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Physics PHYS 356
Quantum Mechanics
Problem Set #6
1.
Angular momentum in position space
  
a)
Starting from the definition angular momentum, L  r  p , show that L̂x
and L̂y can be written in spherical coordinates as


 
Lˆ x    sin 
 cot  cos 
 and
i

 


 
Lˆ y   cos 
 cot  sin 
.
i

 
b)
Now using these write down Lˆ  Lˆ x  iLˆ y and Lˆ  Lˆ x  iLˆ y in position
space.
c)
Show using the position space operators and wave functions that
Lˆ , m     1  mm  1 , m  1 and
Lˆ , m     1  mm  1 , m  1
2.
Quantum Mechanical Rotator
Consider the following model of a perfectly smooth cylinder. It is a ring of
equally spaced, identical particles, with mass M N so that the mass of the ring is
M and its moment of inertia MR2 with R the radius of the ring. Calculate the
possible values of the angular momentum. Calculate the energy eigenvalues.
What is the energy difference between the ground state of zero angular
momentum and the first rotational state? Show that this approaches infinity as
N  . Contrast this with the comparable energy for a “nicked” cylinder, which
lacks the symmetry under rotation through 2 N radians. This example implies
that it is impossible to set a perfectly smooth cylinder in rotation, which is
consistent with the fact that for a perfectly smooth cylinder such a rotation would
be unobservable.
3.
Angular Momentum Eigenstates
A particle in a spherically symmetric potential is in a state described by the wave
function
 x, y, z   C xy  yz  zxer .
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What is the probability that a measurement of the square of the angular
momentum yields zero? What is the probability that it yields 6 2 ? If the value of
 is found to be 2, what are the probabilities for measuring m  2,1,0,1,2 ?
4.
Expectation Values
A particle is in an eigenstate of L̂2 and L̂z . Show that the average value of L̂x
and L̂y are both zero. Calculate Lˆ2x and Lˆ2y .
5.
Angular Momentum Eigenstates
a)
Townsend, problem 9.20 (9.19 in 1st ed.) Remember that
1,0
b)
6.
x

1
2
1,1 
1
2
1,1 .
Townsend, problem 9.20 (9.20 in 1st ed.)
Clebsch-Gordan Coefficients
Calculate the Clebsh-Gordan coefficients for finding the states of total angular
momentum of two spin-1 particles.
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