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Physics 264L Second Midterm Exam
Professor Greenside
Monday, November 9, 2015
This exam is closed book and no electronic devices are allowed. The test will last the entire time from
10:05 am to 11:20 am. Please note the following:
1. Unless indicated otherwise, please answer all questions on extra blank pages so that you will have
plenty of space to write clearly.
2. Please write your name and the problem number at the top of each extra page. Before handing in
your exam, please sort your pages in order of increasing problem number and then staple all the pages
together.
3. When writing on the extra blank pages, please write clearly and logically. You will lose points if the
grader and I can not easily read and understand what you write.
4. Unless otherwise stated, you need to justify your answers, at least briefly with short phrases or sketches.
In particular, algebra without any verbal explanation or description will be given little or no credit,
even if the reasoning and answer are correct.
5. Do not hesitate to ask for help during the exam if you need clarification of a problem.
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Shorter Questions
All questions in this section require a brief justification or explanation, unless otherwise indicated. For true-false or multiple choice questions, please circle the answer on these pages.
1. (4 points) True or false: If an experiment is carried out many times and the experiment is
always prepared exactly the same way, then the measured result must always be the same (to within
experimental error).
2. (4 points) True or false: Only the bound states of a quantum system are quantized (have discrete
well-separated values).
3. (4 points) Explain briefly what is a fermion in terms of the properties of a many-particle quantum
mechanical wave function Ψ.
4. (4 points) Which is bigger in order of magnitude, the spacing ∆E between energy levels of an electron
in a 1 cm long piece of wire, or the change in potential energy when an electron is raised a height of
one hydrogen atom diameter (≈ 0.1 nm) in Earth’s gravitational field at sea level?
5. (4 points) True or false: The Schrodinger equation is capable in principle of predicting the
experimentally measured value of the mass me of an electron.
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6. (4 points) A free particle with initial kinetic energy K and with initial de Broglie wavelength λ enters
a region of space of constant non-zero potential V0 . The new de Broglie wavelength is given by
(a) λ(1 + K/V0 )
(c) λ(1 − K/V0 )−1
(b) λ(1 − V0 /K)
(d) λ(1 + V0 /K)1/2
(e) λ(1 − V0 /K)−1/2 .
7. (4 points) True or false: The Nobel-prize winning Davisson-Germer experiment established that
electrons had a wave-like behavior in accord with de Broglie’s relation λ = p/h by examining the
interference pattern on a screen caused by shooting a monoenergetic beam of electrons at two narrow
slits.
8. (4 points) Consider a one-dimensional potential V (x) and some solution ψ(x) of the time-independent
Schrodinger equation with energy E. What condition must the four numbers V−∞ = limx→−∞ V (x),
V∞ = limx→∞ V (x), Vmin = minx V (x), and E satisfy for ψ(x) to be a bound state?
Longer Questions
Unless stated otherwise, please write all of your answers on the extra blank pages.
9. (10 points) Explain briefly how the photoelectric effect confirmed the fundamental relation E = hf
for photons, and describe one feature of this effect that cannot be explained by classical physics.
10. (10 points) Show that the time-dependent one-dimensional Schrodinger equation conserves probability in that, if a solution Ψ(t0 , x) at some initial time t = t0 is normalized so that ⟨ Ψ(t0 , x) | Ψ(t0 , x) ⟩ = 1,
then for all later times t > t0 , the future state Ψ(t, x) that evolves from Ψ0 is also normalized.
Hint: first show that, if Ψ(t, x) satisfies the one-dimensional Schrodinger equation, then
(d/dt)⟨ Ψ(t, x) | Ψ(t, x) ⟩ = 0, and then complete your argument.
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11. When a stone is dropped into the middle of a deep pond, waves propagate away from the disturbance
according to the dispersion relation
σ
(1)
ω 2 = gk + k 3 ,
ρ
where ω is the wave’s angular frequency, k is the wave’s wavenumber, g is the acceleration due to
gravity at the Earth’s surface, σ is the surface tension of the water, and ρ is the mass density of the
water.
(a) (6 points) For these water waves, determine whether the phase velocity vϕ is bigger, the same,
or smaller than the group velocity vg
i. for the regime in which the water wave has a tiny wavelength.
ii. for the regime in which the water wave has a large wavelength.
(b) (8 points) Derive and give a one-dimensional partial differential equation (a wave evolution
equation) for a function u(t, x) that describes waves that are governed by the dispersion relation
Eq. (1).
12. (10 points) The lower curve in this figure is the stationary wave function ψ5 (x) of the fifth energy
level E5 above the lowest energy level E1 , as calculated by some computer code for the potential V (x)
drawn in the upper part of this figure. By drawing arrows or by circling parts of the wave function,
indicate with brief explanations all the ways that this plot is incorrect. Note: to get full credit, you
need to explain why some detail is wrong, not just indicate where there is a wrong detail.
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13. Consider a particle of mass m in a one-dimensional infinitely tall well of width L, whose energy states
consists of the orthonormal wavefunctions ψn (x) with corresponding energies En . At some initial
time t = 0, an experimentalist creates an initial state Ψ0 (x) of the particle described by the function
√
Ψ0 (x) = (3 − i) ψ2 (x) + i 6 ψ3 (x).
(2)
(a) (4 points) Normalize this wave function.
(b) (4 points) Give an explicit mathematical expression for the normalized state Ψ(t, x) of the
system at some later time t > 0.
(c) (4 points) For all integers n ≥ 1, what is the probability pn (t) that, at a later time t > 0, an
energy measurement of the system will yield the energy value En ?
(d) (4 points) In terms of the energies En , what is the average energy ⟨E⟩t of this system at time t?
(e) (4 points) Write down an explicit expression involving some integral in terms of sin and cos
functions (all complex exponentials eliminated so only real-valued terms appear in the integrand)
that gives the probability at some later time t > 0 of finding the particle to be in the interval [0, L/4] or in the interval [3L/4, L], and discuss briefly whether this probability varies with
time. (You do not need to evaluate your integral, just write it down.)
(f) (4 points) An energy measurement at time t1 produces the value E2 . If another energy
measurement is then carried out at a later time t2 > t1 , what is the probability of observing the
energy value E3 ?
(g) (6 points) At time t in the future, a measurement of the location of the particle produces the
value x = L/2. If an energy measurement is carried out a very short time afterwards, what is
the probability of observing the particle to have energy E1 ? (Your answer should be expressed
in some closed form, in terms of parameters such as m, L, and ℏ.) Discuss briefly whether your
answer is consistent with your answer for p1 (t) in part (c) above.
14. A particle of mass m moves in a one-dimensional potential V (x) that is a sum of an attractive delta
function at x = 0 with strength α > 0 and of a repulsive delta function at x = L with strength β > 0:
V (x) = −αδ(x) + βδ(x − L),
A schematic drawing of V (x) is given in panel (a) of this figure:
(a)
V
(b)
β
x=L
E1
-α
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(3)
(a) (5 points) Draw a qualitative bound-state wave function ψ1 (x) for the lowest energy state E1 ,
where E1 < 0.
Note: you might find it useful first to consider a related potential like panel (b) on the right in
the above figure, in which the attractive delta function is replaced by a finite narrow deep well
with depth V = −V0 , and in which the repulsive delta function at x = L is replaced by a finite
narrow tall rectangular peak with height V = V0 centered on x = L. Then consider how the wave
function for panel (b) must change into the wave function for panel (a) as the well and rectangular
peak become taller and narrower.
(b) (5 points) Draw a qualitative wave function ψ2 (x) for the second lowest energy bound state
with energy E2 (assuming it exists), with E2 satisfying the inequalities E1 < E2 < 0.
(c) (9 points) The potential Eq. (3) naturally suggests three different spatial regions to consider
when preparing to solve the time-independent Schrodinger equation
i. region I with x ≤ 0, with wave function ψI (x).
ii. region II with 0 ≤ x ≤ L, with wave function ψII (x).
iii. region III with L ≤ x, with wave function ψIII (x).
Give general mathematical forms for the wave functions ψI , ψII and ψIII .
(d) (12 points) Using part (c), deduce and give a transcendental equation whose solutions determine
the bound state energies E for the particle in the potential Eq. (3).
Note: you do not need to analyze or solve this equation, just obtain and give it.
A hint is that, by taking ratios of appropriate equations, you can first eliminate the constants
describing the wave functions in regions I and III. This will lead to two equations that will allow
you to eliminate the remaining constants. The final equation will contain only the parameters m,
α, β, and L, and the unknown energy E.
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Points lost:
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• Problem 14:
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