Name _________________
Test 3
November 10, 2014
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
Reflection off a step:
 E  E  V
0

R   E  E  V0


1

2

 if E  V0


if E  V0
Barrier penetration:
E
E
T  16 1   e 2 L
V0  V0 

2m V0  E 

Harmonic Osc.
En    n  12 
n  0,1, 2,
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
Hydrogen
En
13.6 eV  Z 2


n2
1D square well:
 2 2 n2
En 
2mL2
2
  nx 
sin 
 n  x 

L
 L 
n  1, 2,3,
1. Which of the following must be true of the wave function   x, t 
A)
B)
C)
D)
E)
It must be normalized (only)
It must vanish in any region where the potential is bigger than the energy (only)
It must be continuous (only)
Both A and B are true, but not C
Both A and C are true, but not B
2. Suppose   x  is a solution of Schrödinger’s time-independent equation with energy E.
What is the corresponding solution   x, t  of Schrödinger’s time-dependent equation?
A)   x  e iE t B)   x  e iE  t C)   x  e iEt  D)   x  e iEt 
E) None of these
3. For the infinite square well with allowed region from x = 0 to L, we included solutions that
went like sin  nx L  but rejected those that went like cos  nx L  . Why?
A)
B)
C)
D)
E)
The function must vanish at the boundaries, so sine works and cosine doesn’t
You can normalize sine, but not cosine
Cosine has negative energy, which is impossible
Sine satisfied the differential equation in the interior, cosine did not
We didn’t have a reason, I just hate cosine.
4. The smallest energy a harmonic oscillator with angular frequency  can have is
C) 
D) 32 
E) None of these
A) 0
B) 12 
5. According to quantum mechanics, if you hit a step potential with less energy than the height
of the potential, E < V0, what happens in the classically forbidden region?
A) The wave continues with the same amplitude as before
B) The wave continues as a wave, but with a much smaller amplitude
C) The wave grows exponentially in the forbidden region
, eventually
becoming infinite
D) The wave damps exponentially in the forbidden region, falling towards zero
E) The wave is automatically zero in the forbidden region
6. Which of the following is lowest energy state in a typical atom?
A) 3p
B) 4p
C) 4f
D) 3d
E) 4d
7. Which of the following corresponds to the momentum as operator, pop?
dx
 
p2
A) mv
B) m
C)
D) k
E)
V
i x
2m
dt
8. If you measured the total spin-squared of an electron, S2, the result you would get would be
A) 0
B) 14  2
C) 12  2
D) 34  2
E)  2
9. Which of the following is true when you do quantum mechanics in three dimensions?
A) The momentum operator gets
replaced by a single radial
momentum operator
B) The wave function becomes a
function of all three positions: x,
y, and z
C) There are three wave functions,
one for each of the dimensions
D) The energy is generally the
product of the energies in each of
the three dimensions
E) The probability density is no
longer represented by the
2
magnitude squared  of the
wave function
10. If we calculate the expectation value
of the Hamiltonian, H , the result
tells you what you would get, on
average, if you measure the
A) Position
B) Momentum
C) Angular momentum
D) Mass
E) Energy
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 we were doing reflection from a step barrier for V0 < E, we found solutions in the
“transmitted” region x > 0 that looked like   x   Ceikx  De  ikx , but we set D = 0. Why?
Later, when we redid it for V0 > E, we found solutions that looked like   x   e  x , but we
threw out e  x . Why?
12. Explain qualitatively how a scanning tunneling electron microscope works.
13. According to the Pauli exclusion principle, you can only have one electron for any given set
of quantum numbers. For the 1s level of hydrogen (or anything), you have quantum numbers
n = 1, l = 0, and m = 0, but you can still fit two electrons into this level. How is this
possible?
Part III: Calculation: [60 points] Choose three of the following four questions and perform the
indicated calculations (20 points each).
14. An electron of mass m  5.11105 eV / c 2 is trapped in a one-dimensional trap of size
L  3.00 109 m
(a) What is the energy of this electron, in eV if it has its minimum possible energy?
(b) What energy of photon would be required to knock it up to the level n = 5?
(c) Infrared photons have energy less than 1.65 eV. To which levels can this electron be
elevated if it absorbs a single infrared photon?
15. A group of 106 electrons with kinetic energy E = 18.00 eV approach a step with positive
potential V0 = +16.00 eV.
(a) How many of the electrons are reflected?
(b) The potential is now changed to a negative but unknown value. It is nonetheless found
that the same number of electrons are reflected. What is the new potential?
2a 3 
. Some possibly useful
x2  a2
  2  2a 2 
16. A particle has normalized wave function given by   x  
integrals appear below. This wave function has p 2
(a) What are x and x 2 for this wave function?
(b) What is the uncertainty x for this wave function?
(c) What is p ? What is the uncertainty p for this wave function?
(d) Check that it satisfies the uncertainty relation.



x n dx
x n dx
dx

n


0
if
is
odd
 x 2  a 2  x 2  a 2 2
 x 2  a 2  a ,


Useful integrals: 




x 2 dx
dx
x 2 dx
,
,



 x 2  a 2
 x 2  a 2 2 2a3  x 2  a 2 2  2a .




17. An electron in hydrogen is in the state n = 3, l = 1, m = –1.
(a) If we were to measure the energy E, the angular momentum squared L2, and the zcomponent of angular momentum Lz, what values would we get?
(b) Write down the explicit form of the wave function  nlm  r ,  ,   . Several radial and
angular wave functions are given below.
(c) For what locations can we be certain the particle cannot be found? You can give your
answers as values of r, , or  as needed.
R1,0 
2
a
Y1,0 
3
e  r a , R3,0 
 2r 2r 2   r 3 a
4 2r
e
, R3,1 
1  
2 
3
3 3a  3a 27 a 
27 3a 5
2
r  r 3a

1   e
 6a 
3
3
21
cos  , Y1,1 
sin  e  i , Y3,1 
sin   5cos 2   1 e  i
4
8
64