Diploma thesis

4 Complex harmonic potential
which coincides with the energy in (4.47). Next we derive an expression for the normalization
constant in the real limit. For this purpose we also set a 1 = a 2 = −n in (4.45) and conclude that
all terms containing Γ (a 1 ) −1 vanish and thus R s = 1 from (4.34). Furthermore all infinite sums
including (−n) ν now have n as their upper summation index, since we have (−n) ν = 0 for all
ν > n. Finally there only remains
|N s |=
{ n
∑
µ,ν=0
[
1
µ!ν!
which reduces to
(−n) ( ν (−n) µ
1
( 1
2)
2)
ν
µ
[ (
Γ µ+ν+ 1 )
−Γ
(µ+ν+ 1 )]
2
2 , ω2
+ (−n) (
ν(−n) µ
1
( 1
2)
2)
ν
µ
Γ
(µ+ν+ 1 ) ]} −1/2
2 , ω2 , (4.53)
⎧
⎪⎨
|N s | =
⎪⎩
n∑
µ,ν=0
Taking into account (4.41) we obtain
1
µ!ν!
(−n) ( ν (−n) ( µ
1 1
2)
2)
ν
µ
(
Γ µ+ν+ 1 ) ⎫ −1/2
⎪ ⎬
. (4.54)
2 ⎪ ⎭
{ ∫ ∞
∣ ( ∣∣∣
|N s | = e −χ2 1F 1 −n, 1 )∣ ∣∣∣ 2 −1/2
dχ}
−∞
2 ; χ2 , (4.55)
∣ (
where we used the symmetry of e −χ2 ∣1 F 1 −n,
1
; 2 χ2)∣ ∣2 with respect to χ = 0 in the last step to
expand the integral over the whole real axis. With the definition of the Hermite polynomials
(4.48) we can write for this integral
|N s | = (2n)!
n!
{∫ ∞
−∞
} √
−1/2
e −χ2 |H 2n (χ)| 2 (2n)! 1
dχ = √
n! π2
2n
(2n)! , (4.56)
where we used the calculation of the integral from [29, Chap. 35].
Taking the real limit a 1 = a 2 = −n of (4.35) yields directly one and the same form in all 3 areas
ψ s 0 (χ) = N s e − 1 2 χ2 1F 1
(
−n; 1 2 ; χ2 )
. (4.57)
Now we insert the absolute value of the normalization constant (4.56) into the general expression
(4.46), so that we can write for the wave function
which reduces with (4.48) to
ψ0 s (χ) = (2n)!
√
(
1
√
n! π2
2n
(2n)! eiϕ e − 1 2 χ2 1F 1 −n; 1 )
2 ; χ2 , (4.58)
√
ψ0 s (χ) = (−1) n 1
√ π2
2n
(2n)! eiϕ e − 1 2 χ2 H 2n (χ) . (4.59)
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