Quantum Mechanics Physics 215
Homework 6.
Not to be handed in. Solutions will be provided. Questions on these topics
may be on the final exam.
The questions on the final exam will be similar to what you had on the
homework.
and
are projection operators, i.e. they obey 
(no sum over j), where I is the identity
operator and the 
are spin- 
operators.
(b) Show that 
and 
project onto the spin-1 and
spin-0 subspaces of the direct product space of the two spin-
spaces. (Note: This is sometimes denoted by :
.)
and 
as
linear combinations of spherical
(irreducible) tensor operators of rank 2.
(b) The expectation value:
is known as the quadrupole moment, where 
refers to unspecified
quantum numbers that characterize the state, and 
. Evaluate:
(where 
) in terms of Q and appropriate
Clebsch-Gordan coefficients.
(c) Using the Wigner-Eckart theorem prove that a spin- particle
cannot possess a quadrupole moment.
respectively, where 
is the (antiunitary) time reversal
operator.
(a) Show that the reduced matrix elements of such an operator must satisfy:
(b) The electric dipole operator is 
.
Prove that if the neutron is observed to have a non-zero electric dipole
moment, then both parity and time-reversal invariance are violated.
where 
is a positive constant (the classical frequency of
the oscillator) and 
is the mass of the particle with charge q. Assume
that the particle has no spin. The particle is placed in a uniform
magnetic field B parallel to the z axis. Define 
, the classical Larmor precession frequency.
(a) Write down the Hamiltonian in the Coulomb gauge in the form:
where 
is the sum of an operator which depends linearly on
(the paramagnetic term) and an operator which depends
quadratically on (the diamagnetic term). First, compute the
energy eigenstates with B=0.
Next, turn on the magnetic field.
Show that the new eigenstates of the system and their degeneracies can be
determined exactly. Compute the energy eigenvalues (and their
corresponding degeneracies) explicitly for arbitrary .
(b) Show that if 
, then the paramagnetic term dominates
over the diamagnetic term.
(c) Consider the first excited states of the oscillator, i.e. the states
whose energies approach 
as
. To first order in 
, what are the
energy levels in the presence of the B-field and their degeneracy?
Sketch the energy levels as a function of B?
(d) Now consider the ground state. How does its energy vary as a
function of ? Is the ground state, in the presence of the
B-field (i) an eigenvector of 
? (ii) an eigenvector of
? (iii) an eigenvector of 
? Give the form of the wave function
and the corresponding probability current. Show that the effect of the
B-field is to compress the wave function about the z-axis in a
ratio 
and to induce a current.