1.1 The Radiation Laws and the Birth of Quantum Mechanics

QUANTUM MECHANICS I (Dr. T. Brandes): Example/Solution Sheets 24
4.17.5 Ground state (20 min)
Use the operator a to calculate the ground state wave function ψ 0 (x) explicitely. Start from
the operation
a|0〉 = 0 aψ 0 (x) = 0, (4.117)
and use the definition of a to derive an ordinary differential equation for ψ 0 (x) that you can
solve.
SOLUTION:
Write a in terms of x and p and solve the resulting differential equation:
√ mω
2 ˆx +
√
2mω
ψ ′ 0 (x) = 0
mω
x + ψ′ 0(x) = 0 ψ 0 (x) ∝ exp ( −mωx 2 /2 ) . (4.118)
4.18 Central Potentials in Three Dimensions
4.18.1 Separations of Variables (20 min)
Show by using the definition of the Laplace operator in polar coordinates and the definition
of the angular momentum square,
[ ( 1
ˆL 2 = − 2 ∂
sin θ ∂ )
+ 1 ]
∂ 2
sin θ ∂θ ∂θ sin 2 (4.119)
θ ∂ϕ 2
that the stationary Schrödinger equation for energy E for the motion of a particle with mass
m in a central potential U(r) can be separated with the Ansatz for the wave function
Ψ(r, θ, φ) = R(r)Y lm (θ, φ). (4.120)
In order to do so, define the radial function χ(r) := rR(r) and show
[ ]
d 2 χ(r) 2m
l(l + 1)
+ (E − U(r)) − χ(r) = 0. (4.121)
dr 2 2 r 2
Which values are possible for l (without proof)
SOLUTION: See lecture notes chapter 4.4 and 4.5.
4.18.2 * Behavior for r → 0 und r → ∞ (10-20 min)
Verify that functions χ(r) with the following properties
lim r→0 χ(r) ∝ r l+1 , lim
r→∞
χ(r) ∝ e −r√ −2mE/ 2 , E < 0 (4.122)
fulfill the radial part of the Schrödinger equation for ‘reasonable’ potentials U(r).