First attempt at a syllabus for

## Physics 402, Fall 2007

Official course description:
PHYS402 Quantum Physics II; (4 credits) Grade Method: REG/P-F/AUD.
Prerequisites: PHYS401, and PHYS374, and MATH240. Credit will be granted for only one of the following: PHYS402 or former PHYS422. Formerly PHYS 422. Quantum states as vectors; spin and spectroscopy, multiparticle systems, the periodic table, perturbation theory, band structure, etc.

0101(55509) Brill, D. (Seats=45, Open=29, Waitlist=0) Books
MWF.......10:00am-10:50am (PHY 1402)
W.........11:00am-11:50am (PHY 1402)

Textbook: David J. Griffiths, Quantum Mechanics (second edition preferred but first edition acceptable), Pearson/Prentice Hall, ISBN 0-13-111842-7

What we will actually do during this term:
We will start with a thorough review of Phys 401. The first assignment will be to re-do problems 1, 3, 4, 7 of the Final Exam of Phys401. (It will actually be a good idea to review all the problems at this time.) The 401 Final is found at the end of this syllabus. View this in Internet Explorer, Firefox distorts the equations.
We will then cover by lectures and problems, and in the order of the textbook, those parts of the Griffith text that were not covered in Phys401. For your reference, the sections considered to be part of Phys401 (but subject to the above-promised review) are 1.1-4.4, 6.1-6.2.

There will be approximately weekly assignments, mainly from the text, that count approximately 20% of your grade.

There will be two exams during the term, one of them a take-home, and a Final Exam that counts about as much as the two in-term exams.

Physics 401                 Final Examination                May 15, 2007

Instructions: Use exam booklets and sign the pledge. You may leave out two of the following eight problems, but at most one from Group II.

 Group I

1. We have a beam of electrons and a Stern-Gerlach apparatus. An electron in the incident beam is described by
 y(y, t) = 1 Ö2 (c+ + c-) e-i(Et-py)
(1)
where c+ and c- are eigenspinors of Sz with eigenvalues +(h/2p)/2 and -(h/2p)/2 respectively. (Also, E is the energy and p is the y-momentum of the incident beam, but this is of no particular importance; the interest is in the spin part of the wavefunction.)

(Picture Omitted, same as FIGURE 4.11)

The problem is to describe what happens when the apparatus is lined up in the z-direction, as shown, and when it is rotated 90o so it is aligned with the x-direction.
A student (not from this course) answers: The state function (1) shows that half of the electrons have spin up (with respect to the z-direction), and the other half have spin down. Therefore when the gradient of the B-field is in the z-direction, it will split the beam into two, the spin-up electrons moving in the +z direction, and the spin-down electrons in the -z direction. When the B-field gradient is in the x-direction, the beam is not deflected, because the spins and magnetic moments of electrons in the state (1) are uniquely in the z-direction, and on such magnetic moments there is no force due to an x-gradient of B.
1. This student apparently has number of misconceptions. Mention everything (s)he says that is wrong about quantum physics.
2. What are the correct answers, and the correct reasons?
3. This student also has the idea that the time-independent Schrödinger equation gives information only about stationary states, and its solutions are of no help in finding time-dependent wave functions. Comment on that idea.

2. In classical mechanics, the reference level for potential energy is arbitrary. What are the effects on
• the wave function
• the probability density
• expectation values of operators x, p, and H
of adding a constant to the potential V in the Schrödinger equation?

3. A free particle of momentum p is represented by a plane wave. A measurement apparatus determines that the particle lies inside a region of length L. The measurement interaction leaves the wavefunction unchanged (except for normalization) for the length L and reduces it to zero outside this region. What are
• the average momentum
• the average energy
of the particle after the measurement has been made?

4. A 1-dimensional particle of mass m is in an infinite square well of size a.
1. What is the particle's energy when it is in its lowest energy state in this well?
2. A later measurement shows that the particle is not in the left half of the well. What is the lowest energy áH ñ compatible with this measurement? Give reasons for your answer!

 Group II

5. A particle of mass m moves in the one-dimensional potential
 V(x) = -a(d(x+a) + d(x-a))
where a is a constant with dimension of energy·length, and a is a constant with dimension of length.
1. Find the ground state wavefunction and energy.
2. Draw a sketch of the ground state wave function.
3. How many bound states exist in this potential?
4. What is the expectation value áxñ for the bound states in this potential?

6.
1. Write a Hermetian operator for the product of the momentum (operator p) and the displacement (operator x) of a 1-dimensional particle of mass m. Call this operator Q.
2. This particle is in a simple harmonic oscillator potential of frequency w. Evaluate all the matrix elements of Q, án|Q|m ñ, between simple harmonic oscillator energy eigenstates. In particular, show that the expectation value áQñ = 0 for any such eigenstate.
Hint: a± = [1/(Ö{2(h/2p) mw})](ip + mwx)
3. This SHO Hamiltonian is perturbed by adding lQ to it. In b you showed that the first order correction to the energy of eigenstates vanishes. Now evaluate the second order correction to the ground state energy,
 E02 = l2 ¥å m=1 |ám|Q|nñ|2 E00 - Em0 .
The answer should be a numerical factor ×  l2(h/2p)w. The numerical factor is the important thing.

7. A rigid rotor is completely described by its moment of inertia I and its angular momentum [L\vec]. (This could be a single particle in 3D space constrained to a constant distance from the origin, whose radial oscillations are not excited; we neglect the constant and large zero-point energy that would be present in any scheme to make the radial distance constant.) Its Hamiltonian is H = L2/2I. Let a typical stationary state of the rotor be described by the usual quantum numbers l, m associated with L2 and Lz respectively.
1. What is the rotor's energy in such a stationary state?
2. Using L± = Lx ±iLy, show that áLxñ = 0 in any of these stationary states.
3. Show that áLx2ñ = áLy2ñ, and evaluate áLx2ñ in terms of (h/2p) and the l, m quantum numbers.
4. A perturbation aLx is applied to the rotor (for example, the rotor has a magnetic moment and is placed in a magnetic field in the x-direction). Does the result of b imply that the first order correction to the energy vanishes? If not, show how to calculate the proper correction

8. Think up a perturbation Hamiltonian for a hydrogen atom with a spinless electron that lifts the degeneracy between levels of the same n but does not change any other degeneracy, to first order (for example, the degeneracy between levels that differ in m only should remain unchanged). Several answers are possible, mention all you can think of. Write your answers in terms of operators (other than spin) we have used in discussing the hydrogen atom. Explain why your answers lift the degeneracy. You do not have to evaluate the first order energy change, but make sure it is not infinite.

File translated from TEX by TTH, version 3.67.
On 14 May 2007, 22:22.