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Physics 264L Final Exam
Professor Greenside
Thursday, December 10, 2015
This exam is closed book and no electronic devices are allowed. The test will last the entire time from 2 pm
to 5 pm. Please note the following:
1. Answers to the shorter questions must be written on the exam itself, in the space provided below each
question, and only answers in these spaces will be looked at. All answers to the longer questions should
be written 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 (possibly
a lot of points) if the graders 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, lots of algebra without any verbal explanation or description will be given little or no
credit, even if you obtain a correct final answer.
5. Do not hesitate to ask for help during the exam if you need clarification of a problem.
1
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) To the nearest power of ten, what is the quantum number n of a 2 mg mosquito that is
flying at its top speed of 0.7 m/s in a sealed tube of length 10 cm?
2. (4 points) Draw a picture of a one-dimensional potential V (x) that has bound states for a classical
particle but that does not have any bound states for a quantum particle.
3. (4 points) True or false: If an energy measurement is carried out on a freely moving particle (i.e,
a particle moving in the absence of any potential so V = 0) whose quantum state Ψ(t, x) is described
by a spatially localized wave packet, then a unique energy value E will be obtained.
4. (4 points) A car sitting still has a string of length L attached to its interior ceiling with a mass m
hanging down vertically from the string. If the car is now driven with constant acceleration a along a
straight road, one finds that the string deviates from the vertical by a constant angle θ. Explain how
to use the equivalence principle to deduce the value of θ in terms of given quantities.
2
5. (4 points) What is the energy level n of the quantum state corresponding to the following expression
(a+ )7 (a− )6 (a+ )4 ψ0 (x),
(1)
where a± are the raising and lowering operators and where ψ0 is the ground state of a quantum
harmonic oscillator?
(a) 5
(b) 6
(c) 7
(d) some other answer.
6. (4 points) Describe briefly what output is produced by this Mathematica command
Manipulate[ Plot[ c t , {t, 0, 2} ], {c, 0, 1} ]
7. (4 points) Explain briefly what is meant by a “transmission resonance” and describe a specific
example of a system that has such a resonance.
3
8. (4 points) A system consisting of many particles must generally be treated with quantum mechanics
(as opposed to being treated with classical mechanics) when the average distance between the particles
is comparable to or smaller than their de Broglie wavelengths. Given this observation, given that a
copper atom has an atomic number of 29 and atomic weight of about 64, and that copper metal has a
number density of about 1029 atoms per cubic meter, then
True or False: the set of all Cu nuclei in a chunk of Cu metal at room temperature
T ≈ 300 K must be treated using quantum mechanics.
9. The following figure
shows a trace on the screen of an oscilloscope from a source that is a superposition of two oscillations.
The screen is graduated in centimeters and the spot on the screen moves horizontally with a constant
speed of 0.5 cm/ms, and the vertical scale is 2 V/cm. Which of the following are most nearly the
observed amplitude and frequency of these two oscillations? (no justification needed).
(a)
(b)
(c)
(d)
(e)
oscillation
oscillation
oscillation
oscillation
oscillation
1:
1:
1:
1:
1:
5
1.5
5
2.5
6.1
V
V
V
V
V
250
250
6
83
98
Hz,
Hz,
Hz,
Hz,
Hz,
oscillation
oscillation
oscillation
oscillation
oscillation
4
2
2:
2:
2:
2:
2.5
3
2
1.25
1.35
V
V
V
V
V
1000
1500
2
500
257
Hz
Hz
Hz
Hz
Hz
10. (4 points) Let E1n denote the energy levels of a particle of mass m in a well of infinite height and
of width L that is centered on the origin x = 0 (so V1 (x) = ∞ if |x| > L/2 and V1 = 0 if |x| < L/2.
Next, let E2n denote the energy levels of the same particle in the same well but to which a repulsive
delta function potential at the origin has been added, V2 (x) = V1 (x) + βδ(x) where β > 0. Which one
of the following statements correctly describes how the energy levels E2n are related to the levels E1n ?
(a)
(b)
(c)
(d)
(e)
E2n = E1n for all n.
E2n > E1n for all n.
For some n, E2n > E1n , for the other n, E2n = E1n .
For some n, E2n > E1n , for some n, E2n < E1n , but for no n does E2n = E1n .
For some n, E2n > E1n , for some n, E2n = E1n , and for the other n, E2n < E1n .
11. (4 points) Consider two radioactive isotopes of a certain element that are observed to undergo alpha
particle decay such that the first nucleus decays with a half-life of t1 and a final alpha particle kinetic
energy of K1 , while the second nucleus decays with a half-life of t2 with a final alpha particle kinetic
energy of K2 . Then if experimentally K1 > K2 , you can conclude that
(a) t1 > t2
(b) t1 ≈ t2
(c) t1 < t2
(d) t1 can be bigger or smaller than t2 .
12. (4 points) Consider five identical particles of mass m inside a one-dimensional sealed tube that is
described by an infinitely tall well of length L, and let Eboson be the smallest possible total energy of
this system if the identical particles are bosons, and let Efermion be the smallest possible total energy
of this system if the identical particles are fermions. What is the value of Efermion − Eboson in terms
of m, L and physical constants?
Note: the total energy of a many-particle system like this is just the sum n1 E1 + n2 E2 + · · · of the
number ni of particles in a given level times the energy value Ei of that level.
5
13. (4 points) Consider a one-dimensional particle of mass m in the following potential
V=∞
V = V2
0
x1
x2
E2
x
x3
E1
V = -V1
that is described mathematically as follows

∞,



−V1 ,
V (x) =
V2 ,



βδ (x − x3 ) ,
if
if
if
if
x ≤ 0,
0 < x ≤ x1 ,
x1 < x ≤ x2 ,
x2 < x,
(2)
where V1 and V2 are positive constants with units of energy, x1 , x2 , and x3 are positive constants with
units of length, and the strength β > 0 of the delta function is positive.
Over the range E2 > 0, draw a qualitative plot of how the reflection coefficient R(E2 ) depends on the
energy E2 for a stationary scattering state with energy E2 that corresponds to particles arriving from
the right (from x = +∞ heading towards the origin).
A big hint: do not try to compute anything for this problem, instead think conceptually and physically
about what is going on.
14. (4 points) For the same potential of the previous problem, write down the mathematical forms of the
stationary wave function ψ(x) with energy E2 in the region 0 ≤ x ≤ x1 , and in the region x1 ≤ x ≤ x2 .
6
15. (4 points) Consider a freely moving proton such that the proton’s kinetic energy equals its rest mass
energy. If λ is the proton’s de Broglie wavelength and if d ≈ 10−15 m is the diameter of the proton,
determine to the nearest power of ten the ratio λ/d.
16. (4 points) (No justification is needed for this problem.) When transmitting high frequency signals
on a coaxial cable, it is important that the cable be terminated at one end with its characteristic
impedance to avoid
(a)
(b)
(c)
(d)
(e)
leakage of the signal out of the cable
overheating of the cable.
reflection of the signals from the terminated end of the cable.
attenuation of the signal propagating in the cable
production of image currents in the outer coaxial conductor.
17. (4 points) Scientists studying diatomic molecules often use a so-called Morse potential energy
(
)2
V (d) = D 1 − e−a(d−d0 ) ,
(3)
with positive parameters D, a, and d0 as a convenient model of how the energy of a diatomic molecule
of mass m depends on the distance d between the two nuclei. Obtain and give an expression (in terms
of D, a, d0 , the reduced mass µ of the diatomic molecule, and physical parameters) for the approximate
energy difference ∆E = E2 − E1 between the two lowest bound-state energy levels of this potential.
18. (4 points) At time t = 0, a position measurement of a particle of mass m in an infinitely tall well
of width L gives the value x0 where 0 < x0 < L. Write down a mathematical expression for the
state Ψ(t, x) of the system at a later time t > 0 in terms of x0 , m, L, the energy levels En of this
system, and physical constants.
7
19. (4 points) If an experimentalist constructs a histogram of measurements for some particular quantity,
recall that the precision of the measurements is determined by the width of the distribution around
the mean of the data, while the accuracy of the data is determined by how close the mean of the data
is to the correct reference value. Given this, consider four classes of students who use four different
methods to measure the height of a building many times, and they generate four different histograms
as shown here:
(a)
(b)
(c)
(d)
Give the letter that best answers each of these three questions (no justification is needed):
• Which method is most precise?
• Which method is most accurate?
• Which method has a systematic error?
20. (4 points) Consider a particle of mass m in the three-dimensional potential
V (x, y, z) =
1
1
1
mω 2 x2 + mω 2 y 2 + mω 2 z 2 .
2
2
2
(4)
What is the third lowest energy level E3 of this system and what is the degeneracy d3 of this level?
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21. (4 points) True of false: For the spherically symmetric bound-state solutions ψn (r) of the hydrogen
atom, the most likely distance to find an electron from the proton is a strictly increasing function of
the quantum number n.
22. (4 points) Let RE ≈ 6, 000 km and ME ≈ 1025 kg denote the radius and mass of the Earth, let RM ≈
(1/4)RE and MM ≈ (1/80)ME denote the radius and mass of the Moon, and let d ≈ 60RE ≈ 4×105 km
be the average distance of the Earth’s center to the Moon’s center. At approximately what location
between the Earth’s surface and the Moon’s surface, along a line connecting the Earths’s center with
the Moon’s center, will time pass as quickly as possible, as measured by a high-precision clock far from
the solar system?
For this problem, assume the Earth and Moon are motionless so you can ignore special relativistic time
dilation and observe that, for weak gravitational fields, general relativistic time dilation depends on
the total gravitational potential at a point.
Longer Questions
Unless stated otherwise, please write all of your answers on the extra blank pages.
23. (10 points) Heide boards a spaceship and travels away from Earth at a constant speed v = (3/5)c =
0.6c (for which γ = 5/4) towards a remote star 500 light-years away. One year later on Earth, her twin
brother Hans boards a second spaceship and follows Heide at a constant speed v = (24/25)c ≈ 0.96c
(for which γ = 25/7 ≈ 3.6).
When Hans catches up with Heidi, what will be the difference in their ages and who will be older?
24. (10 points) What is the value of the commutator
[ r2 , Lz ],
(5)
and discuss briefly the physical significance of its value. Here r2 = x2 + y 2 + z Z is the operator
corresponding to the distance squared from the origin, and Lz = xpy −ypx is the operator corresponding
to the z-component of the angular momentum.
Hint: use commutator identities like [A + B, C] = [A, C] + [B, C] and [AB, C] = A[B, C] + [A, C]B.
9
25. (10 points) Consider a particle of mass m in an harmonic oscillator potential V (x) = (1/2)mω 2 x2 ,
and consider the normalized initial state
1
i
2
Ψ(t = 0, x) = − √ ψ2 + √ ψ6 (x) + √ ψ7 (x)
6
6
6
(6)
where ψ2 (x), ψ6 (x) and ψ7 (x) are normalized stationary energy states, and ψ0 (x) denotes the ground
state. What is the average value of the position of the particle in the state Ψ(t, x) at time t > 0:
⟨x⟩t = ⟨ Ψ(t, x) | x Ψ(t, x) ⟩ ?
Hint: x =
(7)
√
ℏ/(2mω) (a+ + a− ).
26. (10 points) A particle of mass m is in the ground state (hence bound state) of a delta-function
well V1 (x) = −αδ(x) with α > 0, that is centered at the origin x = 0. At time t = 0, the delta-function
is moved instantly to the right a distance L so that the new potential is V2 = −αδ(x − L). What is
the probability p(t) that, at a later time t > 0, an energy measurement of the particle will find the
particle to be in the ground state of the potential V2 ?
27. A particle of mass m and kinetic energy E1 approaches from the right (so starting at x = +∞ and
moving to the left towards the origin) a potential energy “cliff” or attractive region such that V = 0
for x > 0 and V = −V0 < 0 for x ≤ 0 as shown in this figure:
V(x)
E1
x
E2
-V0
(a) (10 points) For E1 = V0 /3, what is the probability that the particle will reflect back to the
right? Your answer should be expressed as a ratio p/q of small integers.
(b) (10 points) Now consider the same particle approaching the origin x = 0 from the left with
energy E2 < 0 as shown in the above figure. Show that the reflection coefficient R = 1 for all
energies satisfying −V0 < E < 0.
10