PHY 4604 Fall 2009 – Homework 7
Due at 5:00 p.m. on Friday, December 4. Please turn in your homework in class before the deadline, or else bring it to NPB 2162.
(You may push it under the door if NPB 2162 is unoccupied.) No credit will be available for homework submitted after the start of class on Monday, December 7.
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
V ( r ) =
1
2
M ω
2
( x
2
+ y
2
+ z
2
) =
1
2
M ω
2 r
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 n x
, n y
, and n z
.
(b) Show that the stationary-state energies are of the form E
N
N = 0, 1, 2,
2) / 2.
. . .
, and that energy E
N
= ( N occurs with degeneracy g
N
+ 3 /
= (
2)
N
~
ω , where
+ 1)( N +
Since the potential is spherically symmetric, it can also be treated by seeking a solution
ψ ( r, θφ ) = R nl ( r ) Y l m
( θ, φ ), where Y l m is a standard spherical harmonic. With considerable effort, one can reproduce the same eigenenergies E
N and degeneracies g
N as one obtains using Cartesian coordinates. The fact that the degeneracies do not follow the pattern 2 l + 1 expected for states R nl ( r ) Y l m
( 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 h x i
= h y i
= h z i
= 0 in any stationary state ψ nlm 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
(b) Calculate h r i
, h r
2 i
, h x
2 i
, and h z
2 i
Y l m
( π
−
θ, φ ) = (
−
1) in the ψ
211 l + m
Y l m
( θ, φ ).
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 − 1 for any function f ( µ ).
3. Griffiths Problem 4.45.
4. Griffiths Problem 4.49.