PHY 335 Exam II
Name ________________________
Instructions:
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Place your name on all of the pages.
Do all of your work in this booklet. Do not tear off any sheets.
Show all of your steps in the problems for full credit.
Be clear and neat in your work. Any illegible work, or scribbling in the margins, will not be
graded.
Put a box around your answers when appropriate..
If you need more space, you may use the back of a page and write On back of page # in the
problem space or the attached blank sheets. No other scratch paper is allowed.
Try to answer as many problems as possible. Provide as much information as possible. Show
sufficient work or rationale for full credit. Remember that some problems may require less work than
brute force methods.
If you are stuck, or running out of time, indicate as completely as possible, the methods and steps you
would take to tackle the problem. Also, indicate any relevant information that you would use. Do not
spend too much time on one problem. Pace yourself.
Pay attention to the point distribution. Not all problems have the same weight.
Be careful to note units and use an appropriate number of significant digits.
Page
Pts
1
22
2
18
3
18
4
12
Total
70
Score
Constants
h  6.626 10
34
Js,  1.055 1034 Js , a0  0.5292 1010 m,
me  9.11 1031 kg, mp  1.67 1027 kg, 1eV  1.6 1019 J
________________________________________________________________________
Bonus:
a. Who introduced the concept of matter waves?
______________________________
b. Enter the quantum numbers (n,l,m) for hydrogenic
probability densities shown on the right.
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PHY 335 Exam II
Name ________________________
1. (12 pts) Evaluate the following:
a. Simplify to standard form:
b. Simplify:
(1  i )3
(1  i ) 2
2i
3  4i
c. Put in exact standard form without trigonometric functions: 6e3 i /4 .
2
d. Evaluate:  , for  ( x, t )  5e2t i (3 x 5t )
2. (6 pts) Let    k 3 . Find the phase and group velocities.
3. (4 pts) It takes 4.0 eV to excite an electron in a one dimensional box from the
ground state to the first excited state. What is the length of the box in nm?
PHY 335 Exam II
Name ________________________
4. (6 pts) A molecule of mass 2.0 1024 kg is moving along at a speed of
100  10 m/sec. The molecule is localized in space and is described by a traveling
wave packet solution to the Schrödinger wave equation centered around x  0 at
t  0 and moving in the positive x direction.
a. What is the minimum uncertainty Δx in the determination of its position?
b. What is the de Broglie wavelength of this molecule?
5. (12 pts) Suppose a particle in a box has an initial wavefunction given by
3 x
 ( x, 0)  A sin
, 0  x  4.
4
a. What must A be for the wavefunction to be normalized?
b. What is the initial probability density at x  2?
c.
What is the probability that the particle is initially in the interval [2,3]?
d. What is the energy of the particle? [Leave in terms of ,  , and m.]
e. What is  ( x,5), i.e., what is the wave function at t  5?
PHY 335 Exam II
Name ________________________
6. (9 pts) Consider the quantum harmonic oscillator.
a. Write the one dimensional time independent Schrödinger equation for
 ( x) for the quantum harmonic oscillator.
b. Sketch the potential energy V ( x ) and wavefunction  3 ( x) for the quantum
harmonic oscillator.
c. What is the energy E3 associated with  3 ( x)?
7. (9 pts) The z-component of the angular momentum operator takes the form
 
 
LZ  i  x  y  .
x 
 y
a. Calculate [ Lz , y] for    ( x, y, z ).
b. Let  322 (r , ,  ) be a hydrogen wave function. Complete the following
eigenvalue problems for this wave function using the appropriate quantum
numbers:
i. Lz 322 
ii. L2 322 
iii. H 322 
PHY 335 Exam II
Name ________________________
8. (12 pts) A free electron of kinetic energy 2.5 eV is traveling in a straight line to
the right. It approaches a region of constant potential of +2.6 eV.
a. What are the wave number k1 (in m-1)and wavelength 1 (in m) of the
electron in the free region on the left?
b. Does the electron propagate as a wave in the region on the right or does
its wavefunction decrease exponentially? ________________________
c. If it propagates, what is the wave number k2 (in m-1) of the electron in this
region on the right? If it decays away exponentially, what is the decay
constant α in the expression e x ?
d. Does the electron undergo any reflection at x  0? Explain in one or two
sentences.