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Quantum Mechanics (I): Homework 5
Due: December 16
Ex.1 Consider the setup of the AB effect as shown in Fig. 1. Instead of setting up a solenoid behind the slits, we
now have a strip of magnetic field behind the slits. If L is very larger than d, we may approximate any two paths
that come out from different slits as parallel lines.
(a) 10 Find the shift ∆x of the diffraction pattern.
(b) 5 The shift ∆x can be viewed as deflection of all trajectories by the small angle α = ∆x/L. Show that by assuming
all incident particles have momentum parallel to y, α can be also understood in terms of the classical Lorentz force
F = qv × B/c.
Fig. 1
L
d
B
W
Ex. 2 10 Consider a particle in a vector potential A(r). Evaluation of the path integral for this particle requires to
consider the following term in the action
j=N
ie X
(xj+1 − xj ) · A(x).
h̄c j=0
The problem is what we should interpret x. Should we use x = xj or xj+1 ? Show that if x = xj+1 + α(xj − xj+1 ),
the resulting Schrödinger equation is
2
∂
h̄
e
1 ih̄e
ih̄ ψ(x, t) =
∇− A ψ(x, t) + V ψ(x, t) + (α − )
ψ(x, t)∇ · A.
∂t
i
c
2 mc
Therefore, one sees that only when α = 1/2, the resulting Schrödinger equation is correct and gauge-invariant.
Ex. 3 10 Consider the nth eigenstate of a harmonic oscillator, what is ∆x∆p?
∂
Ex. 4 10 By re-expressing the creation operator a† in terms of p and ∂p
, show that the Fourier transformation of
the nth eigenstate of a harmonic oscillator is
1/4
1
p
hp|ni = (−i)n
exp(−p2 /2mh̄ω)Hn ( √
).
πh̄mω22n (n!)2
mh̄ω
Ex. 5 Charged Harmonic Oscillator
(a) 5 Consider a charged 1D harmonic oscillator (mass = m, charge = q, frequency = ω) placed in a constant electric
field E = − dφ
dx . If we set φ(0) = 0, what is the Hamiltonian?
(b) 5 Rewrite the Hamiltonian in terms of the creation and annihilation operators (i.e., a† and a).
(c) 10 The Hamiltonian obtained in (b) is not ready to be solved. Can you redefine new creation and annihilation
operators ( b† and b ) in terms of a† and a so that the Hamiltonian is purely quadratic in b† and b?
(d) 5 Identify the zero-point energy from the results of (c).
(e) 5 The physics behind the results of (c) is actually quite simple: a constant electric field only shifts the equilibrium
position of a harmonic oscillator to a new position. What is the new equilibrium position?
Ex.6 5 From the expression
n
d
2
Hn (x) = exp(x /2) x −
exp(−x2 /2),
dx
2
show that the Hermite polynomials can be also rewritten as
Hn (x) = (−1)n exp(x2 )
d
dx
n
exp(−x2 ),
and the coefficient of the maximal power is 2n .
Ex.7 10 Page 212, ex. 7.4.3.
Ex.8 (a)10 Ex.8.6.2. (b)10 Ex.8.6.3.
Ex.9 The Landau Levels
When a particle of mass m and charge q is placed in a uniform magnetic field B, its energy spectrum will change
from continuous one to discrete spectrum. The resulting discrete spectrum is known as the Landau levels.
(a) 5 If we align B along z direction, show that the vector potential A can be chosen to be
A=
B
(−yx̂ + xŷ).
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This is known as the symmetry gauge. You can verify that other choices such as A = Bxŷ (known as the Landau
gauge) are also good.
(b) 10 When we align B along z direction, the particle is free along z direction. Therefore, we can ignore the z
direction. Let us thus consider the Schrödinger equation in two dimension (x and y). Using the symmetry gauge,
show that the Hamiltonian can be rewritten as that of a simple harmonic oscillator
1
P2
+ mωc2 Q2 .
2m 2
H=
Here ωc = qB/mc is the cyclotron frequency, and Q ≡ (cp̂x + qyB/2)/qB = cmvx /qB and P ≡ p̂y − qBx/2c = mvy .
P and Q satisfy the commutation relation [Q, P ] = ih̄ (i.e., vx and vy no longer commute in the presence of magnetic
field). Therefore, the energy spectrum of the particle is the same as that of a simple harmonic oscillator. The resulting
spectrum is known as the Landau Levels.
(c)10 Follow (b), find the normalized ground state wavefunction.
(d) 15 Suppose now that in addition to the above magnetic field, one applies an electric field E in the x direction.
By using the Landau gauge, find the energy spectrum.
Ex. 10
(a) 10 The denisty matrix operator at temperature T is defined by
ρ̂ =
e−Ĥ/kB T
,
Z
where Z is the partition function and is equal to T r(e−Ĥ/kB T ). To find the matrix element of ρ̂, one needs to find
ρ(x0 , x) ≡ hx0 |e−Ĥ/kB T |xi. Consider a special case in which Ĥ = p̂2 /2m + V (x̂) describes a single particle in the
potential V (x) = ax, find the corresponding ρ(x0 , x).
(b) 10 Neutron interferometry due to gravity
g
Using results of (a), consider now an monoenergetic beam of neutrons whose wavelength of matter wave is λ =1.445Å.
As shown in the above figure, the beam is incident at point A and is splitted into two parts that travel along two
different paths ABD and ACD and finally meet at D. Here ABCD is a rectangular with AB being 2.709cm and AC
being 3.885cm. It is found that by changing the angle φ, the intensity of neutron at D changes as shown in the right
figure. Deduce from these data the value of acceleration g due to gravity .
~ r) is given by
Ex. 11 (a) 5 Suppose that the propagator of a particle in the presence of a vector potential A(~
0
0
~
~
~
K(~r, t ; ~r, t). How does K(~r, t ; ~r, t) change if A(~r) undergoes the gauge transformation, A(~r) → A(~r) + ∇Λ(~r) ?
(b) 5 Suppose that the particle in (a) carries mass m and charge q. If the particle now moves in the outside of a cylinder
of radius R. The magnetic field Bis uniform inside the cylinder and vanishes outside the cylinder. Furthermore, the
electron is blocked from moving into the cylinder. What are allowed magnitudes of B?
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Bonus (+1) In this problem, I will guide you to find the propagator for the harmonic oscillator.
(a) Let us first re-analyze the free particle case. Show that one can rewrite
Z
Z
Z
m (N +1)/2
i
exp −ψ T M ψ
hxb tb |xa ta i = e h̄ Scl lim
dx1 dx2 · · · dxN
N →∞
2πih̄∆t
i
≡ A(tb − ta ) e h̄ Scl ,
where Scl is the classical action that we found in class [= im(xb − xa )2 /2h̄(tb − ta )], ∆t = (tb − ta )/(N + 1),
x1
 x2

ψ= ·
 ·
xN




,

and ψ T is the transpose of ψ. M is a N × N matrix. Find M and show that for any general real symmetric matrix
M
Z
Z
Z
π N/2
dx1 dx2 · · · dxN exp −ψ T M ψ = √
,
det M
(Hint: diagonalize M first.) Let IN ≡ det M when M is a N × N matrix. By finding the recursion relation for IN ,
calculate IN . From here, one concludes
r
m
.
A(tb − ta ) =
2πih̄(tb − ta )
(b) Following the same procedure outlined in (a), show that for the harmonic oscillator
r
mω
A(tb − ta ) =
.
2πih̄ sin ω(tb − ta )
Here ω is the natural frequency of the harmonic oscillator. Note that Scl was already found in Hw.4 Ex.4 and ω → 0,
one recovers the result for free particles.
(c) Now, consider a particle described by Ĥ = p̂2 /2m − Fx̂. By rewriting the path integral in the momentum space,
find the evolution operator U (x, t; x0 , t0 ) ≡ hx| exp[ h̄i H(t − t0 )]|x0 i.