PHY 4604 Spring 2012 – Homework 7
Due at the start of class on Friday, April 20. Please turn in No credit will be
available for homework submitted after the start of class on Wednesday, April
25.
Answer all four questions. Please write neatly and include your name on the front page of
your answers. You must also clearly identify all your collaborators on this assignment. To
gain maximum credit you should explain your reasoning and show all working.
1. The isotropic harmonic oscillator. A particle of mass M moves in the potential
1
1
V (r) = M ω 2 (x2 + y 2 + z 2 ) = M ω 2 r2 .
2
2
The stationary states of this problem can be found in Cartesian coordinates by the
method of separation of variables.
(a) Based on your knowledge of the one-dimensional harmonic oscillator, write down
the general form of the normalized stationary states ψ(x, y, z) and their energies
in terms of three non-negative integer quantum numbers nx , ny , and nz .
(b) Show that the stationary-state energies are of the form EN = (N + 3/2)~ω, where
N = 0, 1, 2, . . ., and that energy EN occurs with degeneracy gN = (N + 1)(N +
2)/2.
Since the potential is spherically symmetric, it can also be treated by seeking a solution
ψ(r, θφ) = Rnl (r)Ylm (θ, φ), where Ylm is a standard spherical harmonic. With considerable effort, one can reproduce the same eigenenergies EN and degeneracies gN as
one obtains using Cartesian coordinates. The fact that the degeneracies do not follow
the pattern 2l + 1 expected for states Rnl (r)Ylm (θ, φ) indicates that the isotropic harmonic oscillator has additional degeneracies beyond those arising purely from spherical
symmetry.
2. The hydrogen atom.
(a) Show that hxi = hyi = hzi = 0 in any stationary state ψnlm of the hydrogen atom.
Hint: Write x, y, and z in spherical coordinates and focus on the angular integrals.
You may find helpful the relation Ylm (π − θ, φ) = (−1)l+m Ylm (θ, φ).
(b) Calculate hri, hr2 i, hx2 i, and hz 2 i in the ψ211 state of hydrogen.
Hint: Integrals over θ are often simplified by the identity
Z π
Z 1
f (cos θ) sin θ dθ =
f (µ) dµ
0
for any function f (µ).
3. Griffiths Problem 4.45.
4. Griffiths Problem 4.49.
−1