Physics 237, Midterm Exam #3
Tuesday
April 24, 2012
8.00 am
– 9.30 am
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. 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 equation sheet, 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: ______________________________________
OneElectron Atoms – Details
The following table lists the n = 1, n = 2, and n = 3
wavefunctions of the oneelectron atom.
In these
wavefunctions, the parameter a_{0}
is defined as
The energy of
each wavefunction is equal to
The radial probability density for the
electron in a oneelectron atom for n
= 1, 2, 3 and various values of l.
Problem 1
(30 points) ANSWER
IN BOOKLET 1
a)
Prove that any
two different nondegenerate bound wavefunctions y_{} and y_{} that are solutions to
the timeindependent Schršdinger equation for the same potential V(x)
obey the orthogonality relation
.
b)
What changes in
part a) when the wavefunctions y_{} and y_{} are degenerate?
Problem 2 (35 points) ANSWER
IN BOOKLET 1
Consider the three lowest energy
levels in Na, shown in the Figure below.
E_{2} 

^{2}P_{3/2} 


E_{1} 

^{2}P_{1/2} 








E_{0} 

^{2}S_{1/2} 

a)
What are the
LandŽ g factors for these levels?
b)
When the atom is
placed in a weak magnetic field, the energy levels split. Draw an energy level diagram showing the
energy levels and determine the corresponding energies (in terms of E_{0}, E_{1}, and E_{2}).
c)
Which
transitions between the ^{2}P and the ^{2}S energy levels are
possible? Explain why you selected
these transitions?
Problem 3 (30 points) ANSWER
IN BOOKLET 2
N
distinguishable atoms are distributed over two energy levels E = 0 and E = E_{1} in a
system that is maintained at temperature T.
a)
What is the
energy of the system?
b)
What is the heat
capacity of the system?
Problem 4 (5 points) ANSWER
IN BOOKLET 2
Who first used the following quote:
Ò May the odds be ever in your favor! Ò
a)
Derek Jeter
b)
Katniss Everdeen
c)
Harry Potter
d)
Frank Wolfs
e)
Effie Trinket
f)
Joe Girardi