Name _________________
Test 2
October 17, 2012
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.
Constants
h  6.626  10 J  s  4.136 1015 eV  s
Black Bodies
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
k  8.988 10 N  m / C
e  1.602 1019 C
1 eV  1.602 1019 J
ke 2
1

 7.29735 103 
137
c
Compton Effect
Wave
h
Relationships

  
1  cos  
mc
2

h
12
k
 2.426 10 m
mc

1
 f 
2
T
Hydrogen Spectrum
9
1 
 1
 2
2
n m 
   91.17 nm  
2
2
1
U
 2  k BT 
15  c 
4
3
maxT  2.898 103 m  K
Rutherford Scattering
kqQ
 
cot  
b
2
m v
2
R
2Ze 2 k
E
Hydrogen-Like Atoms
c2  2 Z 2

k 2e4  Z 2

E
2 2 n 2
2n 2
 13.6 eV  Z 2
E
n2
Reduced Mass
mM

mM
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
1. If the wave function is given by   x   2ie   x , what is the probability density?
A) 4e 2  x
B) 4ie 2  x
C) 4e 2  x
2. The formula for group velocity is
A) f 
B)  k
C)  k
D) 4e   x
2
D) d  dk
E) 4e   x
2
E) f 
3. In the Frank-Hertz experiment, what is the source of the light that is being emitted?
A) The voltage is getting large enough to cause arcing, which is causing the glow
B) The accelerated electrons are getting converted directly into photons
C) Accelerated electron pairs collide to produce light
D) The electrons collide with the anode, which becomes warm and glows
E) The atoms that have been impacted by electrons fall back to a lower energy state
4. Why do different isotopes of hydrogen have actually very slightly different spectral
lines?
A) The charges of the nuclei are very slightly different
B) You have to work with the reduced mass , rather than the electron mass, which
is a tiny bit different for different isotopes
C) The photons emitted are partially absorbed by the nucleus, losing some energy
D) The atoms are in motion, causing Doppler shift, but the heavier isotopes are
moving more slowly
E) Magnetic interactions with the nucleus cause a shift in the energies
5. Suppose you know the speed of an electron. What can you learn by studying the
curve it makes in a magnetic field?
A) Its mass m
B) Its charge e
C) The ratio of its mass and its charge, e/m
D) The product of its mass and its charge, me
E) None of these can be learned from this experiment
6. The fundamental assumption Planck had to make to explain the black body spectrum
was
A) Light acts as if it comes in chunks of energy proportional to the frequency
B) Light acts as if it comes in chunks of energy proportional to the wavelength
C) The black body spectrum must have a peak wavelength max
D) The angular momentum of photons was always a multiple of 
E) Light is actually waves, not particles as previously thought
7. Rutherford scattering with -particles allowed Rutherford to
A) Measure the angular momentum of the electrons in an atom
B) Measure the energy levels of the electrons in an atom
C) Measure the charge of the electrons in an atom
D) Measure the size of atoms
E) Measure the size of the nucleus
8. Which of the following was a success of the Bohr model of the atom?
A) It correctly predicted the approximate radius of hydrogen (only)
B) It correctly predicted the wavelengths of the light emitted from hydrogen (only)
C) It correctly predicted the wavelength of light emitted from other atoms, like
neutral helium (only)
D) A and B are true, but not C
E) A, B, and C are all true
9. What is the correct relationship between h and  ?
h
1
2
A)   2 h B)  
C)  
D) h 
2

h
E)  
1
2 h
10. How did deBroglie’s relation p  h help
explain the Bohr model?
A) It explained diffraction for electrons,
which was assumed by Bohr
B) It explained why only circular orbits
worked in the Bohr model
C) It explained why energy always was
emitted as a single photon
D) It explained why the angular momentum
was always multiples of 
E) It explained the isotope effects of
different isotopes of hydrogen
Part II: Short answer [20 points]
Choose two of the following questions
and give a short answer (2-3 sentences) (10
points each).
11. Explain qualitatively how one can use the
spectrum of light from a distant star to determine the star’s temperature, assuming
that spectrum is a black body distribution.
12. When a photon scatters from an electron at rest, how does the scattered photon’s
wavelength, momentum, and energy differ (increase or decrease) from the initial
photon?
13. Classically, it was not understood why the electron did not simply fall to the position
of the nucleus in an atom. Explain, in terms of the uncertainty principle, why this
doesn’t happen.
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each).
14. Ytterbium (Yb) has a work function   2.60 eV .
(a) What is the lowest frequency f that can liberate an electron from ytterbium?
(b) Suppose light with wavelength   257 nm impacts some ytterbium. What is the
voltage Vmax that the liberated electrons from ytterbium can overcome?
(c) When an unknown light is shone on ytterbium, it is found that the electrons can
only overcome a voltage of Vmax = 1.75 V. What is the frequency of the light?
15. A certain element, when all but one electron has been removed, emits light with a
wavelength of 7.60 nm when an electron falls from n = 2 to n = 1.
(a) What is the energy of this photon?
(b) What is the Z-value of this element?
(c) If this atom had an electron fall from n = 9 to n = 8, what would be the energy of
the resulting photon?
16. Neutrons with a mass of
m  1.675 1027 kg are being fired at a

neutrons
narrow slit of width w  14.0 nm with a
kinetic energy of E  4.86 1023 J .
(a) What is the momentum of these
neutrons?
(b) By Carlson’s rule, what is the uncertainty in their vertical position based on the
fact that they went through the slit? Based on the uncertainty principle, what is
the corresponding uncertainty in their vertical momentum p ?
(c) Assume the total momentum p is the same after they pass through the slit, but
they have now acquired a vertical momentum p . How much has their direction
changed by passing through the slit?
17. The wave function of a particle is given by
 N  a 3 x  x 4  0  x  a,
  x  
0
otherwise .


Na 4
where N and a are positive constants. The
wave is sketched at right.
x a
(a) What is the most likely place to find the
particle?
(b) What is the normalization constant N?
(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 3 2 ?