Physics 237, Midterm Exam #1

Thursday February 26, 2009

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, nor each other.

 

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

 

2.     Problem 3 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, 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 (40 points)                                                                       ANSWER IN BOOKLET 1

 

Consider a one-dimensional region where the potential can be described by the potential of a simple harmonic oscillator:

 

 

a)     Verify that the wave function is a solution of the Schršdinger equation for this potential.

 

b)    What is magnitude of A?

 

c)     What are the expectation values of x and x2?

 

d)    What are the expectation values of p and p2?


Problem 2 (30 points)                                                                       ANSWER IN BOOKLET 1

 

The Wilson-Sommerfeld quantization rule states that

 

For any physics system in which the coordinates are periodic functions of time, there exists a quantum condition for each coordinate.  These quantum conditions are

 

 

where q is the one of the coordinates, pq is the momentum associated with that coordinate, nq is a quantum numbers which taken on integral values, and means that the integration is taken over one period of the coordinate q.

 

a)     Show how the Bohr quantization of angular momentum follows from the Wilson-Summerfeld rule.

 

b)    Show how PlanckÕs quantization law follows from the Wilson-Summerfeld rule.

 

Note: the area of the ellipse x2/a2 + y2/b2 = 1 is ¹ab.


Problem 3 (30 points)                                                                       ANSWER IN BOOKLET 2

 

The energy of a linear harmonic oscillator is equal to .  The angular frequency of this oscillator is .

 

a)     Show, using the uncertainty relations, that the energy of the linear harmonic oscillator can be written as

 

 

b)    Show that the minimum energy of the oscillator is hv/2 where