Name _________________
Test 2
October 15, 2014
This test consists of three parts. Please note that in parts II and III, you can skip one
question of those offered. The equations below may be helpful with some problems.
Black Bodies
Constants
h  6.626  10 J  s  4.136 1015 eV  s
34
  1.055 1034 J  s  6.582 1016 eV  s
k B  1.3807 1023 J/K  8.6173 105 eV/K
U
 2  k BT 
15  c 
4
3
maxT  2.898 103 m  K
k  8.988 10 N  m / C
Rutherford Scattering
e  1.602 1019 C
kqQ
 
1 eV  1.602  1019 J
cot  
b
2
m v
2
ke 2
1

 0.00729735 
2 Ze 2 k
137
c
R
E
Compton Effect
Hydrogen-Like Atoms
h
c2  2 Z 2
Wave

k 2e4  Z 2
   
1  cos  
E

2 2
2
mc
Relationships
2 n
2n
h
2
12
 13.6 eV  Z 2
 2.426 10 m

E

mc
k
n2

1
Hydrogen Spectrum
 f 
Reduced
Mass
2
T
1
1 
 1
mM
   91.17 nm   2  2 

n m 
mM
9
2
2
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
1. If   ei , what is  ?
2
A) e 2i
B) e  i
C) e
2
D) e 
2
E) 1
2. For light in vacuum, the phase velocity is __ and the group velocity is __?
A) c, c/2
B) c/2, c
C) c, c
D) c, 2c
E) 2c, c
3. The Millikan oil drop experiment allowed Millikan, for the first time, to measure
A) The size of the atom
B) The size of the nucleus
C) The wavelength of electrons
D) The charge of electrons
E) The size of the electrons
4. When an X-ray scatters from an electron at rest in an atom, the final photon will have
a _____ wavelength and a ________ frequency
A) longer, lower
B) longer, higher
C) shorter, lower
D) shorter, higher
E) unchanged, unchanged
5. Which assumption did Planck have to make to account for blackbody radiation?
A) The electron orbits the nucleus with angular momentum a multiple of 
B) The energy of electromagnetic waves comes in multiples of hf
C) The wavelength of an electron of momentum p is h/p
D) An electron is described in terms of a wave function, not position and velocity
E) When electrons move from one level to another in an atom, the energy comes out
as a single photon
6. An expression for  is
A) 2 h
B) h  2 
C) 2 h
D)  1  2 h  E) none of these
7. Which of the following is approximately the radius of a typical atomic nucleus?
A) 3 106 m B) 3 109 m C) 3 1011 m D) 3 1015 m E) 3 1018 m
8. What evidence convinced Rutherford that the positive charge and most of the mass of
the atom was concentrated in a small region, rather than spread out
A) Alpha particles, when scattered off of atoms, sometimes rebounded at large angles
B) He could measure the change in charge when a single nucleus left or was added to
a small piece of gold foil
C) He was able to focus the alpha particles as they scattered from the nucleus to
produce an image of the nucleus
D) He could see the nuclei because they were knocked out of the gold foil by the
alpha particles
E) The electrons knocked out of atoms had angles indicating they were coming from
a point charge
9. When electrons in the innermost level of an atom change levels, you can get the
approximate energy by treating it as if the nucleus is Z  1 instead of Z. Why is this
appropriate?
A) The electron is moving at high speed, which due to relativistic effects, reduce the
effective nuclear charge
B) The electron is partly inside the nucleus, which means some of its charge is
cancelled
C) The other electrons partly shield and cancel out the nuclear charge
D) The Coulomb force gets weaker at short distance, effectively reducing the charge
E) There are magnetic effects as well as electric forces, which partly overcome the
attraction of the nucleus
10. To make a wave packet, what must you do?
A) Include both cosines and sines in the wave
B) Use a complex wave, eikx
C) Include one wave moving to the right and one to the left
D) Include a potential that traps the particle in a small region
E) Add up many waves with different wavelengths
Part II: Short answer [20 points]
Choose two of the following questions and give a short answer (2-3 sentences)
(10 points each).
11. There is background light left over from the big bang event that formed the universe.
Explain, in principle, how one could measure the temperature of this light. Include at
least one equation
12. Explain qualitatively why an electron confined to a small region automatically has a
lot of energy. At least on equation or inequality is appropriate.
13. Explain impact parameter b for a collision between two objects is. It might help to
sketch a diagram.
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each).
14. When silicon is irradiated with light, it is observed that electrons are released, but
only if the light has a frequency f > 1.1121015 s-1
(a) Explain why there is a minimum frequency. Find the work function  for silicon.
(b) Silicon is irradiated with light with frequency f = 2.431015 s-1 and electrons are
found to escape from the surface. Predict the maximum resulting voltage Vmax
that these electrons can overcome after escaping the surface
(c) Light of unknown frequency and wavelength is found to produce electrons that
have 1.60 eV of kinetic energy after having been ejected from silicon. What is
the frequency and wavelength of this light?
15. An atom with unknown atomic number Z has a single electron in the orbit n = 5. The
electron then falls down the smallest possible amount to the next level.
(a) Find a formula for the initial energy and for the final energy, in eV, in terms of Z,
as well as their difference
(b) The photon released is found to have an energy of 2.75 eV. What is Z?
(c) The atom is returned to the n = 5 state, and we wish to now increase its energy by
the smallest possible amount, so it goes up one level. What is the energy
difference this time?
(d) What wavelength of light should you bathe the atom in if you want it to increase
its level as described in part (c)?
16. A one-dimensional crystal has sound waves with dispersion relation   0 sin  kd  .
(a) Find a formula for the phase velocity and the group velocity in terms of k.
(b) Evaluate both velocities if k  1.80 109 m 1 , d  2.00 1010 m ,
0  6.00 1012 s 1 . Don’t forget to work in radians!
(c) Just as light waves are quantized, so also are sound waves. Find the energy of
one phonon (quantum of sound) in eV using the same numbers given in part (b).
17. The wave function for a particle is given by
 2N

  x    3 N
 0

0  x  3a
3a  x  4a
otherwise
2N
N
 ( x)
x
-N
a
2a 3a 4a
-2N
where N and a are positive constants. The wave
-3N
is sketched at right.
(a) What is the normalization constant N?
(b) Which of the regions has the highest probability density of finding the particle?
(c) If the particle’s position is measured, what is the probability that it will be found
to have a value in the range 0  x  a ?