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PHY4605–Introduction to Quantum Mechanics II
Spring 2004
Test 1 Solutions
February 6, 2004
1. Short Answer. Must attempt (only) 3 of 4. Circle answers to be graded.
(a) Explain the Einstein-Podolsky-Rosen ”paradox”. How does the Copenhagen school of quantum mechanics escape the apparent violation of
causality?
Simplest example: spin 0 particle decays in lab frame into two spin–1/2
particles which recoil in opposite directions. Quantum Mechanics says
spin state must be |s = 0i = √12 (↑↓ − ↓↑). When observer at point A
far from B measures spin of particle 1 to be up, wave function colapses
⇒ 2 is down with probability 100%, i.e. 1 influences 2 instantaneously
apparently contradicting relativistic causality. Copenhagen: no info can
be transferred due to statistical nature of measurement.
(b) What’s the DC Stark effect? When is it linear and when quadratic in the
applied field?
Stark effect is the shift of the spectral lines in an atomic gas subject to an
external constant electric field. In the case of a nondegenerate level in an
atom, the first-order shift is zero, so the shift is quadratic in the field. If
the line is degenerate, a linear term may result.
(c) 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
2
need to move through region of B 6= 0 to feel an effect. Not true in
Quantum Mechanics.
(d) In terms of what fundamental ratio is the conductivity quantized in the
integer quantum Hall effect? Sketch the energy eigenstates for a 2D electron gas in a perpendicular magnetic field B, identify the spacing, and
comment on the significance of a completely filled level for observation of
the effect.
σxy
E
4
3
2
n=1
µ 2¶
e
=n
h
(1)
hω
If level completely filled, no final states available for electron scattering if temperature is low. Therefore scattering time τ → 0.
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2. Landau Levels/Ring. A particle of mass m, charge e moves on a rectangle
of length Lx , width Ly in the x − y plane in a constant magnetic field B k ẑ.
(a) Write down the Hamiltonian in Landau gauge, A = Bxŷ.
H=
¡ e ¢2 2 2 e ¡
¢¢
1 ¡
e ¢2
1 ¡ 2
p̂ − A =
p̂x + p̂2y +
B x − py Bx + Bxpy
2m
c
2m
c
c
since [py , x] = 0
=
¡ e ¢2 2 2 ¢
2e
1 ¡ 2
p̂x + p̂2y − Bxp̂y +
B x
2m
c
c
(b) Show that this problem is equivalent to a simple harmonic oscillator, and
use what you know about the solution to the SHO problem to calculate
the energy levels. Express your answer in terms of ω, the analog SHO
oscillation frequency; what is ω and what is its physical interpretation?
Note H doesn’t depend
¡ 2 on 2 y2 ⇒2e free motion
¡ e ¢2 2 in2 ¢ y-direction, ψ =
1
iky y
e X(x) ⇒ Hψ = 2m p̂x + ~ ky − c ~ky + c B x . Complete square:
µ
¶
¡ eB ¢2 ¡
¢2
p̂2x
y
1
H → 2m + 2 m mc x − x0
, x0 = ~ck
. SHO Hamiltonian with effeceB
tive ω = eB/mc = cyclotron frequency of classical orbit in perpendicular
magnetic field B.
(c) Now instead of solving the problem exactly, treat A as weak. Show H '
p̂2 /2m + (e/2mc)B · L. [Hint: symmetric gauge A = r × B/2 is more
convenient.]
¶2
µ
·
µ
¶¸
1
1
e
e
H=
p̂ − A '
p̂ −
p̂ · A + A · p̂
2m
c
2m
c
since
(2)
4
·
¸
1
2e
A ⊥ r, ∇ · A = A · ∇, so H '
p̂ − A · p
2m
c
(3)
symmetric gauge
·
¸
¡
¢
¢
1
2e 1 ¡
e
p̂2
e
p̂
2
H=
p̂ −
r × B · p̂ =
+
B · r × p̂ =
+
B · L̂. (4)
2m
c 2
2m 2mc
2m 2mc
(d) Use the result (1) to discuss a particle of mass m and charge e constrained
to move on a ring of radius a. Find
eigenvalues and eigenfunc³ the
´ energy
∂ψ
1 ∂
1 ∂2ψ
2
tions. [Hint: in 2D , ∇ ψ = ρ ∂ρ ρ ∂ρ + ρ2 ∂φ2 ]
Constrained to ring ⇒ ψ = ψ(φ), note Bkẑ
µ 2 2
¶
µ
¶
−~ ∇
e
−~2 ∂ 2
e
+
B · L̂ ψ = 2ma2 ∂φ2 + 2mc B L̂z
ψ
|{z}
2m
2mc
−i~∂/∂φ
¶
µ
−~2 ∂ 2
i~eB ∂
= 2ma2 ∂φ2 − 2mc ∂φ ψ = Eψ
(5)
(6)
Solution:
ψ = eibφ ,
~2 2 ~ω
b +
b = E, b = 0, 1, 2, · · · for single–valuedness.
2ma2
2
(7)
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3. Stern-Gerlach effect. A Stern-Gerlach setup is defined to be a region of
inhomogeneous magnetic field, with field gradient pointing in the same direction
as field itself.
(a) Show that a neutral particle with spin S k x̂ entering the SG setup shown
feels a force in the x̂ direction. Explain briefly how you could create such
a field, and what is wrong with the picture.
F = −∇U = ∇(µ · B) ∼ Sx (∇Bx ) ∼ Sx |∇Bx |x̂
sign depends on Sx . Figure indicates B, ∇B point in same direction.
Could do this approximately e.g. with coil whose winding density increases
in x direction. However there must be some small transverse gradient to
satisfy ∇ · B = 0.
(b) The 1st particle entering the SG setup shown is a neutron (spin 1/2),
prepared in a pure spin state in accelerator A with S k x̂. What is the
probability it is recorded in particle detector D1 ? Detector D2 ?
Since particle has Ŝkx̂, ⇒ it is in pure eigenstate of Ŝx with eigenvalue
+~/2 ⇒ it will be observed in detector D2 with 100% probability.
(c) The 2nd particle is a ρ meson (charge 0, spin 1) prepared in a spin up
(m = 1) state (the spin quantization axis is the ẑ direction, as usual).
What is the probability the ρ is detected in D1 ? D2 ? (Note any particle
whose trajectory lies on the y-axis is missed by both detectors).
 
1
 
 
Write |z1i = 0 in terms of eigenstates of Ŝx for S = 1 particle.
 
0


0 1 0




Work out mattrix elements hm|Ŝx |m0 i for S = 1, find Sx = √~2 1 0 1


0 1 0
6

 
 
1
1
√ 
 
√1 
 
 
 
eigenvectors |x1i = 12  2, |x0i = √12  0 , |x, −1i = 21  2
 
 
 
1
−1
1
P
1
1
1
√ |x, 0i + |x, −1i ⇒ probability
|z1i =
m hx|m|z1i|xmi = 2 |x1i +
2
2
2
to be detected at D1 or D2 is (1/2) = 1/4.

(d) Now a camera with a very narrow-angle lens is placed at position C pointing along the −y axis and another ρ in the same state as (c) is injected into
the SG region. The camera does not observe the particle during the time
it takes for the particle to pass through the setup. What is the probability
it is recorded in D1 ? D2 ?
If particle observed not to travel a long y-axis, probability must be 1/2 to
be either in |x1i or |x, −1i ⇒ wave function collapses.
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4. Molecule. A molecule is made up of three identical X atoms arranged in
an equilateral triangle. An electron is added to the molecule to form an X3−
ion. Given that the elctron has zero energy when it sits on any one of the i
X-atoms, hi|H0 |ji, with i = 1, 2, 3, · · · and that it hops between atoms with
matrix element hi|H0 |ji = −a, for i 6= j, and taking a > 0,
(a) Find the eigenvalues of H0 . State the degeneracy of each level.

 0

H0 = −a

−a

−λ

£
¤

det H0 − λ = 0 = −a

−a
−a
0
−a
−a
−λ
−a

−a

−a ⇒

0

−a


2
2
2
−a = λ(λ − a ) − 2a(aλ − a )

−λ
= (λ − a)[λ(λ + a) − 2a2 ] = (λ − a)2 (λ + 2a) ⇒
E0 = −2a, E1 = a (degeneracy 2)
(b) Find the eigenvectors of H0 . [Be sure to express your eigenvectors in terms
of a basis consisting of states where the electron is located with probability
1 on a given X-atom.]
Ground state |ψ0 i:
 
 
 
y + z = 2x
x
x
1
 
 

 
1
 
 

 
−a 0 −a  y  = −2a y  ⇒ x + z = 2y ⇒ |ψ0 i = √ 1 .
 
 

3 
x + y = 2z
z
z
−a −a 0
1

0 −a −a
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Excited states |ψ1 i, |ψ2 i:

 
 
y + z = −x
 0 −a −a  x
x

 
 
−a 0 −a  y  = a y  ⇒ x + z = −y ⇒

 
 
−a −a 0
z
z
x + y = −z
 
 
0
−2


1 
1 
 
 
|ψ1 i = √  1  , |ψ2 i = √  1 
2 
6 
−1
1
Note chosen so hψ0 |ψ1 i = hψ0 |ψ2 i = hψ1 |ψ2 i = 0.
(c) Suppose an electric field in the ẑ-direction is now applied as shown, so that
the potential energy for the atom on top is lowered by an amount b, with
b ¿ a (this is the only effect of the E-field!). Calculate the shifts in the
energy of the three zero-field eigenstates.


−b 0 0


V̂ =  0 0 0


0 0 0
 
1
´ 1   1
1 ³
 
(1)
δE0 = √ 1 1 1 V̂ √ 1 =
3
3  3
1
 
0
³
´ 1  
1


(1)
δE1 = √ 0 1 −1 V̂ √  1  = 0
2
2 
−1
 
−2
³
´ 1  
1
2b
 
(1)
δE2 = √ −2 1 1 V̂ √  1  = −
3
6
6 
1
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N.B. if now chose eigenvectors for |ψ1 i, |ψ2 i such that hψ1 |V̂ |ψ2 i 6= 0, you
have to diagonalize. With the choice above it’s not necessary.
(d) Calculate the ground state wave function in the perturbed system to 1st
order.
0
z }| {
√
√
2b/ 18
hψ1 |V̂ |ψ0 i |ψ1 i hψ2 |V̂ |ψ0 i|ψ2 i
− 2b
δψ0 =
+
=
|ψ2 i =
|ψ2 i ⇒
E0 − E1
E0 − E1
(−2a) − a
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√
2b
new ψ̃0 = c(|ψ0 i −
|ψ2 i)
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where c is a normalization factor. Your answer might look different depending on how you chose ψ1 , ψ2 .