Im f(z) = − 2xy + 2qy − 1 2 tan−1 y x = = Im f(z ±)
C ± , 2
C R → ∞
C +
z +
0
H n ( √ 2n + 1 q ) = 1 n!
2πi (2n + 1) (I n/2 − + I + )
∫
I ± = dz e (2n+1)f(z)− 1 2 log z
C ±
−R
C −
z −
R

15 282
θ − = φ 2 , θ + = π − φ 2 n → ∞ z = z ±
I ± ≈
∫ ε
(
dr e iθ± exp (2n + 1)f ± − 2(2n + 1)r 2 cos φ − 1 )
2 log z ±
−ε
(
≈ exp iθ ± + (2n + 1)f ± − 1 ) ∫ ∞ (
)
2 log z ± dr exp −2(2n + 1)r 2 cos φ
(
= exp iθ ± + (2n + 1)f ± − 1 2 log z ±
)√
−∞
π
2(2n + 1) cos φ
iθ + − 1 2 log z + = iπ − i 2 φ − 1 ( )
1
2 log 2 e−i(φ−π/2) = iπ + log √ 2 − iπ 4
iθ − − 1 2 log z − = log √ 2 + iπ 4
, f ±
(
exp iθ ± − 1 )
2 log z ± = √ 2 i e ±iπ/4
f ± = f(z ± ) = 1 4 ± i 2 e∓iφ sin φ + 1 2 log 2 ± i 2
(
φ − π )
= f 1 ± i f 2
2 2
f 1 = 1 2 + log 2 + sin2 φ , f 2 = sin φ cos φ + φ − π 2
(
(
I ± ≈ i exp (n + 1/2)f 1 ± i (n + 1/2)f 2 + π )) √
π
4 (2n + 1) cos φ
H n ( √ 2n + 1 q ) ≈ √ 1 n! e
(
(n+1/2)f1
√ cos (n + 1/2)f π (2n + 1) (n+1)/2 2 + π )
cos φ
4
= D n
e (n+1/2) sin2 φ
√ cos φ
( (
)
cos (n + 1/2) sin φ cos φ + φ − nπ 2
(
e (n+1/2)q2 2n + 1
(
= D n
(1 − q 2 ) cos q √ )
1 − q 1/4 2
2 + sin −1 q − nπ 2
)
)
n ≫ 1
√
2
D n =
n (n!) 2
π(n + 1/2) n+1 e −(n+1/2)
(
(n + 1) log n + 1 ) (
≈ (n + 1) log n + 1 )
≈ (n + 1) log n + 1 2
2n
2
(n + 1/2) n+1 ≈ n n+1 e 1/2 n n e −n ≈ n!/ √ 2πn
D n ≈
( 2 n+1/2 n!
√ nπ
) 1/2
(15.104)

15 283
q > 1 ,
, z − C −
H n ( √ 2n + 1 q ) = 1
∫
n!
dz e (2n+1)f(z)− 1
2πi (2n + 1) n/2 2 log z
C −
C −
z − z +
z = z −
f(z) = f(z − ) + µr 2 e 2iθ ,
z − z − = re iθ
µ = 1 (
2 f ′′ (z − ) = 4z + (z + − z − ) = 2 q + √ √
q 2 − 1)
q2 − 1
e 2iθ = −1 θ = π/2
H n ( √ 2n + 1 q ) ≈ 1
(
n!
2πi (2n + 1) exp (2n + 1)f(z n/2 − ) − 1 ) ∫ ∞
2 log z − i dr e −(2n+1)µr2
−∞
= 1
√
n!
π
2π (2n + 1) n/2 e(2n+1)f(z −)
(2n + 1)µz −
µz − = √ q 2 − 1
f(z − ) = 1 4 + qz − − 1 2 log z − = 1 4 + qz − + 1 2 log z + + log 2
H n ( √ 2n + 1 q ) ≈ D n
2
= D n
2
(2z + ) n+1/2 (
)
(q 2 − 1) exp (2n + 1)qz 1/4 −
(
q + √ ) n+1/2
q 2 − 1
( 2n + 1
exp
(q 2 − 1) 1/4 2
(
q 2 − q √ ) )
q 2 − 1
, D n (15.104)
√ 2n + 1 q q z = q/
√ 2n + 1
( 2n + 1
⎧⎪
exp(q 2 /2) ⎨ cos
H n (q) = D n
|1 − z 2 | × 2
1/4 ⎪ ⎩
1
2
(
z + √ z 2 − 1
(
z √ )
1 − z 2 + sin −1 z − nπ 2
)
, |z| < 1
) (
n+1/2exp
− 2n + 1 z √ )
z
2
2 − 1 , z > 1
(15.105)

284
94, 229
α 157
246
247
18
185
(1 ) 25
(3 ) 120
112
1, 6, 16, 217
32, 277
141
4
S 186
7
7, 210
59, 248, 280
159
107
74
99
3, 6
2
79
2
() 258
157
9, 72, 213, 250
213
263
174
3
9
81, 117
() 8
69
260
260
120, 260, 272
84, 255
111
85, 214
81
121, 144, 199
120, 124
228
21, 67
103
19
209
178
1, 4, 12, 60, 74
1
265
242
241
68
8
8, 211
246
176
1
162
224
97
115
114
112
22, 76, 110, 118, 122, 126
143
122
163
12
148

285
79
79
168
61
34, 131, 140, 146, 207
109
90
88
1 105
3 105
61
241
94
23
134
62
165
271
164
WKB 148
170
175
(1 ) 57
(3 ) 125
9, 72, 249, 254
14
6, 86
164
238
42
118
75
47, 156
34
163
90
19, 49, 64
2, 218
159
42
221
222
175
16, 63, 120, 127
239
166
12, 20, 62, 65, 80
223
185
186
185
209
19
18
264
109
172
260
153
248, 252, 275
4
180
250
8
196
96
() 123, 274
201
97
243
243
84, 250
84, 250
237

286
216