University of California at San Diego – Department of Physics – Prof. John McGreevy
Quantum Mechanics C (130C) Winter 2014
Assignment 7
Posted March 3, 2014
Due 11am Thursday March 13, 2014
This is the last problem set.
Problem Set 7
1. Stationary phase. [extra credit]
P
inθ
? Consider both the case when θ is an integer multiple of 2π/N ,
(a) What is N
n=1 e
and when it is not.
P
iϕi
where {ϕi } are chosen randomly
(b) Add together some random phases, i.e. N
i=1 e
and uniformly in the interval [0, 2π). (The Mathematica command RandomReal
is useful here.) How does the magnitude of the answer depend on N as N grows?
(c) Evaluate numerically the integral
Z
I(N ) =
dxeiN V (x)
with V (x) = (x2 − 1)2 − 12 x3 for several values of N , and compare to the answer
from the stationary-phase approximation.
[The range of integration shouldn’t matter too much, but you can take x ∈
(∞, ∞). If you are worried about convergence of the integral, you can define
it by
Z
∞
dxee
I(N ) =
i( π
2 −)
N V (x)
−∞
for arbitrarily small positive epsilon.]
2. Path integral for a free particle
Consider the path integral description of the quantum mechanics of a free particle in
one dimension. The action is
Z tf
m
S[x] =
dt ẋ2 .
2
0
(a) What is the equation of motion 0 =
δS
?
δx(t)
(b) Find the classical solution x(t) with x(t = 0) = x0 and x(t = tf ) = xf .
1
(c) Evaluate the action for the classical solution S[x], and evaluate the stationaryphase approximation to the path integral for the quantum propagator
Z
U (xf , tf ; x0 , 0) = hxf |U(tf )|x0 i = [dx]eiS[x] ' Usc ≡ eiS[x] .
R
(d) Derive the Hamiltonian associated to the action S = dtL above.
[That is, find p = ∂L
and eliminate ẋ in H(x, p) = pẋ − L.]
∂ ẋ
(e) Treating this Hamiltonian quantum mechanically, evaluate the exact quantum
propagator,
U (xf , tf ; x0 , 0) = hxf |U(tf )|x0 i = hxf |e−itf H |x0 i.
[One way to do this is to use the fact that U here solves the ODE
i∂t U (x, t; x0 , 0) = −
~2 2
∂ U (x, t; x0 , 0)
2m x
with the initial condition U (xf , 0; x0 , 0) = δ(xf − x).]
Compare with the semiclassical approximation Usc defined above.
3. A charged particle, classically. [Extra credit]
This problem is an exercise in calculus of variations, as well as preparation for our
discussion of particles in electromagnetic fields.
Consider the following action functional for a particle in three dimensions:
Z
m
e
~
S[x] = dt
~x˙ 2 − eΦ(~x) + ~x˙ · A(x)
.
2
c
(a) Show that the extremization of this functional gives the equation of motion:
e
δS[x]
= −mẍi (t) − e∂xi Φ(x(t)) + ẋj Fij (x(t))
i
δx (t)
c
where Fij ≡ ∂xi Aj − ∂xj Ai . Show that this is the same as the usual CoulombLorentz force law
~
v
~
~ + ×B
F~ = e E
c
with Bi ≡ ijk Fjk .
(b) Show that the canonical momenta are
Πi ≡
∂L
e
i
=
m
ẋ
+
Ai (x).
∂ ẋi
c
R
Here S = dtL. (I call them Π rather than p to emphasize the difference from
the ‘mechanical momentum’ mẋ.) Show that the resulting Hamiltonian is
2
X
1 e
i i
H≡
ẋ Π − L =
Πi − Ai (x(t)) + eΦ.
2m
c
i
2
4. Coherent states.
Consider a quantum harmonic oscillator. The creation and annihilation operators a†
and a satisfy the algebra
[a, a† ] = 1
and the vacuum state |0i satisfies a|0i = 0.
Coherent states are eigenstates of the annihilation operator:
a|αi = α|αi.
(a) Show that
|αi = e
−|α|2 /2 αa†
e
−|α|2 /2
|0i = e
∞
X
αn
√ |ni
n!
n=0
is an eigenstate of a with eigenvalue α. (a is not hermitian, so its eigenvalues
need not be real.)
(b) Coherent states with different α are not orthogonal. (a is not hermitian, so its
2
eigenstates need not be orthogonal.) Show that |hα1 |α2 i|2 = e−|α1 −α2 | .
(c) Compute the expectation value of the number operator n = a† a in the coherent
state |αi.
(d) Time evolution acts nicely on coherent states. The hamiltonian is H = ~ω a† a + 21 .
Show that a coherent state evolves into a coherent state with an eigenvalue α(t):
e−iHt |αi = e−iωt/2 |α(t)i
where α(t) = e−iωt α.
5. Two coupled spins. [based on Le Bellac problem 6.5.4]
Consider a four-state system consisting of two qbits,
H = span{|1 i ⊗ |2 i ≡ |1 2 i, =↑z , ↓z }.
(a) For each qbit, define σ ± ≡ 21 (σ x ± iσ y ). (These are raising and lowering operators for σ z : [σ z , σ ± ] = ±2σ ± . Show this.) Show that
~1 · σ
~ 2 = 2 σ1+ σ2− + σ1− σ2+ + σ1z σ2z .
σ
~1 · σ
~ 2 on the basis states
(b) Determine the action of the operator σ
| ↑↑i, | ↑↓i, | ↓↑i, | ↓↓i.
(c) Show that the four vectors
1
1
|0, 0i = √ (| ↑↓i − | ↓↑i) , |1, 1i ≡ | ↑↑i, |1, 0i ≡ √ (| ↑↓i + | ↓↑i) , |1, −1i ≡ | ↓↓i
2
2
~1 · σ
~ 2 with eigenvalues 1 or −3.
are orthonormal and are eigenvectors of σ
3
~ 2 )2 and Jz ≡ σ1z + σ2z and
(d) Show that they are also eigenvectors of J2 ≡ (~
σ1 + σ
find their eigenvalues.
(e) Consider the operator
1
~1 · σ
~ 2)
(1 + σ
2
acting on the two spins. Show that P1,2 acts by exchanging the states of the two
spins:
P1,2 |1 2 i = |2 1 i .
P1,2 ≡
6. Spin chains and spin waves. [Related to Le Bellac problem 6.5.5 on page 200]
A one-dimensional ferromagnet can be represented as a chain of N qbits (spin-1/2
particles) numbered n = 0, ...N − 1, N 1, fixed along a line with a spacing ` between
each successive pair. It is convenient to use periodic boundary conditions (as in HW 2
problem 2), where the N th spin is identified with the 0th spin: n + N ≡ n. Suppose
that each spin interacts only with its two nearest neighbors, so the Hamiltonian can
be written as
N −1
1 X
1
~n · σ
~ n+1 .
σ
H = N J1 − J
2
2 n=0
where J is a coupling constant determining the strength of the interactions.
(a) Show that all eigenvalues E of H are non-negative, and that the minimum energy
E0 (the ground state) is obtained in the state where all the spins point in the same
direction. A possible choice for the ground state |Φ0 i is then
|Φ0 i = | ↑z in=0 ⊗ | ↑z in=1 ⊗ ... ⊗ | ↑z iN −1 ≡ | ↑↑ ... ↑i.
(b) Show that any state obtained from |Φ0 i by rotating each of the spins by the same
angle is also a possible ground state.
P
~ n commutes with the Hamilto[Hint: the generator of spin rotations ~J ≡ n σ
nian.]
[Cultural remark: the phenomenon of a ground state which does not preserve a
symmetry of the Hamiltonian is called spontaneous symmetry breaking. ]
(c) Now we wish to find the low-energy excitations above the ground state |Φ0 i. Show
that H can be written
H = N J1 − J
N
−1
X
Pn,n+1 = J
n=0
where
N
−1
X
(1 − Pn,n+1 ) .
n=0
1
~n · σ
~ n+1 ) .
(1 + σ
2
Using the result of the previous problem, show that the eigenvectors of H are
linear combinations of vectors in which the number of up spins minus the number
of down spins is fixed. Let |Ψn i be the state in which the spin n is down with all
the other spins up. What is the action of H on |Ψn i?
Pn,n+1 ≡
4
(d) We are going to construct eigenvectors |ks i of H out of linear combinations of the
|Ψn i. Let
N
−1
X
|ks i =
eiks n` |Ψn i
n=0
with
2πs
, s = 0, 1, ...N − 1 .
N`
Show that |ks i is an eigenvector of H and determine the energy eigenvalue Ek .
Show that the energy is proportional to ks2 as ks → 0. This state describes an
elementary excitation called a spin wave or magnon with wave-vector ks .
ks =
7. A damped quantum harmonic oscillator [extra credit] [from Preskill]
In this problem we extend the amplitude damping channel which we described for
a qbit to a harmonic oscillator. This arises if we couple a harmonic oscillator with
creation operator a to an environment via an interaction like
X
∆H =
λi ab†i + h.c.
(1)
i
Here we have modeled the environment by a collection of oscillators whose modes are
created by b†i .
For the first part of the problem, let’s instead model the environment by two-state
system. Consider a situation where evolution over a time dt is given by
p
√
U|ni ⊗ |0iE = 1 − p|ni ⊗ |0iE + pa|ni ⊗ |1iE .
In this expression we suppose that p 1 and we will work only to leading order in p.
(a) Note that a|ni ∝ |n − 1i. Find the proportionality constant.
(b) Find the Kraus operators M0 , M1 for the evolution above and show that they
satisfy the unitary condition 1 = M0 M†0 + M1 M†1 to leading order in p.
(c) Find the evolution of the density matrix.
To do better than the above, let’s introduce a little more technology. Suppose we are
studying an environment which forgets very rapidly. Then we may take the limit where
dt → 0 and write a differential equation for the density operator. For sure we have
ρ(dt) = Mµ ρ(0)M†µ = ρ(0) + O(dt)
and so√one of the Kraus operators is M0 = 1 + O(dt), and the others are small, of
order dt, describing transitions that may occur with probability proportional to dt.
Let’s make the ansatz
√
Mi =
dtLi , i = 1, 2, 3
M0 = 1 + dt (−iH + K)
5
where H is the (hermitian) Hamiltonian for the subsystem (H = ~ω a† a + 12 for the
SHO) – this term describes the unitary evolution that would happen if the system
were closed – and K is some new Hermitian operator describing the damping. Each of
H, Li , K is zeroth order in dt.
(d) Determine K in terms of the Li using the unitarity condition.
You should find that
X
1 †
1 †
†
∂t ρ = L[ρ] ≡ −i[H, ρ] +
Li ρLi − Li Li ρ − ρLi Li .
2
2
i
(2)
The superoperator on the RHS is called the Lindbladian.
(e) Now we return √
to the situation above in (1). In this case there is one lindblad
operator, L1 = Γa. (Think of Γ as the rate for one of the environment oscillators
b to decay from the first excited state to the groundstate.) Apply the evolution in
(2) to the density matrix, and show that the occupation number of the oscillator
n ≡ a† a satisfies
∂t hn(t)i = tra† aρ̇ = −Γhn(t)i .
Integrate this equation and show that the oscillator is damped.
(f) [Note: It is possible to do this part starting from eqn (2), even if you did not do
the previous parts.]
Now we will think about decoherence. The fact that the oscillator is coupled to
the environment via the operator a suggests that it is eigenstates of a that will
be the classical states. This turns out to be correct. As we saw in problem 4,
eigenstates of a are coherent states.
Suppose we begin in a Schrödinger’s cat state of the form √12 (|α1 i + |α2 i), with
|α1 − α2 | 1. The initial density matrix is
1
(|α1 i + |α2 i) (hα1 | + hα2 |) .
2
Show that, as a result of the coupling to the environment above, the off-diagonal
terms decay rapidly:
ρ(0) =
ρ(t) =
1
1 1
2
(|α1 ihα1 | + |α2 ihα2 |) + e− 2 Γt|α1 −α2 | (|α2 ihα1 | + h.c.) .
2
2
So the decoherence rate is Γdec = 21 |α1 − α2 |2 Γ, much faster than the damping
rate, if the coherent states are very different.
[First hint: It may be useful to write
X
?
†
ρ(0) =
βi βj eαi a |0ih0|eαj a
ij=1,2
with βi =
1
2
√1 e− 2 |αi | ,
2
and to make the ansatz
X
?
†
ρ(t) =
βij (t)eαi a |0ih0|eαj a .
ij=1,2
6
The claim is that for tΓ 1,
βii (t) ∼ |αi |2 ,
but
1
β12 (t) ∝ e− 2 |α1 −α2 |
]
7
2 Γt
α1 α2? .