Physics 237, Final Exam

Tuesday May 8, 2018

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 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.

 

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

 

_____________________________________________________________________________

 

_____________________________________________________________________________

 

_____________________________________________________________________________

 

Name:  ______________________________________________________________________

 

Signature:  ____________________________________________________________________


 

ConstantsAndFactors.jpg

 

 


 

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 a0 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.

 

 


 


Table17-3.jpg

 

 

Ground-state Properties of the Deuteron

 

      Binding energy: DE = 2.22 MeV

      Nuclear spin: 1

      Nuclear parity: even

      Magnetic dipole moment: = +0.857n

      Electric quadrupole moment: q = +2.7 _ 10-31 m2

      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 - capture by a deuteron.  A slow 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:

 

- + 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 e1 = 0 and e2 = e.

 

a)     What is the energy of this system?

 

b)    What is cV 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 j quantum number that can be assigned to these states?

 

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

 

c)     What is the s 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 V0 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 V0.  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 > V0 or E < V0) 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 = x0.  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

 

 


 


 


 


 


 

LEAD Technologies Inc. V1.01

Good Luck!