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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.

  1. (a) Show that

    displaymath78

    and

    displaymath80

    are projection operators, i.e. they obey tex2html_wrap_inline82 (no sum over j), where I is the identity operator and the tex2html_wrap_inline88 are spin- tex2html_wrap_inline90 operators.

    (b) Show that tex2html_wrap_inline92 and tex2html_wrap_inline94 project onto the spin-1 and spin-0 subspaces of the direct product space of the two spin- tex2html_wrap_inline90 spaces. (Note: This is sometimes denoted by : tex2html_wrap_inline98 .)

  2. (a) Write tex2html_wrap_inline100 and tex2html_wrap_inline102 as linear combinations of spherical (irreducible) tensor operators of rank 2.

    (b) The expectation value:

    displaymath104

    is known as the quadrupole moment, where tex2html_wrap_inline106 refers to unspecified quantum numbers that characterize the state, and tex2html_wrap_inline108 . Evaluate:

    displaymath110

    (where tex2html_wrap_inline112 ) in terms of Q and appropriate Clebsch-Gordan coefficients.

    (c) Using the Wigner-Eckart theorem prove that a spin- tex2html_wrap_inline90 particle cannot possess a quadrupole moment.

  3. An irreducible tensor is even or odd under time reversal if:

    displaymath118

    respectively, where tex2html_wrap_inline120 is the (antiunitary) time reversal operator.

    (a) Show that the reduced matrix elements of such an operator must satisfy:

    displaymath122

    (b) The electric dipole operator is tex2html_wrap_inline124 . Prove that if the neutron is observed to have a non-zero electric dipole moment, then both parity and time-reversal invariance are violated.

  4. Consider the isotropic three-dimensional oscillator:

    displaymath126

    where tex2html_wrap_inline128 is a positive constant (the classical frequency of the oscillator) and tex2html_wrap_inline130 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 tex2html_wrap_inline138 , the classical Larmor precession frequency.

    (a) Write down the Hamiltonian in the Coulomb gauge in the form:

    displaymath140

    where tex2html_wrap_inline142 is the sum of an operator which depends linearly on tex2html_wrap_inline144 (the paramagnetic term) and an operator which depends quadratically on tex2html_wrap_inline144 (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 tex2html_wrap_inline144 .

    (b) Show that if tex2html_wrap_inline152 , 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 tex2html_wrap_inline154 as tex2html_wrap_inline156 . To first order in tex2html_wrap_inline158 , 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 tex2html_wrap_inline144 ? Is the ground state, in the presence of the B-field (i) an eigenvector of tex2html_wrap_inline168 ? (ii) an eigenvector of tex2html_wrap_inline170 ? (iii) an eigenvector of tex2html_wrap_inline172 ? 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 tex2html_wrap_inline178 and to induce a current.




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Peter Young
Tue Mar 7 21:23:41 PST 2000