1.1 The Radiation Laws and the Birth of Quantum Mechanics

QUANTUM MECHANICS I (Dr. T. Brandes): Example/Solution Sheets 17
3.11.3 * Expansion into eigenmodes (40 min)
Consider the vector f ∈ H, f(x) = cx(L − x).
a) Calculate the constant c such that f is normalized, i.e. ‖f‖ = 1. Show that c = √ 30/L/L 2 .
∫ L
∫ L
1 = ‖f‖ 2 = dxf ∗ (x)f(x) = c 2 x 2 (L − x) 2 =
0
0
∫ L
[ 1
= c 2 dx[x 4 − 2Lx 3 + L 2 x 2 ] = c 2 L 5 5 − 2 4 + 1 ] √
30
c =
3
L
0
b) Show that f can be expanded in the basis ψ n as
1
L 2 .
∞∑
f = c n ψ n ,
n=1
c n = 2 √ 60 1 − (−1)n
n 3 π 3 (3.90)
We have
√
2
c n = 〈ψ n |f〉 = c
L
= √ 60
∫ 1
0
∫ 1
∫ 1
c) Use b) to prove the formula
We have
√
30
L
0
0
∫ L
0
dxx(L − x) sin
dy(y − y 2 ) sin(nπy).
dyy sin(nπy) = − cos(nπ)
nπ
dyy 2 sin(nπy) = − 2
n 3 π 3 + 2 − n2 π 2
n 3 π 3
1
L 2 x(L − x) = √ 60
(1/2)(1/4) =
π3
32
=
∞∑
n=1
∞∑
k=0
3.11.4 * Scalar product (20 min)
∞∑
n=1
∫ 1
0
( nπx
)
= [y = x/L) =
L
cos(nπ)
dy(y − y 2 ) sin(nπy) = 2 1 − (−1)n
n 3 π 3
π 3 ∞
32 = ∑ (−1) k
(2k + 1) . 3
k=0
2 1 − (−1)n
n 3 π 3
2 1 − (−1)n
n 3 π 3
(−1) k
(2k + 1) 3 .
sin
√
2
( nπx
)
L sin , setx = L/2
L
∞∑
( nπ
)
= [n = 2k + 1] =
2
k=0
4(−1) k
π 3 (2k + 1) 3
a) Use the bra and ket notation to show that for an orthonormal basis {|ψ n 〉} and two Hilbert
space vectors |ψ〉 and |χ〉, one has
〈ψ|χ〉 =
∞∑
〈ψ|ψ n 〉〈ψ n |χ〉. (3.91)
n=0