Physics 237, Midterm Exam #1

Thursday
February 18, 2010

12.30
pm – 1.45 pm

**Do not
turn the pages of the exam until you are instructed to do so.**

**Exam
rules:** You may use *only* a writing instrument while taking
this test. You may *not* consult any calculators, computers,
books, or each other.

1. Problems 1 and 2 must be answered in booklet # 1.

2. Problems 3 and 4 must be answered in booklet # 2.

3. The answers need to be well motivated and expressed in terms of the variables used in the problem. You will receive partial credit where appropriate, but only when we can read your solution. Answers that are not motivated will not receive any credit, even if correct.

At the end of the exam, you need to hand
in your exam, your Òcheat sheetÓ, and the two blue exam booklets. All items must be clearly labeled with
your name, your student ID number, and the day/time of your workshop.

Name:
__________________________________________________

ID number:
______________________________________________

Workshop
Day/Time: ______________________________________

**Problem 1 (35 points) ANSWER
IN BOOKLET 1**

Consider a particle of mass *m* moving with a linear momentum *p*.
Assume the velocity of the particle is much less than the speed of light
and relativistic effects do not need to be considered. In order to describe the particle in
terms of a matter wave, we first consider the following matter wave:

a)
What is the
propagation velocity of this matter wave?
Specify both the magnitude and the direction of the propagation velocity. Express your answer in terms of *k* and *n*.

b)
How does the
propagation velocity of the matter wave compare with the velocity of the
particle?

Now consider that we describe the
particle by the following matter wave:

where *d**k*
<< *k* and *d**n*
<< *n*.

c)
This matter wave
has a low- and a high-frequency component. What are the propagation velocities associated with the low-
and the high-frequency components?
Specify both the magnitude and the direction of these propagation velocities. Express your answers in terms of *k*, *n*, *d**k*, and *d**n*.

d)
How do the
propagation velocities of the matter wave obtained in c) compare with the
velocity of the particle?

Your answers need to be well
motivated. A correct answer
without any motivation will not receive any credit.

**Problem 2 (30 points) ANSWER
IN BOOKLET 1**

Consider a
photon with an energy *E** _{g}* travelling in a vacuum. The energy of the photon is larger than 2 times the rest
energy of the electron (

a)
Can the photon
convert all of its energy by creating an electron-positron pair? If you answer is yes, calculate the
total kinetic energy of the electron and the positron. If your answer is no, show why this
process cannot happen in a vacuum.

Now consider a
photon with an energy *E** _{g}* producing an electron-positron pair in the vicinity
of a nucleus of mass

b)
What was the
energy of the pair-producing photon?

c)
What fraction of
the photonÕs linear momentum is transferred to the nucleus?

**Problem 3 (30 points) ANSWER
IN BOOKLET 2**

The graph shows the voltage dependence
of the current you measure in the Franck-Hertz experiment you carry out in the
advanced laboratory.

a)
Based on the
information provided in the Figure, construct an energy-level diagram of the
atoms used in the experiment.

b)
What are the
energies of the photons that are emitted by the atoms used in the experiment when
the experiment is operated with an accelerating potential of 7 V?

Consider the two-slit experiment used to observe
electron diffraction shown in the Figure.
The condition for constructive interference is . The
distance between adjacent maxima on the screen is .

c)
After observing
the interference pattern we install a monitor system that determines the
position of the electron just behind the slit screen with an accuracy so that
we can tell through which slit each electron went. Show that this measurement will wipe out the interference
pattern.

**Problem 4 (5 points) ANSWER
IN BOOKLET 2**

How many times did the Yankees win the world series?

1. 4

2. 25

3. 27

4. 45