Download Massachusetts Institute of Technology

Massachusetts Institute of Technology
Physics Department
Physics 8.321
Quantum Theory I
Fall 2006
October 9, 2006
Assignment 6
Due October 20, 2006
Announcements
• There will be a “makeup” lecture on Friday, October 13, from 12:30–2:00pm in 2-105.
• The 8.321 midterm exam will take place in class on October 31 (Hallowe’en). It will
be an hour and a half exam.
• Here’s some information about the schedule for the rest of the term:
As previously announced, there will be additional makeup lectures on Nov. 3 and 17.
There will be no lecture on Tuesday, Nov. 21 (the Tuesday before Thanksgiving).
There will be no lecture on Thursday, Nov. 30.
Problem set 11 will be distributed on Nov. 13 and due on Dec. 1. The last problem
set, # 12, will be distributed on Nov. 27, and due Dec. 8.
Reading topics for this period
• Harmonic oscillator; coherent states; energy-time uncertainty relation; the propagator.
Reading Recommendations 6
• The harmonic oscillator is discussed in almost every textbook. Sakurai §2.3; Shankar
§7; and Gottfried & Yan §4.2.
• Coherent states are (for some unknown reason) completely omitted from Sakurai.
They are discussed in Gottfried & Yan §4.2(d).
• Coherent states are also discussed in the 8.321 lecture notes on time evolution in
quantum mechanics (posted on the website).
• The propagator in quantum mechanics is discussed in Sakurai §2.5 (which continues
on to derive the path integral — next week’s subject in 8.321); similar material is
covered in Gottfried & Yan §2.6.
1
MIT
8.321 Quantum Theory I
Fall 2006
2
Problem Set 6
Topics covered in the problems
• Two state systems in isolation and in a time dependent external field.
• The harmonic oscillator: two dimensions.
• The harmonic oscillator: ladder operator algebra.
1. Benzene — A Two State System
Benzene is a cyclic hydrocarbon with composition C6 H6 . The six carbon atoms are
arranged in a ring with alternating single and double bonds. Each carbon has a single
hydrogen attached to it. There are two distinct arrangements of the single and double
bonds as shown in the figure below (the hydrogen bonds are not shown). Call these
states |Ai and |Bi
and
denote an arbitrary state as |ψi = ψA |Ai + ψB |Bi, or for
ψA
simplicity |ψi →
ψB
A
B
Figure 1: A cartoon of the benzene molecule
We will ignore any electronic, vibrational, or rotational excitation of the benzene
molecule, and concentrate on the dynamics of |Ai and |Bi. The expectations of the
Hamiltonian in the states A and B are equal: hA|Ĥ|Ai = hB|Ĥ|Bi = E0 .
It is an experimental fact that a molecule prepared in the state |Ai, will after a short
time ∆t, have a small amplitude proportional to ∆t, to be found in the state |Bi.
Suppose this amplitude is −ic∆t (to first order in ∆t), where c is a real, positive
constant.
(a) Write the (2 × 2 matrix) Hamiltonian describing the benzene molecule two state
system in the AB basis.
(b) Find the eigenstates and eigenenergies of the benzene molecule.
cos θ
(c) Suppose a molecule is initially in the state |ψ(0)i =
(in the AB basis) at
sin θ
time t=0. What is the expectation value of its energy? What is the probability
that a measurement of its energy would yield the lower of the two eigenvalues
you found in part (b)? Is this probability time dependent?
MIT
8.321 Quantum Theory I
Fall 2006
3
(d) Suppose a molecule is prepared in the |Bi state at time t = 0. What is the
probability that it will be measured to be in the |Ai state as a function of time.
Graph your answer.
2. Two Dimensional Harmonic Oscillator
~ =
Consider a particle moving in two dimensions, bound in a harmonic potential, V (X)
1
2
2
~
2 mω X .
~ and P~ ) and show that
(a) Introduce operators a1 and a2 (define them in terms of X
the Hamiltonian can be written
H = ~ω(a†1 a1 + a†2 a2 + 1)
(b) Show that any operator of the form a†i aj (i, j = 1, 2) commutes with H.
So there are four Hermitian combinations of a†i aj that commute with the Hamiltonian.
They can be diagonalized along with H and are therefore observable attributes of
energy eigenstates. The aim of this problem is to interpret them.
To enumerate them is convenient to define a two component “vector”
a1
a=
a2
~ ≡ a †~σ a, where ~σ are the Pauli matrices.
and consider the operators A
(c) Which is of the Ak is related to the angular momentum L = ~x × p~?
(d) Apparently L and H can be simultaneously diagonalized. Write
their eigenstates
as polynomials in a†1 and a†2 (acting on the ground state 0 ) for E = 2~ω and
E = 3~ω.
(e) So far we have a physical interpretation of only one of the operators Ak , the one
associated with L. Show that the other two are proportional to components of
the operator Qij defined by
Qij =
mω 2
1
Pi Pj +
Xi Xj , i, j = 1, 2
2m
2
Clearly the trace of Qij is the Hamiltonian, and it is symmetric. So the parts of Q
that still need interpretation are Q11 − Q22 and Q12 = Q21 . To gain insight, let’s
study the dynamical variable Qij classically.
(f) The general solution to the classical oscillator is an ellipse. First assume that
the ellipse is aligned with respect to the x − y axes:
X1 ≡ x(t) = ξ cos ωt
X2 ≡ y(t) = η sin ωt
(1)
MIT
8.321 Quantum Theory I
Fall 2006
4
2ξ and 2η are the lengths of the axes of the ellipse. Now rotate the ellipse by
an angle θ so that it is no longer oriented in a special direction. Show that the
classical trajectory is now
x(t) = ξ cos θ cos ωt − η sin θ sin ωt
y(t) = η cos θ sin ωt + ξ sin θ cos ωt
(2)
Show by explicit construction that Qij is time independent, and that the components depend on the lengths and directions of the symmetry axes of the ellipse.
The fact that the orientation of the orbit of an oscillator is a constant of the classical
motion is a signal of a “dynamical symmetry” that we’ll study further in section 3 of
the course.
3. Time Dependent Perturbation of a Two-State System
I won’t have time in lecture to cover one of the important features of a two state
system: the response to a time dependent potential of the form V cos ωt and ~ω ≈ ∆E
(where ∆E is the splitting between the two levels). This problem is discussed and
solved using the interaction (Dirac) picture of time evolution in Sakurai’s §5.5. Please
read that section of Sakurai carefully. Then
Consider the Hamiltonian given in Sakura’s eq. (5.5.18), however use a more realistic
time dependent perturbation,
V (t) = cos ωt(1 2 + 2 1)
instead of Sakurai’s choice.
Starting from the time dependent Schrödinger equation in the Schrödinger picture,
derive Rabi’s formula, eq. (5.5.21a) in Sakurai, for the population of the state that is
initially unoccupied at t = 0. Be sure to state clearly what approximations you make
(Sakurai’s choice yields Rabi’s formula as an exact result.).
4. Harmonic Oscillator Algebra
The operators a and a† obey the simplest, non-trivial commutation relation, [a, a† ] =
1. This makes it possible to evaluate many important matrix elements without resorting to integrals over coordinate space wavefunctions (which are products of Gaussians
and Hermite polynomials).
(a) Suppose you wanted to “normal order” [an , a†m ], which means writing
Pit in a form
where all raising operators are to the left of all lowering operators, kl ckl a†k al .
Explain how this is related to the problem of computing
n
dn m
md f
(z
f
(z))
−
z
dz n
dz n
for an arbitrary function, f (z), which is not a polynomial of order less than n.
Use this relationship to compute [a, a†m ] and [a2 , a†m ]. [Hint: Remember that
p → −i~d/dx in coordinate representation.]
MIT
8.321 Quantum Theory I
Fall 2006
5
(b) Suppose a harmonic oscillator in its ground state is perturbed by a non-linearity
of the form V ′ (x) = λxn , where n is an even integer. Show that 0V ′ 0 is
dn −z 2 /2
proportional to dz
|z=0 . Use this to evaluate the expectation value of
ne
4
6
λx and λx . [Hint: Remember the Baker-Campbell-Hausdorff Lemma, eA eB =
1
eA+B e 2 C , if C = [A, B] is a c-number.]
5. Properties of a Coherent State
[The operator identities derived in Problem Set 2 will come in handy for this problem.]
This problem is all about the coherent state of the one dimensional harmonic oscillap2
tor, H = 2m
+ 21 mω 2 x2 , that is defined as follows,
|x0 i = e−iP x0 /~|0i
where |0i is the harmonic oscillator ground state.
(a) Show that
1
1
hx0 |H|x0 i = mω 2 x20 + ~ω
2
2
(b) Show that |x0 i is a minimum uncertainty state with h∆X 2 i = ~/2mω and
h∆P 2 i = mω~/2 and that it remains a minimum uncertainty state at future
times with the same variance in both P and X.
(c) Show that |x0 i is a special case of the general coherent state,
|x0 i = N |zi
†
where N is a normalization constant and |zi = ea z |0i. Find both N and z.