Name _________________
Test 3
November 14, 2012
This test consists of three parts. Please note that in parts II and III, you can skip one
question of those offered. Some possibly useful formulas can be found below.
h  6.626  1034 J  s  4.136 1015 eV  s
  1.055 1034 J  s  6.582 1016 eV  s
1D square well:
 2 2 n2
En 
2mL2
2
  nx 
sin 
 n  x 

L
 L 
n  1, 2,3,
Harmonic Osc.
En    n  12 
n  0,1, 2,
Barrier penetration:
E
E
T  16 1   e 2 L ,
V0  V0 

2m V0  E 

Hydrogen
13.6 eV  Z 2
En  
n2
Reflection off a step:
 E  E  V
0


R   E  E  V0


1

2

 if E  V0


if E  V0
Part I: Multiple Choice [20
points]
For each question, choose the best answer (2 points each)
1. Suppose an electron has n = 4, l = 2, and m = 0. Such an electron would be a ___
electron.
A) 4p
B) 4s
C) 4d
D) 2s
E) 2d
2. Suppose a particle is in a spherically symmetric potential. What is the total angular
momentum squared L2 if l = 4?
B) 12 2
C) 16 2
D) 17 2
E) 20 2
A) 4
3. The operator whose expectation value is the average of the total energy is called the
A) Schrodinger B) Lagrangian C) Hamiltonian
D) Momentum E) Potential
4. Suppose after a lot of work, I found a wave function   x  which satisfies
Schrodinger’s equation, but the normalization is wrong, so that    x  dx  4 .
2
What should I multiply the wave function by to fix this problem?
E)
A) 2
B) 4
C) 16
D) 12
1
16
5. What distinguishes a bound state from an unbound state?
A) Bound states have less energy than the potential at infinity
B) Bound states have more energy than the potential at infinity
C) Bound states have less energy than any barriers they might encounter
D) Bound states have more energy than any barriers they might encounter
E) Bound states are wave functions that cannot be normalized
6. In a complicated atom like Uranium, with 92 electrons, the reason you only get two
electrons in the 1s state is because
A) The Pauli Exclusion Principle says you can only have one particle per state, and
there are only two states (spin up and spin down) in the 1s orbital
B) The Pauli Exclusion Principle says you can only have two particles per state
C) Electrical repulsion makes it unfavorable for more than two electrons to be in this
state
D) Putting multiple electrons into this small space means they have enormous
momentum, according to the uncertainty principle
E) The US constitution states that just as you can only have two Senators from each
state, you can also only put two electrons in each quantum state. It wasn’t true
before 1791
7. If I send photons one at a time through a half-silvered mirror, and then put ideal
perfect detectors to see which way the photon went, what will the detectors see?
A) Each detector will register half a photon each time
B) Half the time one detector will see it, half the time the other will see it, but never
both and never neither, and the results will be random
C) Photons will alternate, one going one way and the next always going the other
way
D) Each detector sees the photon 50% of the time, so 25% of the time they will both
see it and 25% of the time neither sees it.
E) No photons will go either way, since quantum mechanical effects only happen if
you do many photons at once
8. Which of the quantum numbers l, m, and ms for an electron in a hydrogen atom can
sometimes be negative?
A) All of them can be negative
B) l and ms can be negative, but not m
C) l and m can be negative, but not ms
D) m and ms can be negative, but not l
E) None of them can be negative
9. When we calculate the expectation value of the position x , what does it tell us?
A)
B)
C)
D)
E)
Where a particle will be found
The average value of the position if you perform many measurements
The most likely place to find the particle
The uncertainty in the position
The root-mean-square value of the position of the particle
10. The radial wave function R(r) for an electron in hydrogen depends on which quantum
numbers?
A) n only
B) l only
C) m only
D) n and m
E) n and l
Part II: Short answer [20 points]
Choose two of the following three
questions and give a short answer (2-4
sentences) (10 points each).
11. When your professor was in high
school, his chemistry teacher taught
him that Schrodinger’s equation was
E  H , and he wondered why you
couldn’t just simplify this to E  H .
Explain why this doesn’t make any
sense.
12. A scanning tunneling microscope, to function, requires that electrons cross a gap
between the tip of the microscope and some object that they are studying. How can
the electrons cross that gap?
13. What is the electronic configuration for Cobalt, with Z = 27 electrons? Your answer
should look like 1s2 2s2 …
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each).
14. A group of quantum mechanical congressmen with kinetic energy E = 400. J are
approaching a fiscal cliff with potential V0 = –9,600 J.
(a) Find the probability that a congressman is reflected from the fiscal cliff.
(b) The potential V0 is now raised until it takes on a positive value, but less than E. It
is nonetheless found that the same fraction of congressmen are reflected from the
fiscal step. Find the new potential V0.
15. A proton of mass m  1.6726 1027 kg is trapped in a one-dimensional infinite
square well of unknown size L . A physicist studying this discovers that he can make
the electron jump from the ground state n = 1 to the state n = 4 when he bathes the
system with microwaves with frequency f  6.13 1010 Hz  6.13 1010 s 1 .
(a) What is the energy of one photon with this frequency?
(b) What is the size L of the infinite square well?
(c) The atom then transitions from n = 4 to n = 3. What is the energy of the resulting
photon that is emitted?
 a
16. A particle has wave function given by
 30  ax  x 2  a 5/ 2
  x  

0
for 0  x  a ,
otherwise .
This wave function has already been properly
normalized. It has expectation values
x  12 a and x 2  72 a 2 .
(a) Find the expectation values p and
x a
p 2 for this wave function.
(b) What are the uncertainties x and p for this particle?
(c) Does this wave function satisfy the uncertainty principle?
17. The wave function for the lowest energy (ground state) of the harmonic oscillator is
 0  x  4
 m x 2 
m
exp  


2 

(a) What is the probability density of the particle being at an arbitrary position x?
(b) What is the probability density (in m-1 or nm-1) of an electron with mass
m  1.6726  1027 kg being at the origin x = 0 if it is in a harmonic oscillator
potential with angular frequency   1.50 1013 s 1 ?
(c) If the wave function of a particle at t = 0 is given by   x, t  0    0  x  , what is
  x, t  at other times?