Physics 237, Final Exam

Wednesday
May 9, 2012

7.15 pm
– 10.15 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
and your equation sheet while taking this test. You may *not* consult any calculators, computers, books, or each other.

1. Problems 1, 2, and 3 must be answered in booklet # 1.

2. Problems 4, 5, and 6 must be answered in booklet # 2.

3. Problems 7, 8, and 9 must be answered in booklet # 3.

4. 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 three 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: ______________________________________

**One-Electron Atoms – Details**

The following table lists the *n* = 1, *n* = 2, and *n* = 3
wavefunctions of the one-electron 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 one-electron atom for *n*
= 1, 2, 3 and various values of *l*.

**Ground-state Properties of the Deuteron**

á
Binding energy: D*E* = 2.22
MeV

á
Nuclear spin: 1

á
Nuclear parity:
even

á
Magnetic dipole
moment: µ = +0.857µ_{n}

á
Electric
quadrupole moment: *q* = +2.7 ×
10^{-31} m^{2}

á
Charge
distribution half-value radius: *a* =
2.1 F

**Problem 1 (20 points) ANSWER
IN BOOKLET 1**

The energy of a linear harmonic
oscillator is equal to . The
angular frequency of this oscillator is . Use the uncertainty principle to determine the
minimum energy of the oscillator, expressed in terms of the frequency *v* where

.

**Problem 2 (20 points) ANSWER
IN BOOKLET 1**

Consider ¹^{ -} capture by a
deuteron. A slow ¹^{ –}
in liquid deuterium looses energy by a variety of mechanisms until it finally
ends up in the lowest Bohr orbit around the (*pn*) nucleus. It is then
captured through the action of the strong force. The result of this capture is the
following reaction:

¹^{ -} + *d* --> *n* + *n*.

a)
What it the
angular momentum of the initial state?

b)
What are the
possible states of the exit channel?
List all possible states with *L*
up to 3 using spectroscopic notation.

c)
Based on the
conservation properties of the strong force, which of the states in part b) do
you expect to be able to observe in this reaction?

d)
How can you use
this information to determine the parity of the pion?

**Problem 3 (30 points) ANSWER
IN BOOKLET 1**

Consider the following wavefunction
describing a single particle in a one-dimensional world:

.

The constant *a* is known.

a)
What is the
value of *N*?

b)
What is the
expectation value of *x*?

c)
What is the
uncertainty in *x*?

d)
What is the
expectation value of *p*?

e)
What it the uncertainty
in *p*?

f)
Do your answers
to part c) and e) agree with the Heisenberg uncertainly principle?

**Problem 4 (30 points) ANSWER
IN BOOKLET 2**

Consider a particle of mass *m* and energy *E*, approaching a step potential of height *V*_{0} at *x* =
0. You do not know in which region
(*x* < 0 or *x* > 0) the potential is non-zero and you do not know from which
direction the particle is approaching the step potential. You also do not know if *E* is larger or smaller than *V*_{0}. But you are able to measure the
probability density distribution in the *x* < 0
region. Three different cases are
shown in the Figure below. For each
case, sketch the step potential and indicate the energy of the particle (*E* > *V*_{0} or *E* < *V*_{0}) and its initial
direction. You need to motivate
your answer.

**Problem 5 (35 points) ANSWER
IN BOOKLET 2**

Consider the first few low-lying
states of the Helium atom. When the
Helium atom is in its ground state, both electrons are in *n* = 1 states. When the
Helium atom is in one of the first excited states, one electron is in an *n* = 1 state and the other electron is in
an *n* = 2 state. The spatial wavefunction of electron *i* can be written as

where *a* contains information about the spatial quantum numbers.

The spin wavefunction of the
two-electron system, , can be written as combinations of the following four
functions

a)
Write down all
possible total wavefunctions for the ground state of Helium (*n*_{1} = 1 and *n*_{2} = 1).

b)
Write down all
possible total wavefunctions for the first low-lying excited states of Helium (*n*_{1} = 1 and *n*_{2} = 2).

c)
Make an energy
diagram showing the location of the low-lying energy levels of Helium (*n*_{1} = 1 and *n*_{2} = 1, 2), assuming there is
no Coulomb interaction between the electrons. Label each level with the *n* and *l* values of each of the two electrons and their total spin. Note: the actual location of these
energy levels is not important, but their relative position is.

d)
Now include the
effect of the Coulomb interaction between the electrons, but ignore the
exchange force. What is the effect
of the Coulomb interaction on the energy of the states shown in the diagram
constructed in c)? In the energy
diagram, indicate the shifts of the energy levels of the low-lying states of
Helium due to the Coulomb interaction. Label each level with the *n* and *l* values of each of the two electrons and their total spin. Note: the relative shifts of the levels
are important and need to be correctly motivated and drawn.

e)
Finally, include
the effect of the exchange force.
What is the effect of the exchange force on the energy of the states
shown in the diagram constructed in d)?
In the energy diagram, indicate the shifts of the energy levels of the
low-lying states of Helium due to the exchange force. Label each level with the *n* and *l* values of each of the two electrons and their total spin. Note: the relative shifts of the levels
are important and need to be correctly motivated and drawn.

**Problem 6 (20 points) ANSWER
IN BOOKLET 2**

Measurements made on the line spectrum
emitted by a certain atom of intermediate *Z*
show that the ratio of the separation energies between three adjacent levels of
increasing energy in a particular multiplet is approximately 3 to 5 (that is
the energy difference between the 2^{nd} and the 3^{rd} member
of the multiplet is 5/3 of the energy difference between the 1^{st} and
the 2^{nd} member of the multiplet.)

a)
What are the *jÕ* quantum number that can be assigned
to these states?

b)
What is the *lÕ* quantum number that can be assigned
to these states?

c)
What is the *sÕ* quantum number that can be assigned
to these states?

**Problem 7 (20 points) ANSWER
IN BOOKLET 3**

Consider multi-electron atoms. To study the properties of the levels
occupied by the electrons, ionization energies and x-ray energies of these
atoms are measured. The following
two figures show the results of the measurements of these energies as function
of the nuclear charge of the atom.

What do we learn about the energy
levels of the electrons in these atoms on the basis of these measurements?

**Problem 8 (20 points) ANSWER
IN BOOKLET 3**

Consider the following potential
energy distribution:

Three acceptable eigenfunctions for
this potential are shown in the figure below.

The three wavefunctions have the same
value at *x* = *x*_{0}. Use this
information to rank the three wavefunctions in order of their energy (lowest
energy, middle energy, highest energy).
Your answer must be well motivated (e.g. node counting is not sufficient
justification).

**Problem 9 (5 points) ANSWER
IN BOOKLET 3**

Who is the
greatest closer in baseball who has saved 608 games, the most in Major League
history, and finished a record 892 games.
He is shown in the picture below, taken by your instructor, when he
broke the Major League record.

a)
Brian Wilson

b)
Mariano Rivera

c)
Jonathan
Papelbon

d)
Scott Downs

e)
Santiago Casilla

f)
Craig Kimbrel

g)
John Axford

h)
Chris Perez

i)
Joel Hanrahan

j)
JJ Putz

GOOD

LUCK!