PHY4605–Introduction to Quantum Mechanics II
Spring 2005
Take-home final exam
April 18, 2005
Due April 22, 2005 5pm
Do all problems. Open book. Do not consult with anyone else.
1. Short answer (10 pts.)
(a) Write the expression hψ|ψi = 1 as an explicit integral equation in three
dimensions, assuming that |ψi represents a wave function ψ(~r). Suppose
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you have |ψi = n cn |ni where the |ni are a complete set of orthonormal
states. What conditions does the above equation impose on the cn ?
(b) How many degenerate levels are there for a hydrogen atom with principal
quantum number n? Are any of these degneracies lifted by the spin-orbit
interaction? Justify your answer.
(c) What is the big difference between a particle with spin quantum number
s = 1/2 and one with s = 1?
(d) Suppose that the operator corresponding to some observable is called Q.
List 2 properties of this operator and/or of its eigenfunctions |ni. The
latter satisfy Q|ni = qn |ni. Suppose further that the quantum-mechanical
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state of a system is given by |ψi = n cn |ni with several of the cn 6= 0. If
you were to make a single measurement of the observable Q, what would
you get as a result?
(e) Two quantum mechanical particles have orbital angular momentum l = 1
and spin angular momentum s = 0. Suppose that there is some coupling
of the two particles. List the values that the total angular momentum j
of the two-particle system may take on. For each j, state what are the
possible values for the z component.
2. Skewed square well (10 pts.).
Consider an infinite well for which the
bottom is not flat, as sketched here. If
the slope is small, the potential V =
²|x|/a may be considered as a perturbation on the square-well potential
over −a/2 ≤ x ≤ a/2.
(a) Calculate the ground-state energy, correct to first order in perturbation
theory.
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(b) Calculate the energy of the first excited state, correct to first order in
perturbation theory.
(c) Calculate the wave function in the ground state, correct to first order in
perturbation theory. (You will probably not be able to get an answer in
closed form, but take the calculation as far as you can–do not evaluate
integrals you encounter here).
(d) At what value of ² does perturbation theory break down? Justify your
answer.
3. p-n scattering (10 pts.)
The scattering amplitude for p − n scattering is modelled by
f (θ) = χ†f (A + Bσp · σn )χi
(1)
where χf and χi are the final and initial spin states of the n − p system. Possibilities are:
χi , χf = | ↑↑i, | ↑↓i, | ↓↑i, | ↓↓i
(2)
where the first spin refers to the proton and the second to the neutron. Note the
wave functions do not have to be antisymmetrized since p, n distinguishable.
Using
p n
n p
σ− + σ+
σ− ),
σp · σn = σzp σzn + 2(σ+
(3)
where

σ+ =


0 1
σx + iσy 

=


2
0 0
; σ− =

0 0
σx − iσy 

=


2
1 0
(4)
Calculate and tabulate all 16 scattering amplitudes. Make a table of results.
4. Selection rules (10 pts.) Recall that the selection rule for electric dipole
transitions in hydrogenic atoms, i.e. those with nonzero matrix elements
hφn`m |d|φn0 `0 m0 i, (d = er) are that that `0 must be ` ± 1 and that m0 must
be m ± 1.
(a) A hydrogen atom is initially in an n = 4, ` = 2 excited state. It decays to
the ground state by a sequence of allowed dipole transitions. How many
steps are required? Write a possible sequence or two.
(b) (b) A hydrogen atom initially in an n = 4, ` = 2 state decays directly
to its ground state. Show that as a dipole transition this is forbidden,
but it is possible as an electric quadrupole transition, with a perturbation
proportional to Qij = e(3ri rj − δij r2 ).
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5. Exchange Symmetry (10 pts.). Suppose you have three particles, one in
state ψa , one in ψb , and one in ψc . Construct a wave function for the three
particles if they are
(a) distinguishable
(b) fermions
(c) bosons
6. Variational principle (5 pts.) Use the variational principle to estimate the
ground state energy of the anharmonic oscillator, H = p̂2 /2m + λx4 , and compare with exact result E0 = 1.06λ1/3 (h̄2 /2m)2/3 .
7. Aharonov-Bohm phase shift (5 pts.) A single electron passes through a
large capacitor of charge Q and capacitance C. Its velocity is v (assume unaltered
by capacitor’s field) and the capacitor has length L. Estimate the change of
phase of the electron’s wave function.
8. Stern-Gerlach (10 pts.). A Stern-Gerlach apparatus is aligned along the z
direction, and a second one is aligned at 45◦ with respect to this in the z − x
plane. A neutral spin-1/2 particle is prepared with spin k ẑ and then passed
through both in succession.
(a) What is the probability a detector located after the two SG setups finds
the particle to have spin up (k ẑ)?
(b) What is the probability a detector measures the spin to be along the ŷdirection?
The initial spin state is now changed to definitely k x̂.
(c) Redo question a)
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