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 : .)
(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.