University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics C (130C) Winter 2013
Non-Assignment 8
Posted March 5, 2013
Due NEVER
Problem Set 8
This problem set is intended to give some examples of problems involving the ideas we
encountered in discovering quantum field theory. Do not hand it in. Do not worry that I
will ask such advanced questions on the final.
1. Landau levels.
Consider an electron moving in the xy plane, in a uniform constant magnetic field
~ = êz B.
B
~ always lies in the xy plane.
(a) Check that the classical Lorentz force F~Lor = ec ~v × B
We will ignore motion in the z direction for the rest of the problem. The (quantum)
Hamiltonian is
1
e
H=
Π2x + Π2y , Πi ≡ pi − Ai .
2m
c
For the vector potential we choose the (Landau) gauge
Ax = −By,
Ay = 0.
(b) Derive the equation of motion for the position expectation value using Ehrenfest’s
theorem, and verify that it gives the Lorentz force law:
m∂t2 h~xi = hF~Lor i.
(c) Show that the momentum px in the x direction is conserved.
(d) Verify that the eigenstates |kx i of px have wavefunctions of the form
ψkx (~x) ≡ h~x|kx i = eikx x φ(y).
(e) Show that when acting on ψkx , H reduces to
2
~
1
1
∂y + mωc2 (y − y0 )2
H=
2m i
2
and find ωc , the cyclotron frequency, and show that y0 =
1
ckx
.
eB
(f) Where have you seen the previous Hamiltonian before? What are the eigenvalues
of H? (In this context, they are called Landau levels.) Explain why they do not
depend on kx . Notice that y0 depends on kx .
(g) Now suppose we restrict the particles to live in a strip of length L in the xdirection and width W in the y direction. We will ignore the boundary conditions
in the y direction and use the wavefunctions above in our estimates below. Place
several electrons in the strip, and assume they only care about each other because
of the Pauli exclusion principle.
What are the possible values of kx ? [Hint: it is constrained by the fact that y
is bounded because y0 = y0 (kx ).] Estimate the maximum number of electrons
that fit in the lowest Landau level and the resulting electron density. Express this
number in terms of the number of flux quanta Φ ≡ WeBL through the sample.
(h) [extra credit] Show that a coherent state of this oscillator describes cyclotron
motion of the electron. That is h~xi moves in a circle of radius equal to the
Larmor radius rL = ωvc , where v is the electron velocity.
2. Heisenberg picture
Here we will try to understand in what sense the field momentum is π ∼ φ̇, and we
will explain the factor of iω by which π and φ differ.
All quarter we have been using what is called Schrödinger picture, where we evolve the
states in time
|ψ(t)i = U(t)|ψ(0)i = e−iHt/~ |ψ(0)i
a nd leave the operators alone. It is sometimes useful to define time-dependent operators by implementing the change of basis associated with U on the operators:
A(t) ≡ U(t)AU(t)† = e−iHt/~ Ae+iHt/~ .
First consider a simple harmonic oscillator,
1
1
p2
2 2
†
+ mω x = ~ω a a +
H=
2m 2
2
with
r
r
~
~
†
x=
a+a =
2Re (a) .
2mω
2mω
r
r
~ω
m~ω
p=i
a − a† =
2Im (a) .
2
2
(a) Show that
r
x(t) ≡ e−iHt xe+iHt =
2
~
2Re e−iωt a .
2mω
(b) Using the expression above, show that
p(t) ≡ e−iHt pe+iHt = m∂t x(t)
in agreement with what you would want from the Lagrangian formulation and
from classical mechanics.
The above was pretty simple. Now we consider a scalar quantum field
d + 1 = 1 + 1 dimensions:
Z
2 X
π(x)2 1 2 ~
d
~
+ µvs ∇φ · ∇φ
=
H= d x
~ωk a†k ak +
2µ
2
k
theory, in say
1
2
.
s
~ i~k·~x
~
e ak + e−ik·~x a†k ,
2µωk
k
r
1 X ~µωk i~k·~x
~
π(x) =
e ak − e−ik·~x a†k ,
i k
2
φ(x) =
X
(c) Show that
s
φ(t, x) ≡ e−iHt φ(x)e+iHt =
X
k
~ i~k·~x−iωk t
~
e
ak + e−ik·~x+iωk t a†k
2µωk
(d) Using the previous result, show that
π(t, x) ≡ e−iHt φ(x)e+iHt = µ∂t φ(t, x)
so that all is right with the world.
3. Complex scalar field and antiparticles
So far we’ve discussed scalar field theory with one real scalar field. The particles created
by this field are their own antiparticles.
To understand this statement, consider a scalar field theory in d+1 dimensions with two
real fields φ1 , φ2 . Organize them into one complex field Φ ≡ φ1 +iφ2 , with Φ? = φ1 −iφ2 .
Z
?
~ · ∇Φ
~ ? − V (Φ? Φ) .
S[Φ, Φ ] = dd xdt µ∂t Φ∂t Φ? − µv 2 ∇Φ
(a) Show that
Z
S[Φ] =
X 1
i=1,2
1
~ i · ∇φ
~ i
µ (∂t φi ) − µv 2 ∇φ
2
2
2
!
− V φ21 + φ22
.
That is, if V = 0, it is just the sum of two copies of the action of the theory we
considered.
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(b) Show by doing the Legendre transformation that the associated hamiltonian is
Z
1
d
?
2~
?
?
~
H= d x
ΠΠ + µv ∇Φ · ∇Φ + V (ΦΦ )
µ
where the canonical momenta are
Π=
with S =
R
∂L
∂L
= µΦ̇? , Π? =
= µΦ̇
∂ Φ̇
∂ Φ̇?
dtdd xL.
(c) This theory has a continuous symmetry under which Φ → eiα Φ, Φ? → e−iα Φ?
with α a real constant. Show that the action S does not change if I make this
replacement.
(d) The existence of a continuous symmetry means a conserved charge – a hermitian
operator which commutes with the Hamiltonian, which generates the symmetry.
Show that
Z
q ≡ i (Φ? Π? − ΠΦ)
generates this transformation, in the sense that
δΦ = iαΦ = iα[q, Φ]
and similarly for Φ? . Show that [q, H] = 0.
(e) For the case where V (ΦΦ? ) = m2 ΦΦ? the hamiltonian is quadratic. Diagonalize it
in terms of two sets of creation operators and annihilation operators. You should
find something of the form
s
~ X 1 ikx
e ak + e−ikx b†k
Φ=
√
2µ k
ωk
(f) Write the canonical commutators
[Φ(x), Π(x0 )] = i~δ(x − x0 ), [Φ(x), Π? (x0 )] = 0
(and the hermitian conjugate expressions) in terms of a and b.
(g) Rewrite q in terms of the mode operators.
(h) Evaluate the charge of each type of particle created by a†k and b†k
(i.e. find [q, a† ]).
I claim that the particle created by a† is the antiparticle of that created by b†
in the sense that they have opposite quantum numbers. This means that we can
add terms to the hamiltonian by which they can annihilate each other without
breaking any symmetries.
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