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 Schrdinger 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
