OPTI 570, Fall 2023 Final Exam Anderson
Untimed exam, available 7pm, Dec 6, 2023.
All students including distance students: unless arranged otherwise, return via D2L
by 7am MST, Thurs. Dec 14, 2023.
Instructions
• This is an untimed take-home exam. You may spend as much time on it as you
choose as long as your solutions are returned by the due date and time above.
• Once you begin the exam or start reading any of the questions, you must adhere to the
following rules:
– Allowed items: Electronic or physical copies of the course notes, Field Guide, any
textbook including Cohen-Tannoudji, OPTI 570 2023 problem sets and solutions (yours
and mine), and any of your own notes that you have personally written or typed.
– Also allowed: Any material currently on the 2023 OPTI 570 D2L site, including all
posted videos.
– Once you begin the exam, you may not ask Carter or anyone else questions about
material from OPTI 570.
– You may, however, ask me questions about how to interpret the exam questions or if
you are stuck in a calculation and do not know how to proceed. I will likely tell you
something that helps to point you in the right direction.
– Calculators and computer-based computation packages are not needed and must not be
used.
• Youmust not accept help from other people, or from printed, electronic, or internet resources
not mentioned above. This includes any information provided by other people such as from
public or private discussions, notes taken or calculations performed by others, and websites
other than D2L. You also must not provide help of any sort to anyone else in the class
using these or other means. You must not discuss specific answers or methods about the
exam questions with anyone else privately or publicly, in person or online, until after the due
date listed above or unless you and the other person have turned in your exams. You are
on your honor to adhere to these rules; violation of these rules will result in a failing grade
if discovered. If you have any questions or uncertainties about what is or isn’t
allowed on this exam, ask me.
• There are 3 problems worth 110 total points. Final scores may be scaled up from raw scores.
• When you are finished with the exam, convert your answers to a single PDF. Return the
PDF via the D2L site as an assignment associated with the exam. You may also email your
solutions to me if there is a problem uploading them to D2L. Your first submission will be
the one graded, so do not submit answers until you are finished (but if you have a problem
uploading and quickly correct your error, the corrected version will be graded).
1
Extra Information
Powers of the 1D position operator Xˆ, expressed as matrices in the usual representation of 1D
harmonic oscillator energy eigenstates, where σ is the usual harmonic oscillator length scale:
Xˆ =
σ√
2

0 1 0 0 0 0 . . .
1 0
√
2 0 0 0
0
√
2 0
√
3 0 0
0 0
√
3 0
√
4 0
0 0 0
√
4 0
√
5
0 0 0 0
√
5 0
...
. . .

Xˆ2 =
σ2
2

1 0
√
2 0 0 0 . . .
0 3 0
√
6 0 0√
2 0 5 0 2
√
3 0
0
√
6 0 7 0 2
√
5
0 0 2
√
3 0 9 0
0 0 0 2
√
5 0 11
...
. . .

Xˆ4 =
σ4
4

3 0 6
√
2 0 2
√
6 0 (remaining elements of this row = 0)
0 15 0 10
√
6 0 2
√
30
6
√
2 0 39 0 28
√
3 0
0 10
√
6 0 75 0 36
√
5
2
√
6 0 28
√
3 0 123 0
0 2
√
30 0 36
√
5 0 183
...

2
Problem 1. [40 pts.] A particle of mass m is trapped in a 2D isostropic harmonic oscillator
potential well 12mω
2 (x2 + y2). H0 is the Hamiltonian of the oscillator, with eigenstates |nx, ny⟩
and eigenvalues Enx ny = ℏω (nx + ny + 1), where nx and ny are integers greater than or equal
to zero that specify the degree of excitation in the x and y directions respectively. The oscillator
length is given by σ =
√
ℏ/(mω). A weak time-independent perturbation W is added to H0 so
that the total Hamiltonian of the trapped particle is H = H0 +W , where
W = λℏω(
√
2X/σ)4.
X is the x-direction position operator, and λ is a positive and real scalar with |λ| << 1. W does
not depend on Y , but since this is a problem with two spatial dimensions, you must still carefully
take the 2D nature of the problem into account in your calculations.
Before proceeding with the calculations below, you may wish to first consider how the matrix
for X4 looks in the {|nx⟩} representation (shown on the previous page), and then figure out how
it looks in the {|nx, ny⟩} representation. If this suggestion confuses you, then pick an order for the
elements of this tensor-product basis, and start placing terms ⟨n′x, n′y|X4|nx, ny⟩ in the appropriate
places in the matrix. Then start simplifying these quantities as needed in the calculations below.
Note that you do not have to follow this suggestion if you understand what is needed in the fol-
lowing calculations, so you may wish to examine the following questions first.
(a) Calculate the ground state energy of H to second order in λ.
(b) Write the ket for the ground state of H to first order in λ, expressing your answer
in terms of the eigenstates of H0. You do not need to normalize your answer.
(c) To first order in λ, carefully calculate the first and second excited state ener-
gies of H.
Problem 2. [30 pts.] A particle of mass m is in a 1D quantum harmonic oscillator potential
of frequency ω0. The Hamiltonian for this system is H0. The particle has been in the ground
state of the oscillator for a long time; we’ll therefore say that at time t = −∞, the particle is in
the ground state. A time-dependent perturbation W (t) is applied that slightly changes the trap
frequency without modifying anything else about the potential:
W (t) =
λ
2
mω20X
2 exp (−|t|/τ),
where τ is a time constant determining the effective duration of the perturbation. The total Hamil-
tonian is therefore H(t) = H0 +W (t), and the constant λ is a real number much less than 1. As
can be seen, the matrix elements of W (t) reach their maximum values at t = 0, and then for times
long after t = 0 the perturbation has essentially been removed.
(a) Given that the particle is in the ground state of H0 at t = −∞, use the methods of first-
order time-dependent perturbation theory to determine the approximate value for the probability
that at t = ∞ the particle will be found in the first excited state of the oscillator. Will this
estimated probability change if the perturbation expansion is taken to second order, and if so, how?
Justify your answer.
3
(b) Given that the particle is in the ground state of H0 at t = −∞, use the methods of first-
order time-dependent perturbation theory to determine the approximate value for the probability
that at t = ∞ the particle will be found in the second excited state of the oscillator. Simplify
your answer as much as possible.
Problem 3. [40 pts.]
Consider a particle (call it “particle A”) of mass m trapped in a 2D isotropic harmonic oscilla-
tor in the (x, y) plane that is centered at coordinate (x, y) = (0, 0). The oscillator is defined by
oscillation frequency ω and a corresponding oscillator length σ =
√
ℏ
mω . A second particle (“par-
ticle B”) that does not feel the oscillator potential is traveling along the y axis with a speed v.
Particle B is initially far away from particle A, but eventually passes by particle A, which is lo-
calized about the center of the potential well. Particle B then continues moving along the y axis
and is eventually again far away from particle A. Suppose that the particles can interact with each
other, and that for particle A the interaction is modeled by a perturbation
W (t) = λℏωσ2 δ(X) δ(Y − vt),
where λ is a real dimensionless scalar with |λ| ≪ 1 and X and Y are position operators for the x and
y directions. Note that these Dirac delta functions each have dimensional units of inverse-length,
and are expressed as δ(x) and δ(y − vt) in the position representation. We will not consider any
changes that the interaction might have on the state of particle B.
Suppose also that at time t = −∞, particle A is in the ground state of the harmonic oscilla-
tor. We will use the notation |nx, ny⟩ to label the energy eigenstates of particle A in the absence of
any perturbation caused by particle B, where nx and ny are the usual quantum numbers associated
with excitation in the x and y directions. So for example |0, 0⟩ is the ground state of particle A in
the oscillator. This state |0, 0⟩ also happens to be the initial state of particle A at t = −∞.
Using first-order time-dependent perturbation theory, calculate the approximate prob-
abilities P
(1)
00→10 and P
(1)
00→01 that the perturbation will induce a transition of particle A
into the states |1, 0⟩ and |0, 1⟩ (respectively) from the ground state |0, 0⟩. Calculate both
of these probabilities for t =∞, corresponding to times long after particle B has passed particle A.
For reference, the normalized energy eigenfunctions of a one-dimensional harmonic oscillator can
be found on page 54 of the Field Guide. You may also choose to use the integral∫ ∞
−∞
du eiaue−u
2
Hn(u) =
√
π(ia)ne−a
2/4
where a is real and Hn(u) is the Hermite polynomial of order n (see Field Guide page 54). If you
use the integral given, you will need to figure out how to relate the Hermite polynomial within the
integral above to the expressions involved in your calculations.
End of exam and semester. Thank you for a great semester and have a wonderful winter break!
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