Physics 214b-2008     Exam 2                  Haverford College  Walter Smith

Exam Guidelines:

1.  IMPORTANT:  Shortly before beginning the exam, you must check your e-mail, and also send an e-mail to wsmith@haverford.edu to notify me that you’re about to start.  This way, if there are last minute corrections, it will be easy to distribute them to the rest of the class.  If you find something on the exam which is erroneous or ambiguous, call me immediately (Office: 610-896-1332, Home 610-896-1565), even if it’s very late at night!  Although I try extremely hard to make everything completely clear, it is sometimes difficult to anticipate the way that others might interpret the problems.  If you call me, we can straighten it out, and using the e-mail mechanism above, we can straighten it out for the other students, also.

2.  This is a 150 minute, take-home exam.  It must be completed in one continuous sitting (i.e., the clock doesn’t stop if you take a break).

3.  No books, notes, etc. of any kind are permitted, with the exception of a “summary sheet” which may contain up to 40 equations. 

4. You should have a calculator to take this exam.  If your calculator has graphing, programmable, or symbolic algebra/calculus features, your may not use these during the exam.  You may not use Mathematica.  You should have a ruler marked in mm and cm.

5.  This exam contains six problems.  When you’re ready to begin, please check that all are present.

6.  For each problem, circle your final answer.

7.  The exam is to be turned in to the envelope outside my office (L110) by 4:00 Thursday.  Please do not turn in this exam itself or your equation sheet; only turn in your examination book.  You should carefully keep this exam for future reference when your graded exam is returned.

8.  Please note your starting and ending times on the front cover of the examination book.

9.  Use three significant digits on all problems, unless otherwise indicated.

Some Words of Advice:

I look forward to giving you partial credit for your work.  To receive credit, present your work in a clear, readable format.  If you find yourself stuck on a problem, don’t panic.  Instead, carefully explain what you do know about the problem, what you think is going on, and how you might proceed if you could somehow get yourself “unstuck” from the part of the problem that is giving you trouble.  Remember, make sure you clearly explain what you write down, since I cannot give partial credit for things I find ambiguous, or for simply copying down equations from your summary sheet. SHOW ALL YOUR WORK!  SHOW ALL YOUR WORK IN THE EXAMINATION BOOKLET!  Be sure to label your work with the problem number.

 

IMPORTANT: You will find some multiple-part problems on the exam.  Even if you can’t get the first part of such a problem, it is almost always possible to go on and do the other parts!!!!!!!!

 

There will be some time pressure on this exam.  Use the time available wisely.

 

There are 100 points total available on the exam.

After You’ve Completed the Exam:

1. Indicate your ending time on the front cover of your examination booklet.

2. Please sign the honor code pledge on the front cover of your examination booklet.

3. Please do not discuss ANY aspect of this exam with your classmates until I tell you it’s okay (this includes, for example, even such things as whether the exam was easy or difficult, long or short, etc.).

 

k = 8.999 Nm2/C2         eo = 8.85-12 C2/Nm2 = 1/(4pk)             Charge of electron = -e, where e = 1.60-19 C

 

c = 3.008 m/s                mo = 4p-7 N/A2                                    Mass of electron = me = 9.11-31 kg

                                                                                                           

h = 6.63-34 J s                = 1.05-34 J s                                    Mass of proton = mp = 1.67-27 kg

 

kB = 1.38-23 J/K            NA = 6.022 atoms/mol

 

 

 

Some integrals you may or may not need:

                                

         

                 

                                  

For the following, the parameter a is assumed >0:

                  

 

                   

 

                      

1. (10 points)  Define the terms eigenfunction and eigenvalue.

 

2. (20 points total)

a. (15 points)

In your solution, you should specify the values of all constants in terms of the energy and the incident amplitude.  It is fine to express your answer in terms of constants such as k and k0, so long as you have defined these constants explicitly.

b. (5 points)  What is the transmission coefficient?

 

3.  (15 points total)

a. (10 points)  A particle of mass m moves in a potential energy . Prove that is an energy eigenfunction, find the value of a, and find the corresponding energy.

b.  (5 points) How can you tell that this must be the ground state?

 

 

 

Text Box: 4.  (15 points) The ground state for the potential energy shown here is called , corresponding to an energy .  The energy  for the sixth state up is shown in the diagram.  Sketch .

 

 

Exam continues on the next page.

 

 

5.  (15 points total)   

a.  (10 points) Show that , where K is the kinetic energy, and we assume , i.e. that the potential energy does not depend on time.  Hint: In one of the practice problems for this exam, we showed that

.  Since , this means that

.  Applying this to this problem means that

.  Also, in class we showed that .

b (5 points)  Explain why the result of part a means that, for an energy eigenfunction, .

6.  (25 points total: 5 points for part a, 10 for part b, 10 for part c)

  Answer: .

End of exam