Physics 237, Midterm Exam #3

Thursday
April 22, 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 (30 points) ANSWER
IN BOOKLET 1**

A system has *N* distinguishable atoms. The atoms are distributed over two energy levels:

a)
What is the
total energy of this system of atoms if the temperature of the system is *T*?

b)
What is the heat
capacity at constant volume of this system at temperature *T*?

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 system of three indistinguishable
particles in a potential well.
Assume that the particles do not interact with each other and can be
described by the following normalized wavefunctions: . The
subscripts *a*, *b*, and *g*
correspond to sets of different quantum numbers that specify the spatial and
spin quantum states of the wavefunctions.

a)
Construct the
total asymmetric wavefunction for this system of particles. Note: make sure that the total
wavefunction is properly normalized.
You must show that the total wavefunction is asymmetric with respect to
the exchange of the labels of any two particles.

b)
Show that the
wavefunction obtained in part a) can describe a system with three fermions.

c)
Construct a
total wavefunction that can describe a system of three bosons. Make sure that the total wavefunction
is properly normalized.

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

**Problem 3 (35 points) ANSWER
IN BOOKLET 2**

Consider
the two optical electrons in a ^{12}C atom. In the ground-state configuration, the electrons are in a 2p^{2}
configuration.

a)
Using proper spectroscopic notation, specify what states can be
formed when the two electrons are in a 2p^{2} configuration?

b)
Rank the states obtained in part a) in terms of energy (starting
with the state with the lowest energy and ending with the state with the
highest energy). Note: ignore
energy shifts due to the spin-orbit coupling.

c)
One of the electrons of the atom is excited to the 3s subshell. Using proper spectroscopic notation,
specify what states can be formed when the two electrons are in a 2p^{1}
3s^{1} configuration?

d)
Rank the states obtained in part c) in terms of energy (starting
with the state with the lowest energy and ending with the state with the
highest energy). Note: ignore
energy shifts due to the spin-orbit coupling.

e)
Make a schematic energy level diagram of all the states obtained in
parts a) and c). The diagram must
show the states, identified by their spectroscopic notation, in the correct
order.

f)
Indicate all transitions that can be observed between the states in
the energy level diagram obtained in part e) and explain why the transitions
you indicated can occur.

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

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

The New York Yankees have played 14
games when you take this exam.
What is their ratio of Wins to Losses after these 14 games?

1. 14:0

2. 13:1

3. 12:2

4. 11:3

5. 10:4

6. 9:5

7. 8:6

8. 7:7

9. 6:8

10. 5:9

11. 4:10

12. 3:11

13. 2:12

14. 1:13

15. 0:14

16. Who are the Yankees?