First attempt at a syllabus for
Physics 402, Fall 2007
Official course description:
PHYS402
Quantum Physics II;
(4 credits)
Grade Method: REG/PF/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.......10:00am10:50am (PHY 1402)
 W.........11:00am11:50am (PHY 1402)
Textbook: David J. Griffiths, Quantum Mechanics (second edition preferred but first edition acceptable), Pearson/Prentice Hall, ISBN 0131118427
What we will actually do during this term:
We will start with a thorough review of Phys 401. The first assignment will be to redo 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 abovepromised review) are
1.14.4, 6.16.2.
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 takehome, and a Final Exam that counts about as much as the two interm 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.
1. We have a beam of electrons and a
SternGerlach apparatus. An electron in the incident beam is
described by
y(y, t) = 
1
Ö2

(c_{+} + c_{}) e^{i(Etpy)} 
 (1) 
where c_{+} and c_{} are eigenspinors of S_{z} with
eigenvalues +(^{h}/_{2p})/2 and (^{h}/_{2p})/2 respectively. (Also, E is
the energy and p is the ymomentum 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 zdirection, as shown, and when it is rotated 90^{o} so it is aligned with the xdirection.
A student (not from this course) answers: The state function (1)
shows that half of the electrons have spin up (with respect to the
zdirection), and the other half have spin down. Therefore when
the gradient of the Bfield is in the zdirection, it will split
the beam into two, the spinup electrons moving in the +z
direction, and the spindown electrons in the z direction. When
the Bfield gradient is in the xdirection, the beam is not
deflected, because the spins and magnetic moments of electrons in
the state (1) are uniquely in the zdirection, and on such magnetic moments there is no force due to an xgradient of B.
 This student apparently has number of misconceptions.
Mention everything (s)he says that is wrong about quantum
physics.
 What are the correct answers, and the correct
reasons?
 This student also has the idea that the
timeindependent Schrödinger equation gives information only
about stationary states, and its solutions are of no help in
finding timedependent 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 1dimensional particle of mass m is in an
infinite square well of size a.
 What is the particle's energy when it is in its lowest
energy state in this well?
 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!
5. A particle of mass m moves in the
onedimensional potential
V(x) = a(d(x+a) + d(xa)) 

where a is a constant with dimension of energy·length, and a is a constant with dimension of length.
 Find the ground state wavefunction and energy.
 Draw
a sketch of the ground state wave function.
 How many bound
states exist in this potential?
 What is the expectation
value áxñ for the bound states in this potential?
6.
 Write a Hermetian
operator for the product of the momentum (operator p) and the displacement (operator x) of a 1dimensional particle of mass m. Call this operator Q.
 This particle is in a simple harmonic oscillator potential
of frequency w. Evaluate all the matrix elements of Q,
ánQm 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)
 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,
E_{0}^{2} = l^{2} 
¥ å
m=1


ámQnñ^{2}
E_{0}^{0}  E_{m}^{0}

. 

The answer should be a numerical factor × l^{2}(^{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 zeropoint energy that would be present in any
scheme to make the radial distance constant.) Its Hamiltonian is
H = L^{2}/2I. Let a typical stationary state of the rotor be
described by the usual quantum numbers l, m associated with
L^{2} and L_{z} respectively.
 What is the rotor's energy in such a stationary state?
 Using L_{±} = L_{x} ±iL_{y}, show that áL_{x}ñ = 0 in any of these stationary states.
 Show that áL_{x}^{2}ñ = áL_{y}^{2}ñ, and evaluate áL_{x}^{2}ñ in terms of (^{h}/_{2p}) and the l, m quantum numbers.
 A perturbation aL_{x} is applied to the rotor (for example, the rotor has a
magnetic moment and is placed in a magnetic field in the
xdirection). 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
T_{E}X
by
T_{T}H,
version 3.67.
On 14 May 2007, 22:22.