Physics 237, Midterm Exam #2

Tuesday March 27, 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: ______________________________________

**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*.

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

Consider the
ground-state wavefunctions of the Hydrogen atom (see Page 2).

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 the following eigenfunction
of the electron in a one-electron atom with atomic number *Z*:

.

a)
What is the
magnetic quantum number of this
eigenfunction?

**Note: you will not receive any credit for a correct
answer if it is not properly justified.
You cannot justify your answer on the basis of the information provided
on page 2 of the exam.**

b)
What is the
azimuthal quantum number of this
eigenfunctions?

**Note: you will not receive any credit for a correct
answer if it is not properly justified.
You cannot justify your answer on the basis of the information provided
on page 2 of the exam.**

c)
How would you
calculate the energy of the electron?

**Note: you do not have to do the actual calculation,
but you should provide sufficient details on how you would carry out this
calculation.**

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

A particle of total energy 9*V*_{0} is incident from the left on a potential given by

Assume the general solution of the
wavefunction of this particle has the following form:

a)
Write down the
general solution for the wavefunction in the *x* < 0 region; that is, determine the functions *f*_{1}(*x*) and *g*_{1}(*x*).
Note: you do not yet have to determine the value of the constants *A* and *B* that appear in the wavefunction.

b)
Write down the
general solution for the wavefunction in the 0 < *x* < *a* region; that is,
determine the functions *f*_{2}(*x*) and *g*_{2}(*x*). Note: you do not yet have to determine
the value of the constants *C* and *D* that appear in the wavefunction.

c)
Write down the
general solution for the wavefunction in the *a* < *x* region; that is,
determine the function *f*_{3}(*x*).
Note: you do not yet have to determine the value of the constant *E* that appears in the wavefunction.

d)
Assume that *A* = 1. Use the boundary conditions at *x* = 0 to determine a relation between
the constants *C* and *D*.

e)
Use the boundary
conditions at *x* = *a* to determine the constant *E*.

f)
What is the
probability that the particle will be transmitted to the positive side of the
potential well (*x* > *a*)?
**Note: you can express your answer
in terms of the constant E.**

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

Which of the following sports team is
the favorite Dutch team of your Phy 237 instructor?

a)
The Yankees

b)
The Buffalo
Bills

c)
The Red Sox

d)
AJAX

Please
enter your answer in your exam booklet.