PHY4605–Introduction to Quantum Mechanics II
Spring 2005
Test 1 Solutions
February 9, 2005
1. Short Answer. Must attempt (only) 3 of 4. Circle answers to be graded.
(a) Sketch an experiment to measure the Aharonov-Bohm effect, and comment
on the significance of the electromagnetic potentials A and φ in quantum
mechanics as opposed to classical physics.
interference
pattern
1
B=0
2
A=0
B=0
Changing B in solenoid changes relative phases of electrons travelling along
“paths” 1+2 shifts interference pattern at screen. Classically e− would
need to move through region of B 6= 0 to feel an effect. Not true in
Quantum Mechanics.
(b) Comment on the significance of unitary transformations in quantum mechanics. Indicate how the Hamiltonian must transform under such a unitary transformation in order to preserve the Schrödinger equation.
Unitary transformations play a special role in quantum mechanics because
they preserve matrix elements of observables. The Hamiltonian, for example, must transform as H → U −1 HU at the same time as states transform
as ψ → U −1 ψ. This means
¡
¢¡
¢
hψ|H|χi → hψ 0 |H 0 |χ0 i = (hψ|U ) U −1 HU U −1 |χi
= hψ|H|χi
(c) a) Explain why, in a Stern-Gerlach (SG) apparatus, a beam consisting of
neutral particles in different spin states are split into different beams. b)
If a single spin-1 particle in state |` = 1, m = 1i (spin quantization axis
ẑ) is sent into a SG apparatus aligned along ŷ, state how many possible
paths the particle can take.
(c) a) In a magnetic field a magnetic moment has (classically) an energy
Edipole = −~µ ·B, and if B is inhomogeneous the particle experiences a force
given by
−∇Edipole = ∇(~µ · B) = ∇(
e
e
S · B) = ∇(Sz Bz ),
m
m
where in last step I arbitrarily assumed Bz was in z-direction. Thus the
force on a particle will be different depending on the value of Sz , splitting the particles with a given initial velocity into different beams. The
quantum mechanical version of this argument, involving calculating the
expectation value of p, is given in Griffiths, and reaches the same conclusion.
(c)b) Since each eigenvalue of the spin experiences a different force, an
unpolarized spin-1 particle beam is split into 3 different beams in general,
corresponding to the 2S + 1=3 independent components with different Sz .
(d) A broad beam of x-rays is incident on an ammonia molecule in its ground
state.
i. Explain why interference fringes are observed on a photographic plate.
ii. Design a gedanken-experiment to determine if the N atom in the
molecule is above or below the plane defined by the three H atoms.
The ammonia atom in its ground state is in a linear superposition of the
quantum states with the H-atom below and above the plane, as shown. A
wide beam of electrons, photons, etc. of short wavelength will therefore
scatter as shown, producing spherical wavelets from each N atom which
would–in principle–interfere at an appropriately placed screen. If one could
on the other hand produce a very narrow beam of particles oriented such
that the N atom would scatter only if it were in the top or bottom state.
Scattering will then constitute measurement of the top or bottom state,
and collapse the wave function.
2. Perturbations to SHO. A 1D oscillator Hamiltonian has the form
H =
p2
1
+ mω 2 x2 + ax3 + bx4
2m 2
(a) Make 2 plots: at left, sketch the potential V (x); at right, sketch the ground
state wave function ψ0 (x). Each plot should have two curves, one for the
a = b = 0 SHO, & one for the anharmonic case with a = 0, b > 0. Indicate
which curve is which, to make qualitative change due to b > 0 clear.
Dashed represents without perturbation, solid with. Note the slight increase near x = 0 in the exact wave function to preserve normalization
due to supression at large |x|.
(b) A priori, in which region of space do you expect the wave functions you
find in perturbation theory to be the best approximations to the exact
eigenfunctions?
As x → ∞, the perturbation becomes arbitrarily large, so the errors in
keeping only corrections to finite order must become large as well. However, since the wavefunctions are exponentially small, this is usually not
terribly important!
(c) Find the energy shift δEn of each of the eigenstates ψn of the a = b = 0
simple harmonic oscillator when both a and b are nonzero,
but may be
R
considered small. Express your answer in terms of cn ≡ dx ψn∗ x4 ψn .
To linear order
Z
3
4
dx ψn∗ (x)(ax3 + bx4 )ψn (x)
Z
= 0+b
dx ψn∗ (x)x4 ψn (x) ≡ bcn
δEn = hn|(ax + bx )|ni =
where the 0 comes from the fact that the SHO eigenfunctions have definite
parity, therefore |ψn (x)|2 is always even, and vanishes when integrated with
x3 over a symmetric interval.
(d) Consider the case a > 0, b = 0. Show that the first-order energy shift
vanishes ∀n. Write down an expression for the leading order energy shift
in terms of the unperturbed SHO eigenfunctions and explain why it doesn’t
vanish.
R
As above, ψn∗ (x)ax3 ψn (x) = 0, so the linear order correction in perturbation theory vanishes. The second order term is
δEn(2) =
X |hn|V̂ |mi|2
En − Em
m6=n
is nonzero because there are nonzero hn|V̂ |mi terms in the sum due to the
∗
fact that the product ψm
ψn can be odd in x if m is even and n is odd, or
vice versa. The product with x3 is then even, giving a nonzero integral.
3. Landau levels. A particle of mass m, charge −e moves on a rectangle of length
Lx , width Ly in a constant magnetic field B k ẑ.
(a) Write down the Hamiltonian in Landau gauge, A = Bxŷ.
H=
¡
¢¢
¢2
1 ¡
1 ¡ 2
p̂ − eA =
p̂x + p̂2y + e2 B 2 x2 − e py Bx + Bxpy
2m
2m
since [py , x] = 0
=
¢
1 ¡ 2
2e
p̂x + p̂2y − Bxp̂y + e2 B 2 x2
2m
c
(b) Show that [H, p̂y ] = 0. As a consequence we may write ψ(x, y) =
eiky y X(x), i.e. electrons have unconstrained motion in y-direction. If we
take periodic boundary conditions on the system, what are the allowed ky ?
Since py doesn’t act on x (commutes with x), all of the terms in H depend
only on powers of py itself and x, so [py , H] = 0. Allowed ky are ky =
2πν/Ly , where ν = 0, ±1, ±2, ..., such that eiky Ly = eiky ·0 = 1.
(c) Show that the Schrödinger equation for X is equivalent to a simple harmonic oscillator equation, and use what you know about the solution to
the SHO problem to calculate the energy levels here (”Landau levels”).
Express your answer in terms of ω, the analog of the SHO oscillation frequency; what is ω and what is its physical interpretation?
Note H doesn’t depend
in¢ y-direction, ψ =
¡ 2 on 2y 2 ⇒ free motion
1
2 2 2
iky y
e X(x)
µ ⇒ Hψ = 2m p̂x + ~ k¶y − 2e~ky B + e B x . Complete square:
¢2
¡ ¢2 ¡
p̂2x
y
H → 2m
, x0 = ~k
+ 12 m eB
x
−
x
. SHO Hamiltonian with effec0
m
eB
tive ω = eB/m = cyclotron frequency of classical orbit in perpendicular
magnetic field B.
(d) What is the degeneracy of each level? Write down an expression for the
magnetic field necessary to exactly fill the lowest level if there are N noninteracting electrons in the system. Neglect spin.
y
The value of ky determines the center of the orbit, x0 = ~k
, which however
eB
can’t be outside the sample, x0 ≤ Lx . Maximum ky , hence maximum ν is
obtained from the equality,
µ
¶
2πνmax
x0 = (~/eB)
= Lx
Ly
eBLx Ly
Φ
⇒ νmax =
=
h
Φ0