Name _________________
Solutions to Final Exam
December 10, 2012
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. When we look at clusters of galaxies, we often see distorted images of galaxies far
behind them. What is causing this distortion?
A) The gravity from the cluster is actually bending the light, like a lens
B) The gas within the cluster causes the light to refract
C) The dust absorbs the light from the distant galaxy, then reemits it to produce a
distorted image
D) We are actually seeing a reflected version of our own galaxy
E) Photon-photon scattering by the light from the intermediate cluster is causing the
image to get distorted
2. Why do astronauts float around inside the space station?
A) There is nothing holding the space station up, so they are falling together
B) They are so far from the Earth the gravity is negligible
C) The Earth’s gravity is screened out by the Earth’s atmosphere
D) The rocket engines constantly hold it up
E) The space station is rotated to create an artificial gravity that counters Earth’s
gravity
3. The way we map out the spiral arms of our galaxy is
A) Using the 21 cm line caused by electron spins flipping
B) Using the light from hot hydrogen regions, where there is ionized hydrogen
C) Mapping the infrared light produced by dust clouds
D) Reflecting light off of planetary nebulas
E) Taking external pictures using space probes that have left the galaxy
4. According to Newton, a planet orbiting a point mass source will move in an ellipse.
According to Einstein, what changes about this prediction?
A) Nothing
B) It still goes around an ellipse, but the speed is different
C) Only circular orbits are possible
D) The ellipse precesses, so the direction of the ellipse is constantly changing
E) The plane of the ellipse moves up and down, so the Sun does not remain in that
plane
5. In what way is the absolute future different from the naïve future?
A) The absolute future has time differences that exceed the uncertainty relation
B) The absolute future is places that are at the same place, but at a different time
C) The absolute future demands only that the absolute value of the time difference is
positive
D) All observers will agree on the absolute future, unlike the naïve future
E) The absolute future is restricted to events that have lightlike separation
6. Probably the most important thing that determines the morphology (shape or
classification) of a galaxy is
A) Its total mass
B) How strong the magnetic fields are
C) Whether it is composed of light or heavy stars
D) Whether it has a source of cool gas flowing in
E) What its hydrogen/helium ratio is
7. According to quantum mechanics, what objects have both particle and wave
properties?
A) Photons, but not electrons
B) Electrons, but not photons
C) Photons and electrons, but not other elementary particles
D) Elementary particles and atoms, but not larger objects
E) Everything
8. What distinguishes a “good” coordinate transformation, one in which the laws of
physics remain unchanged, from a “bad” coordinate transformation?


A) Good ones leave the equation F  ma unchanged
B) Good ones are restricted to space translations and rotations around an axis
C) Good ones leave the four-dimensional distance formula unchanged
D) Good ones leave volumes unchanged
E) Good ones do not mix time with space
9. Antimony (Sb) is element number 51 on the periodic table. How many neutrons
does 121Sb have?
A) 19
B) 51
C) 70
D) 121
E) 172
10. In the photoelectric effect, why does (typically) visible light not eject any photons
from some metals, but ultraviolet light does?
A) The light has to match the resonance frequency for extracting electrons
B) Ultraviolet lights are much brighter than visible light
C) The photons of ultraviolet light have enough energy to extract an electron;
visible light photons do not
D) Metals reflect visible light but are transparent to ultraviolet
E) Ultraviolet light has a wavelength short enough it can slip between atoms and pull
out the electrons
11. Which of the following might be a good summation of Einstein’s general theory of
relativity?
A) Things are even more out of whack than you think
B) Matter tells space how to curve; space tells matter how to move
C) Light tells objects how fast they can go; objects determine the speed of light
D) Gravitational force is proportional to one over the distance squared; force equals
mass times acceleration
E) Planetary orbits precess in gravitational fields; time slows down near a black hole
12. The best evidence that galaxies contain large amounts of dark matter spread
throughout galaxy haloes is
A) We often cannot see much of the galaxy, since it is blocked by dark objects
B) Gravitational lensing shows most of the mass is in the form of massive compact
halo objects (MACHOs)
C) The total mass as measured by gravity is much less than the total mass as
measured by light
D) There are too few stars to create all the light, so there must be a dark source of
light
E) The rotation curves measured by Doppler shift do not fall off with distance
13. How do the three types of radioactive decay rank, from most penetrating to least
penetrating?
A) ,,
B) ,,
C) ,,
D) ,,
E) ,,
14. The central engine that powers an active galaxy is almost certainly a very massive
A) conventional star
B) black hole
C) neutron star
D) supernova
E) star cluster
15. The momentum operator in quantum mechanics is
 
2 d 2
B) x
C) 
D) V  x 
A)
i x
2m dx 2
E) 
2 d 2
V  x
2m dx 2
16. What is the approximate ratio of the size of an atom divided by the size of a nucleus?
A) 300,000 B) 3,000
C) 30
D) 0.03
E) 0.0003
17. When you excite the electrons in hydrogen, why do you only get very specific
wavelengths back out?
A) Hydrogen forms a lattice of atoms, and only those wavelengths that fit evenly
between the atoms are possible
B) The electrons can only take on discrete energy values, and when the electron
moves from one level to another, it emits a definite amount of energy
C) Hydrogen is very good at absorbing almost all wavelengths, so you only see those
wavelengths that aren’t absorbed
D) The interaction takes a very specific amount of time to take place, and the time it
takes times the speed of light c yields the wavelength
E) Actually, all wavelengths come out, but the diffraction gratings used to measure
the wavelengths cause interference for most other wavelengths
18. Which of the following formulas is still true in special relativity?

 



A) F  ma B) W  F  d C) E  12 mv 2 D) p  mv
E) None of these
19. What fraction of a heavy nucleus (A ~ 200) might be protons?
A) 10%
B) 40%
C) 50%
D) 60%
E) 90%
20. In general relativity, the gravitational source (cause of gravity) is
A) Mass (only)
B) Energy (only)
C) Momentum (only)
D) The stress-energy-momentum tensor
E) Invisible elves that push everything around
21. Radio galaxies typically produce most of their radio power
A) Right in the center, at the nucleus
B) Spread throughout the disk of the galaxy
C) Coming from one or two lobes that stick well out of the galaxy
D) From a sphere that completely surrounds the visible galaxy; the halo
E) Coming from a ring in the plane of the galaxy that is near the rim
22. The name of the galaxy we live in is
A) Andromeda
B) Coma C) Milky Way D) Fornax
E) Whirlpool
23. If a wave function satisfies Schrodinger’s equation, but doesn’t satisfy the
normalization condition, what do you do to fix it?
A) Try changing the energy, and see if that can fix the problem
B) Adjust the potential function V to make it work out
C) Rescale the distance x to make it work out
D) Multiply (or divide) by a constant to make the normalization work out
E) Normalization is not ever necessary or useful, so ignore the problem
24. Which of the following can be found in significant quantities in spiral galaxies like
ours?
A) Cool gas (only)
B) Dust (only)
C) Young stars (only)
D) Cool gas and dust, but not young stars
E) Cool gas, dust, and young stars
25. Which of the following is the correct statement of the uncertainty principle?
x 1
p 1
A) x  12  B) p  12  C)
 2  D)
  E) xp  12 
p
x 2
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. Why is it that you would expect there to be a fourth component of momentum p
in special relativity? What does the fourth component of the conservation of
momentum rule turn out to be?
Since spacetime is four-dimensional, we would expect there to be a fourth
component of momentum, since a three-dimensional vector in four-dimensional space
makes no sense. The fourth component of momentum turns out to be energy (up to a
factor of c or so), and the fourth conservation law is nothing more than conservation of
energy.
27. Explain in terms of the uncertainty principle why it is that confining a particle
into a small space gives it energy.
If you put a particle in a small space, you are specifying its position, so x is
small. Since xp  12  , this means that the uncertainty in the momentum must be large,
which in turn implies that it is likely to have a high momentum, and hence a high energy.
28. Under what circumstances does it make sense to describe the wave function of a
particle as the product of a radial and angular part,   R  r  Y  ,   ?
Dividing the wave function into an angular and a radial part makes sense when
the potential V(r) is spherically symmetric, so it depends on r but not  or  .
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 (in terms of the neutron/proton ratio) which sort of nuclei
are likely to undergo - decay and which undergo + decay.
When a nucleus undergoes - decay, it loses a neutron and gains a proton,
whereas when it undergoes + decay it loses a proton and gains a neutron. Hence the
former decay tends to happen in nuclei with too high a proportion of neutrons, and the
latter in those with a proportion of protons. The ideal proton/neutron mix is about a
50/50 mix for light nuclei, and about a 40/60 (more neutrons) for heavy nuclei.
30. Explain qualitatively how it might be possible that different categories of active
galaxies might actually be more similar than they appear.
It is possible that the same active galaxy will look different from different angles.
Viewed edge on, it might be a radio galaxy, from an angle it might look like a radio-noisy
quasar, and looked straight along the axis where jets are shooting out, it might be a
blazar.
31. What are the four different categories of galaxies?
The four categories are spiral (pinwheel shaped), barred spiral (pinwheel with a
bar across the middle), elliptical (featureless ellipse) and irregular (none of the above.)
32. What is a geodesic? Under what circumstances does an object follow geodesics,
according to general relativity?
A geodesic is the longest proper time path between two points in spacetime.
Objects that have no forces other than gravity on them follow geodesics.
Part IV: Calculation (review material) [40 points]
Choose two of the following three questions and perform the indicated
calculations (20 points each)
33. A Borg Collective Cube ship is in the shape of a cube 3040 m on a side. The ship
is then accelerated to a speed such that at least one of its dimensions is now 2000
m, as viewed by a stationary observer.1
(a) What are the other dimensions of the ship?
Lorentz contraction is only in the direction of motion, so the dimensions will be
2000 m  3040 m  3040 m .
(b) How fast is it traveling?
The Lorentz factor is

1
1  v2 c2
1
Lp
L

3040 m
 1.520 ,
2000 m
 1.520 ,
1
v2

 0.4328 ,
2
c 1.5202
v2
 1  0.4328  0.5672 ,
c2
v  c 0.5672  0.753c  2.26 108 m/s .
(c) How long would it take, in years, at this speed, to travel the 4.2 light years
from -Centauri C to the Earth?
This is just distance over speed, so
t
d 4.2 c  y

 5.58 y .
v 0.753c
(d) According to a clock on the Borg ship, how long would it take to go this
distance?
Because of the high speed, time will be dilated, and we find

1
t


5.58 y
 3.67 y .
1.520
Assume that “warp drive” does not exist. In other words, assume relativity applies.
34. A harmonic oscillator with angular frequency   1.20 1014 s 1 is in the state n
= 11.
(a) What is the energy of this electron?
We use the formula for the energy of the harmonic oscillator,
E11   11  12    6.582 1016 eV  s 1.20 1014 s 1  11.5   0.9083 eV
(b) If the electron shifts to the lowest possible energy state, what is its new
energy? Will it emit or absorb a photon to make this transition?
This is just the same formula, but we have to remember that the minimum energy
is n = 0, so we have
E0    0  12    6.582 1016 eV  s 1.20 1014 s 1   0.5   0.0395 eV .
Since the energy decreased, the excess energy must be emitted, not absorbed.
(c) What is the energy of the photon in part (b)?
The energy of the photon is just the difference between the two energies, so
E  E11  E0  0.9083 eV  0.0395 eV  0.8688 eV .
(d) What is the frequency and wavelength of the photon in part (c)?
The frequency can be found from E  hf , so we have
f 

E 11.5  0.5  
11
11

 11  
1.20 1014 s 1   2.1011014 s 1 .


h
h
h
2
2
We then get the wavelength from E  f   c , so that

c 2.998 108 m/s

 1.427 106 m  1427 nm .
f 2.101 1014 s 1
35. A neutral hydrogen atom is not in the ground state, but rather it has principal
quantum number n = 4.
(a) What are the possible values of l? What would be the corresponding possible
values of L2?
The value of l runs from 0 to n – 1. The value of L2 is given by L2   2  l 2  l  .
We therefore have
l  0,1, 2,3 , L2  0, 2 2 , 6 2 ,12 2 .
(b) What are the possible values of m? What would be the corresponding
possible values of Lz?
The value of m runs from –l to +l. The value of Lz is given by Lz  m . We
therefore have
m  3, 2, 1, 0,1, 2,3 , Lz  3, 2, , 0, , 2,3 .
(c) What are the possible values of s? What would be the corresponding
possible values of S2?
Electrons have s  12 , and therefore S 2   2  s 2  s    2  14  12   34  2 .
(d) What are the possible values of ms? What would be the corresponding
possible values of Sz?
The value of ms runs from –s to +s. The value of Sz is given by S z  ms . We
therefore have
ms   12 , 12 , S z   12 , 12  .
Part V: Calculation (new material): [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each)
36. The world produces about 400 g of tritium, 3H, per year, which has a half-life of
12.32 y.
(a) Approximately how many atoms of tritium are there in 400 g?
The mass of one atom is approximately 3 u, so the number of atoms is
N
M 400 g 400
400


NA 
6.022 1023   8.029 1025 .

m
3u
3
3
(b) What is the decay rate  in s-1 for this isotope? A year is 3.156107 s.
We use the formula relating the decay rate and the half-life, which is

ln 2 0.6931
y

 0.05626 y 1 
 1.783 109 s 1
7
t1 2 12.32 y
3.156 10 s
(c) How many decays per second are there when this much tritium is fresh?
Each decay produces 18,590 eV of energy. What is the power, in W, of all
this tritium? 1 eV = 1.60210-19 J.
We simply use the formula
R   N  1.783 109 s 1  8.029 1025   1.43 1017 s 1
We then get the power by simply multiplying by the energy per decay, so
P  RE  1.43 1017 s 1  18,590 eV  1.602 1019 J/eV   426 W .
(d) How long do you have to wait until you only have one atom of tritium left?
We use the formula N  N 0 e  t , solving this for t, wehave
et  N 0 N ,
t  ln  N 0 N   ln  8.029 1025   59.65 ,
t
59.65


59.65
 3.34 1010 s  1060 y .
9 1
1.783  10 s
37. Photocopied with the equation on the next page is a portion of Appendix A from
the text. 7Be is an unstable atom which decays with a half-life of 53.3 days.
(a) What would be the resulting isotope if this isotope underwent  decay?
What if it underwent electron capture? What if it underwent + decay?
Under  decay, the value of Z decreases by two, and A decreases by 4, so the
isotope that would be left over would be 3He (though in this case, it is unclear which is
emitting which!). Under both electron capture and + decay, A is unchanged, while Z
decreases by one, which leaves us with 7Li.
(b) What is the Q-value for each of these processes?
For  decay, the Q-value is given by
Q   M P  M D  M 4 He  c 2   7.016928  3.016029  4.002602  uc 2
  0.001703 931.494 MeV   1.586 MeV
For electron capture, the formula is simply
Q   M P  M D  c 2   7.016928  7.016003 uc 2
  0.000925  931.494 MeV   0.862 MeV
For + decay, the formula is the same except we subtract twice the rest energy of
an electron
Q   M P  M D  c 2  2me c 2  0.862 MeV  1.022 MeV  0.160 MeV
(c) Cosmic rays from deep space, many light years away, sometimes contain
7
Be+4 nuclei, which have no electrons. With such a short half-life, how is this
possible?
Looking at the three decays, we see that only electron capture is possible.
However, 7Be+4 is a bare nucleus, and therefore has no electrons to capture. So this
nucleus is in fact stable; it is only the atom (which includes electrons) which can decay
by electron capture.
38. A certain galaxy, viewed edge on,
has the H- line measured, which
normally has a wavelength of
  656.3 nm . It is found to vary
as a function of distance from the
center, as sketched at right.
(a) How fast is each side of the
galaxy moving, compared to
us, in km/s? In each case, is it
towards us or away from us?
We use the equations:
v

and z  r ,
c
0
vr 
  1.
c 0
1 z 
As far as I can tell, the asymptotic wavelength on the left side is about 650.1 nm and on
the right about 652.7 nm. We therefore have
vL 650.1 nm

 1  0.00945 ,
c 656.3 nm
vL   0.00945  299,800 km/s   2830 km/s ,
zL 
vR 652.7 nm

 1  0.00549 ,
c 656.3 nm
vR   0.00549  299,800 km/s   1640 km/s .
zR 
Since they came out negative, both sides of the galaxy are moving towards us.
(b) How fast is the galaxy moving towards or away from us, on average?
The average speed is
vA 
vL  vR 2830 km/s  1640 km/s

 2235 km/s .
2
2
Again, since it is negative, it is moving towards us.
(c) How fast is this galaxy rotating around its center, and which side is rotating
towards us or away from us?
The rotational velocity is the difference between the mean velocity and either
side, which is about 595 km/s. Since the left side is moving towards us faster, it must be
rotating towards us, and the right side away from us.
39. A black hole of mass M  20.0 M  ( M   1.989 1030 kg ) has a clock lowered to
a distance R from the black hole, and left there for one hour. It is then found
that the clock has the incorrect time by four minutes.
(a) What is the Schwarzschild radius for this star, in km?
We use the formula for the Schwarzschild radius, specifically
11
3
2
30
2GM 2  6.673 10 m / kg / s   20.0 M   1.989 10 kg/M  
RS  2 
2
c
 2.998 108 m/s 
 5.91104 m  59.1 km .
(b) Will the clock be four minutes fast or four minutes slow?
Time slows down near a mass, so it will be four minutes slow, so   56 min .
(c) What is the radius R, in km?
We use the formula
R
2GM
 t 1 s ,
2
cr
r
 R 
 2  t 2 1  s  ,
r 

  t 1
Rs  2  56 min 
 2 
1
  0.8711 ,
r
t
 60 min 
Rs
 1  0.8711  0.1289 ,
r
Rs
59.1 km

 458 km .
r
0.1289 0.1289
2
Equations
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
Constants:
G  6.673 1011 m3 / kg / s 2
N A  6.022 10
u  1.6611027 kg
2me c 2  1.022 MeV
23
M He  4.002602 u
1 y  3.156 107 s
Harmonic Oscillator: E    n  12  , n  0,1,
Schwarzschild radius: RS  2GM c 2
Gravitational time dilation:
2GM
  t 1 2
cr
Red Shift:
Isotope Masses
2GM
  0  1  2 
cr 

1/ 2