Physics 237, Final Exam

Tuesday
May 8, 2018

7.15 pm
Ð 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 and 5 must be answered in booklet # 2.

3. Problems 6, 7, and 8 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, the blue exam booklets, and the
equation sheet. All items must be
clearly labeled with your name, your student ID number, and the day/time of
your recitation. **If any of these items are missing, we will
not grade your exam, and you will receive a score of 0 points.**

**You are required to complete the following Honor Pledge for Exams.
Copy and sign the pledge before starting your exam.**

ÒI affirm that I will not give or receive any unauthorized help on this exam, and that all work will be my own.Ó

_____________________________________________________________________________

_____________________________________________________________________________

_____________________________________________________________________________

Name:
______________________________________________________________________

Signature: ____________________________________________________________________

**One-Electron Atoms Ð 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 (30 points) ANSWER
IN BOOKLET 1**

Consider the
ground-state wavefunctions of the Hydrogen atom.

a)
What is the
expectation value <*V*> of
the potential energy of the hydrogen atom when it is in its ground state?

b)
Express the
energy of the ground state of the hydrogen atom in terms of the expectation
value <*V*> of the potential energy.

c)
What is the expectation value of the kinetic energy of the ground state?

**Problem 2 (30 points) ANSWER
IN BOOKLET 1**

Consider ¹^{ -} capture by a
deuteron. A slow ¹^{ Ð} 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 total
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 (20 points) ANSWER
IN BOOKLET 1**

Consider a system of *N* distinguishable atoms, maintained at a
temperature *T*, which are distributed
over two energy levels *e*_{1} = 0
and *e*_{2} = *e*.

a)
What is the
energy of this system?

b)
What is *c _{V}* for
this system?

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

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 uncertainty principle?

**Problem 5 (30 points) ANSWER
IN BOOKLET 2**

The correspondence principle can be
used to justify the selection rules observed in experimental studies of the
atom. In this problem, we will
consider the Bohr model of the hydrogen atom.

a)
What is the
correspondence principle, enunciated by Bohr in 1923?

b)
In the Bohr
model of the hydrogen atom, we assume that the electron of mass *m* is moving in a circular orbit. Classically, we expect that the electron
will radiate electromagnetic waves with a frequency equal to the frequency of
the orbital motion of the electron.
Determine the frequency of the orbital motion of the electron after applying
the Bohr quantization condition of the orbital angular momentum.

c)
In quantum
mechanics, we assume that radiation can only be emitted when transitions occur
between the quantized energy levels of the atom. Assuming that the electron of the
Hydrogen atom undergoes a transition from an energy level characterized by the
quantum number *n* + _*n* to an energy
level characterized by the quantum number *n*,
what is the frequency of the emitted radiation?

d)
Comparing the
results obtained in part b) and part c) when *n* becomes large, what do these two results tell you about the
selection rules that govern the transitions that can be observed in the
hydrogen atom?

**Problem 6 (30 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 second and the third member of the multiplet
is 5/3 of the energy difference between the first and the second 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**

a)
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 7 CONTINUED ON NEXT PAGE!**

b)
Consider the potential
energy distribution, shown in the following Figure.

Three
acceptable eigenfunctions for this potential are shown in the following Figure.

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 8 (10 points) ANSWER
IN BOOKLET 3**

a)
Which of the following
airlines would you take if you want to reach destination? Only one answer is correct.

1.
KLM, Royal Dutch
Airlines

2.
Air France

3.
United Airlines

4.
American
Airlines

5.
Delta Airlines

b)
Which of the
following airlines would you take if you want to be stranded, at a airports such as CDG, JFK, ATL, DUL, PHL? Multiple answers may be correct.

1.
KLM, Royal Dutch
Airlines

2.
Air France

3.
United Airlines

4.
American
Airlines

5.
Delta Airlines

Good Luck!