Name _________________
Solutions to Final Exam
December 9, 2013
This test consists of five parts. Please note that in parts II through V, you can skip one
question of those offered.
Part I: Multiple Choice (mixed new and review questions) [50 points]
For each question, choose the best answer (2 points each)
1. The force that binds quarks together is carried by particles called
A) Stickons B) Gluons C) Forceons D) Mesons E) Baryons
2. Which of the following is not a quark?
A) Up
B) Charm C) Muon D) Bottom E) Top
3. Suppose a particle impacts on a step potential (infinite in extent) with an energy too
low to make it over the potential, E < V0. What does the wave function   x  look
like in the classically forbidden region, where it has insufficient energy to go?
A) It doesn’t go into the classically forbidden region at all
B) It continues, still an oscillating wave but much smaller
C) It continues, still an oscillating wave, with a magnitude equal to before
D) I continues, and increases exponentially with distance
E) It continues, but falls off exponentially with distance
4. Which of the following is not a prediction of general relativity?
A) Masses orbiting each other should generate gravitational waves
B) Objects orbiting a point mass should not be exact ellipses; instead, these
approximate ellipses should gradually rotate, or precess
C) Time speeds up when you are near a gravitational source
D) A rotating mass will cause a twisting of space and time near it
E) If you squeeze a massive object small enough, it will form a black hole
5. Which combination of quarks or anti-quarks never occurs?
A) Two quarks
B) Three quarks
C) Three anti-quarks
D) One quark and one anti-quark
E) Actually, all of these do occur
6. The expectation value of the Hamiltonian, H , tells you the average value of the
A) Momentum
B) Position
C) Energy
D) Mass
E) Time
7. A spaceship going at v  34 c emits a light ray that, in its view, is traveling sideways at
c. What is the speed of the light ray as viewed in a frame not moving with the
spaceship?
A) 14 c
B) 47 c
C) c
D) 54 c
E) 74 c
8. Momentum is a three-dimensional vector, but in four dimensional spacetime, all
vectors should have four components. What is the corresponding fourth component
or momentum?
A) Time
B) Distance C) Space D) Mass
E) Energy
9. Which of the following roughly sums up general relativity?
A) Matter causes gravity, gravity causes matter
B) Matter tells space how to curve; space tells matter how to move
C) Energy causes motion through time; momentum causes motion through space
D) Spacetime can be curved because of a change in coordinates, even when there is
no gravity
E) The distance formula should generally be calculated using infinitesimal distances,
not long distances
10. Our text lists the following as the masses (in u) of some isotopes of Mg. Which one
is obviously wrong?
A) 24.985838 B) 25.982594 C) 26.984341 D) 27.983876 E) 28.375346
11. If a man travels at half the speed of light to the star Sirius and back, he would be
A) The same age as when he left
B) Older than when he left, but younger than if he had stayed on Earth
C) Older than when he left, and older than if he had stayed on Earth
D) Older than when he left, and the same age as if he had stayed on Earth
E) Older or younger than if he had stayed on Earth, depending on exactly what path
he follows
12. Which types of interactions conserve strangeness?
A) Weak (only)
B) Electromagnetic (only)
C) Strong (only)
D) Strong and electromagnetic, but not weak
E) All interactions conserve strangeness
13. Which of the following is an impossible amount of energy to have in photons of
frequency f (zero photons is a possibility)?
A) 0
B) 12 hf
C) hf
D) 7hf
E) All of these are possible
14. If you have a perfectly thermal distribution of light, which of the following is false
A) The total amount of power is finite
B) There is a wavelength where the light is strongest, inversely proportional to the
temperature
C) The power at any given wavelength is finite
D) The contribution from extremely short wavelengths is always much more
than all other wavelengths (the ultraviolet catastrophe)
E) The total amount of power grows as the temperature increases
15. Which of the following is a pretty good statement of the equivalence principle in
general relativity?
A) The effects of gravity are equivalent to the effects of being in an accelerated
reference frame
B) Mass and energy are equivalent in terms of their gravitational effects
C) Calculating motion in a curved coordinate system is equivalent to calculating it in
standard Cartesian coordinates
D) All coordinate systems are equivalent in general relativity
E) A geodesic in curved coordinates is equivalent to a straight line in flat coordinates
16. For which of the following types of radioactive decay will there not necessarily be
any even slightly dangerous radiation at all from a radioactive atom?
A)  decay B) + decay C) - decay D)  decay
E) electron capture
17. Which of the following assumptions of the Bohr model actually hold up when you
have a full quantum theory of the atom?
A) The angular momentum of an electron about the z-axis is an integer multiple
of 
B) The electron has a specific location at all times as it orbits the atom
C) The electron follows a circular orbit around the atom
D) The electron is at one of several well-defined distances from the atom
E) In fact, all of these are invalid in the quantum theory
18. The only type of radioactive decay that changes neither Z nor A is
A)  decay B) + decay C) - decay D)  decay E) electron capture
19. Under what circumstances can light falling on a metal dislodge electrons from the
metal?
A) Only if the frequency of the light is high enough
B) Only if the frequency of the light is low enough
C) Only if the intensity of the light is high enough
D) Only if the intensity of the light is low enough
E) Only if the metal is heated to a certain critical temperature
20. How many different colors does any given quark come in?
A) One
B) Two
C) Three D) Four
E) Five or more
21. A geodesic in four-dimensional spacetime is
A)
B)
C)
D)
E)
A straight line, just possibly expressed in curved coordinates
The path followed by a particle feeling external forces on it
A line following a constant coordinate value, in any set of coordinates
The path that takes the longest proper time between two points in spacetime
The path of an object moving in a circle, as written in arbitrary coordinates
22. How many neutrons are there in 235U (Uranium is element 92)?
A) 92
B) 143
C) 235
D) 327
E) None of the preceding
23. Another name for an -particle would be
A) 4He nucleus B) electron C) proton
D) neutron
E) photon
24. Which of the following is the correct formula for the momentum of a particle with
mass m and speed v in special relativity?
mv
mv
B) mv 1  v 2 c 2  C) mv 1  v 2 c 2 D)
E)
A) mv
2
2
1 v c
1  v2 c2
25. Which sorts of particles are believed to have wave-like properties in quantum
mechanics?
A) Electrons
B) Electrons and any other fundamental particles
C) Fundamental particles and very small objects like atoms, but not molecules
D) All particles
E) No particles
Part II: Short answer (review material) [20 points]
Choose two of the following three questions and give a short answer (1-3
sentences) (10 points each).
26. Suppose I have two equal masses M connected by a string that are orbiting each
other at high velocity. Explain why one cannot simply add their masses, so that
the composite object can be thought of as a single object of mass 2M.
The two masses are moving, and hence have an energy  Mc 2 . They also have
momenta, which presumably cancel. The total energy is 2 Mc 2 , with no total
momentum, which would represent an object with mass 2 M , not 2M. It’s actually
more complicated than this, since one can show the string also generally has mass, but we
will ignore that fact.
27. Classically, the electron in a hydrogen atom would tend to fall directly into the
nucleus, ending up at r = 0. Explain using the uncertainty principle why this
doesn’t happen.
Quantum mechanically, there is a relationship between the uncertainty in the
position and the uncertainty in the momentum,  x  p   12  . By specifying the
position with arbitrary accuracy, you are then assuring that the momentum is completely
undetermined, and therefore the electron would have enormous kinetic energy.
28. Give at least three restrictions on the wave function   r, t  that it must satisfy.
Equations are allowed but not necessary.
The wave function must satisfy Schrodinger’s equation, it must be continuous, it
must have a continuous derivative, and it must be normalized, so that the square of its
magnitude integrates to one. The last condition implies that the wave function must fall
off at infinity as well.
Part III: Short answer (new material) [30 points]
Choose three of the following four questions and give a short answer (1-3
sentences) (10 points each).
29. Explain qualitatively why very heavy nuclei (anything heavier than lead) are
unstable.
The strong force is a short range force. In contrast, electromagnetism is long
range. Hence in a large nucleus, the protons will feel the attractive force of only their
nearest neighbors, while feeling the electrostatic repulsion of all of the other protons in
the nucleus. As the nucleus gets more heavy, eventually the repulsions wins, and a
nucleus can spit out an alpha-particle to reduce the repulsion.
30. For a stable light nucleus (A < 50), approximately what fraction of the nucleons
tends to be protons? For a stable heavy nucleus (A > 150), approximately what
fraction of the nucleons tends to be protons?
Light nuclei are about 50% protons, heavier nuclei are more like 40% protons,
with intermediate mass nuclei somewhere in between.
31. What are the three most common/lightest types of quarks, and give their spins
and charges. A neutron is a baryon that has no strangeness. Which quarks (or
anti-quarks) is it made of?
The up (charge +2/3), down (charge -1/3) and strange (charge -1/3) are the most
common quarks. They are all spin ½. A neutron is a baryon, so it is made of three
quarks, but since it has no strangeness, none of them are strange. The only combination
of three quarks that can then be neutral is up-down-down.
32. Explain qualitatively the difference between curved coordinates and actual
curvature.
Curved coordinates are simply coordinates that are not straight, such as spherical
coordinates. The apparent weird behavior of particles in curved coordinates can be
eliminated by switching to Cartesian coordinates. Actual curvature, on the other hand,
occurs when the spacetime itself is curved, and cannot be removed by switching to
Cartesian coordinates.
Part IV: Calculation (review material) [40 points]
Choose two of the following three questions and perform the indicated
calculations (20 points each)
A
33. An alien triangle ship is in the shape of a 30-60-90
triangle, with angle A being 60. This triangle ship is
then accelerated to high speed, such that it still looks
like a 30-60-90 triangle, but now angle A looks like 30.
(a) How fast must the triangle be moving? In which
C
direction? Give your answer as a fraction of c.
Contraction occurs in the direction of motion. To decrease the angle A, we must
shrink the distance BC. This can only be done if the triangle is moving to the right or to
the left.
Initially, the ratio of the two legs of triangle ABC is
 BC   tan 60 
 
 AC 
3.
When we perform a boots, BC changes but not AC, so that
 BC   tan 30 
 
 AC 
3
3
It follows that
 BC   tan  30  
 BC  tan  60 
3 1 1

 .
3
3 3
It follows from L  LP  that
1  BC  L 1


  1  v2 c2 ,
3  BC  LP 
v 2 c 2  1   13   89 ,
2
v c
2 2
3
 0.9428 .
(b) The moving ship is discovered to have a total energy of E  5.40  1019 J .
What is the rest mass of the ship in kg?
From previous work it is pretty obvious that   3 . We therefore have
E   mc 2 ,
m
E
5.40 1019 J

 200.3 kg .
 c 2 3  2.998  108 m/s 2
B
34. Joe Atom is driving his molecular car, which has a mass of 1.0010-19 kg. He
wants to park it in a space that is 6.4010-8 m long.
(a) Estimate the minimum uncertainty in the car’s position, if the owner wants
to make sure it is within the parking space.
By Carlson’s rule, the uncertainty is about one-fourth the size of the space, or
x  14 L  1.60 108 m .
(b) Estimate the corresponding uncertainty in its momentum. Assuming the
owner TRIED to put it in park (v = 0), estimate the ACTUAL velocity caused
by this uncertainty in momentum.
The uncertainty principle says that  x  p   12  , and therefore
p 

1.055  1034 J  s

 3.297 1027 kg  m/s .
8
2x 2 1.60  10 m 
This implies a corresponding uncertainty in its velocity of
v 
p 3.297  1027 kg  m/s

 3.297  108 m/s .
m
1.00 1019 kg
(c) Estimate how long Joe can safely leave his car parked before the quantum
velocity found in (b) will cause his car to drift out of its parking space.
This question is actually a bit vague, since we don’t know what “safely” means.
Let’s assume the car is in the center of the parking space. To escape the space, it must
then move half the length of the space. The time it would take to do so would be
8
1
L 2  6.40 10 m 
t

 0.97 s .
v 3.297 108 m/s
1
2
Since the question is vague, our answer should be vague. The car will drift out of the
space in about one second.
35. A particle in the first excited state of the harmonic oscillator has wave function
  x   Nxe  Ax
2
2
,
where A is real and positive, and N is a normalization factor. Some useful
integrals can be found below.
(a) What is the correct normalization factor N for this wave function?
The normalization condition is


N2
2
1     x  dx  N 2  x 2 e  Ax dx 
2

2

A3
N 2

,
A3
.

(b) If the position of the particle is measured, what is the average value x that
would be obtained? If we measure the average value of the position squared,
what is the average value x 2 ?
To calculate these, we simply throw in another factor of x or x2.

x  

*

 x  x  x  dx  N  x e
2

3  Ax 2
dx  2

A3

x 2    *  x  x 2  x  dx  N 2  x 4 e  Ax dx  2

2

0  0 ,
A3 3

 4

5
A

3
2 A2

3
.
2A
(c) What is the uncertainty in its position x ?
This is straightforward:
x 
Possibly helpful integrals:



x 2 e  x dx 
2



x2  x
2

3
3
.
 02 
2A
2A
x n e  x dx  0 if n is odd ,
1 
,
2 3
2



x 4 e  x dx 
2
3 
,
4 5

e  x dx 

,

x 6 e  x dx 
15 
.
8 7

2




2
Part V: Calculation (new material): [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each)
36. Most modern smoke detectors contain Americium, typically, about 0.28 g of
241
Am, which has a half-life of 432.2 years
(a) What is the approximate weight, in u, of 241Am? How many atoms of 241Am
would be in 0.28g?
The approximate weight of 241Am is 241 u. This should be accurate to about one
part in a thousand or better, so it is more accurate than some of the other numbers in the
problem. The total number of atoms is
N0 
6
23
M 0.28  106 g  0.28  10  6.022  10 


 7.00  1014 .
241 u
241
m
(b) What is the decay rate  in y-1 and in s-1 for 241Am (1 y = 3.156107 s)?
The decay rate is

ln 2 0.6931
1.606  103 y 1

 1.606  103 y 1 
 5.082 1011 s 1 .
7
3.156  10 s/y
t1/ 2 432.2 y
(c) How many radioactive decays per second would occur for 0.28g of 241Am?
This is straightforward. We simply use
R0   N 0   5.082 1011 s 1  7.00 1014   35, 600 s 1 .
(d) How long would we have to wait for the decay rate to drop to 1,000. s-1?
We use the formula R  R0 e  t and solve for t:
R  R0 e  t ,
e t 
R0 35,571 s 1

 35.57 ,
1, 000 s 1
R
 t  ln  35.57   3.57 ,
t
3.57


3.57
 2220 y .
1.606 103 y 1
37. Photocopied with the equations on the next page is a portion of Appendix A
from the text. 146Pm is an unstable nucleus
Daughte
mode
Q (MeV)
with half-life of 5.5 yr. You may use the
r
table at right if you wish.
142
Pr

1.915
(a) What would be the resulting isotope if
146
–
Sm

1.542
this isotope underwent – decay, +
146
+
Nd

0.454
decay, electron capture, and -decay?
?
yes
yes
yes
146
For – decay, Z increases by one while A stays the same, so we end up with
Sm. For + decay and electron capture, Z decreases by one while A stays the same, so
we end up with 146Nd. Finally for -decay, Z decreases by two and A decreases by four,
so we end up with 142Pr.
146
(b) What is the Q-value for each of these processes?
For – decay and electron capture, the formula is the same, but the daughter mass
is different. We find
Q       M  146 Pm   M  146 Sm   c 2  145.914698 u  145.913043 u  c 2
 0.001655 uc 2  0.001655  931.494 MeV   1.542 MeV ,
Q  e.c.   M  146 Pm   M  146 Nd   c 2  145.914698 u  145.913113 u  c 2
 0.001585 uc 2  0.001585  931.494 MeV   1.476 MeV .
For + decay, the daughter is the same as for electron capture, but the formula is
different, so we have
Q       M  146 Pm   M  146 Nd   c 2  2me c 2  1.476 MeV  1.022 MeV  0.454 MeV .
Finally, the -decay is another formula entirely:
Q     M  146 Pm   M  142 Pr   M  4 He   c 2
 145.914698 u  141.910040 u  4.002602 u  c 2
 0.002056 uc 2   0.002056  931.494 MeV   1.915 MeV .
(c) Which of these modes is allowed or excluded?
Since all four decays came out with positive Q – values, all four are possible.
38. The 0 baryon decays almost always to one of the particle pairs listed below:
 +e-  0 
p+K–
K – e+
p+
p+–
The reasons, and which ones are impossible, are
 Charge must be conserved, so the right side must
be neutral, but this is violated by p+.
 Baryon number must be conserved, and there is
one baryon on the left, so there must be one on the right, but this is violated by
K–e+.
 The total number of fermions on the left (one) plus right must be odd, so we must
have an odd number (one) on the right as well. This is violated by +e-.
 For a decay, the energy on the right must not exceed the energy on the left. This
is violated by p+K–, with a total mass of 938 MeV + 494 MeV = 1,432 MeV >
1,193 MeV.
baryons mesons
A table of particles is listed at right. The charges
are implied by their names (the photon  is
neutral).
(a) Four of these are impossible. Give a reason in
each case why they cannot occur
All masses in MeV/c2
Name Mass Spin Strange
0
1193
½
-1
+
1189
½
-1

0

1116
½
-1
938
½
0
p+
K
494
0
-1
–
135
0
0

e
0.5 ½
0

0
1
0
This leaves only the two remaining decays, 0 and p+–.
(b) For the remaining two, determine which type of interaction (strong,
electromagnetic, weak) is likely to be responsible
For 0, strangeness is conserved, the  is not strongly interacting, and there are
no neutrinos involved, so it is electromagnetic.
For p+–, strangeness is not conserved, since there is no strange particle on the
right, so it must be weak.
Because the 0mass was given incorrectly in the problem, it was accepted as an answer
if 0 was given as an impossible decay mode.
39. A white dwarf star of one solar mass (M = 1.9891030 kg) is estimated to have a
radius of about R = 5730 km. The Hydrogen-alpha line normally has a
wavelength of 0 = 656.28 nm.
(a) What would be the wavelength  of H- light coming from the surface of this
white dwarf?
We use the formula for wavelength shift, namely


  0 1 
2GM 

c2r 
1/2
 2  6.673 1011 m3 / kg / s 2 1.989 1030 kg  

 0 1 
2
8
6

 2.998 10 m/s   5.730 10 m  

  656.28 nm  1  0.0005154
1/ 2
1/2
 656.45 nm .
(b) The star is now increased to two solar masses, and becomes a neutron star.
The H- line is now observed at  = 1038.00 nm. What is the radius of the
neutron star?
We now take the same formula and rearrange it to solve for the radius, so that


  0 1 
1
2GM 

c2 R 
1/2
,
2GM 0
 ,
c2 R

02
2GM
656.282
1
1




 0.6003 ,
c2 R
1038.002
2
2  6.673 1011 m3 / kg / s 2  2 1.989 1030 kg 
2GM
R

2
0.6003c 2
0.6003  2.998 108 m/s 
 9841 m  9.841 km .
Due to an error in the problem, 14.76 km was also accepted as an answer.
(c) The star’s mass is increased to three solar masses and becomes a black hole.
What is the Schwarzschild radius of this black hole?
The Schwarzschild radius is just given by
11
3
2
30
2GM 2  6.673 10 m / kg / s  3 1.989 10 kg 
RS  2 
 8,860. m  8.860 km .
2
c
 2.998 108 m/s 
Equations
Constants:
h  6.626  1034 J  s  4.136 1015 eV  s
u  931.494 MeV / c 2
  1.055 1034 J  s  6.582 1016 eV  s
u  1.661 1027 kg
G  6.673 1011 m3 / kg / s 2
2me c 2  1.022 MeV
M He  4.002602 u
N A  6.022 1023
Gravitational time dilation:
2GM
  t 1 2
cr
Red Shift:
2GM
  0  1  2 
cr 

1/ 2
Schwarzschild radius: RS  2GM c 2
Compton Effect:
Expectation values:
   
h
1  cos   ,
mc

h
 2.426 pm
mc
    *  x    x  dx

Uncertainty:
Isotope Masses
  
2
 2  
2