The final exam produces a pretty wide spread of results. Out of 120 possible points, the grades were 103, 91, 79, 74, 74, 69, 56, 55, 51, 46, 43. I suspect the results might have been better had you had a little more time between final exams (and perhaps a little more sleep!). Although I had hoped for a somewhat better result, taking into account the effort over the spring quarter, everyone has earned a passing grade for the course. Grades will be posted before I leave for Germany on Saturday.

The graded homework and final exams will be left in the ISB mailbox of the grader, Eddie Santos. Feel free to retrieve your work the next time you stop by the Physics Department.

Final exam solutions have been posted to Section III of this website.

Have a great summer break---you earned it!!!

The General Information and Syllabus handout is available
in either PDF or Postscript format
[PDF | Postscript]

Some of the information in this handout is reproduced here.

## General Information | ||
---|---|---|

Instructor | Howard Haber | |

Office | ISB 326 | |

Phone | 459-4228 | |

Office Hours | Mondays 2--4 pm | |

haber@scipp.ucsc.edu | ||

web page | http://scipp.ucsc.edu/~haber/index.html | |

Lectures: Tuesdays and Thursdays, 12:00--1:45 pm, ISB 235

* Principles of Quantum Mechanics*, 2nd Edition,
by Ramamurti Shankar
[Errata for
the 13th printing (dated 2006) can be found here in
PDF format.]

* Lectures on Quantum Mechanics*, by Gordon Baym

It is assumed that you are familiar with the material from Shankar, Chapters 1,4,5,6,7,9,11,12,13,14,15.1 and 15.2; and Baym, Chapters 3,4,5,6,7,14,15.

45% Homework (5 problem sets)

20% Midterm Exam (take-home exam; Monday May 14--Wednesday
May 16, 2012, due at 5 pm)

35% Final Exam (Wednesday June 13, 2012, 8--11 am)

Homework assignments will be due on every second Thursday of the
academic quarter, with the exception of the first
assignment, which is due on Tuesday April 17, 2012.
The homework problem sets are *not* optional.
You are encouraged to discuss the class material and homework
problems with your classmates and to work in groups, but all
submitted problems should represent your own work and understanding.

The final exam will be held in ISB 235. This exam will be three hours long and cover the complete course material. You must take the final exam to pass the course. You will be permitted to consult the class textbook, your own handwritten notes, and any class handout during the final exam.

- Advanced topics in angular momentum theory [Shankar, Chapter 15.3; Baym, Chapter 17]
- Path integral formulation of quantum theory [Shankar, Chapter 8]
- The WKB approximation [Shankar, section 16.2]
- The variational method [Shankar, section 16.1]
- Time-independent perturbation theory [Shankar, Chapter 17; Baym, Chapter 11]
- Quantum theory of scattering [Shankar, Chapter 19; Baym, Chapters 9--10]
- Time-dependent perturbation theory [Shankar, sections 18.1--18.3; Baym, Chapter 12]
- Quantum theory of radiation [Shankar, section 18.5; Baym Chapter 13]
- Identical particles [Shankar, Chapter 10; Baym, Chapter 18]
- Second quantization [Baym, Chapter 19]

Problem sets and exams are available in either PDF or Postscript formats.

- Problem Set #1--due: Tuesday, April 17, 2012 [PDF | Postscript]
- Problem Set #2--due: Friday, April 27, 2012 [PDF | Postscript]
- Problem Set #3--due: Monday, May 14, 2012 [PDF | Postscript]
- Midterm exam--due: 5 pm Wednesday, May 16, 2012 [PDF | Postscript]
- Problem Set #4--due: Tuesday, May 29, 2012 [PDF | Postscript]
- Problem Set #5--due: Monday, June 11, 2012 [PDF | Postscript]
- Final exam--8--11 am on Wednesday, June 13, 2012 [PDF | Postscript]

The problem set and exam solutions are available in either PDF or postscript formats.

- Solution Set #1 [PDF | Postscript]
- Solution Set #2 [PDF | Postscript]
- Solution Set #3 [PDF | Postscript]
- Midterm Exam Solutions [PDF | Postscript]
- Solution Set #4 [PDF | Postscript]
- Solution Set #5 [PDF | Postscript]
- Final Exam Solutions [PDF | Postscript]

1. This handout examines the properties and derives an explicit form for the most general 3x3 proper rotation matrix R(

2. This handout defines the tensor spherical harmonics as the
simultaneous eigenfunctions of **J**^{2},
J_{z} , **L**^{2} and **S**^{2}
in the coordinate representation. The cases of s=1/2 and s=1 are
treated explicitly. A useful table of Clebsch-Gordon coefficients is
provided for the analysis. The Clebsch-Gordon series and its
applications are treated in Appendix A, and the
vector spherical harmonics are treated explicitly in Appendix B.
[PDF | Postscript].

3. The section on time reversal in Shankar does not provide much detail. So, I will take advantage of a wonderful collection of notes that were prepared for the graduate quantum mechanics course at Berkeley for the 2011--2012 academic year by Professor Robert Littlejohn. Links to his notes on time reversal in quantum mechanics are provided here: [PDF | Postscript]

4. For a very nice treatment of the mathematics of antilinear and antiunitary operators, have a look at the following set of notes by C.M. Caves: [PDF].

5. A summary of the important properties of the Airy functions can be found in this handout. [PDF | Postscript].

6. A nice set of notes by Frank Porter of Caltech
discusses the fundamental role of the electromagnetic vector potential
and the Aharonov-Bohm effect. These note can be found here:
[PDF | Postscript]

Solutions to the problems given at the end of Porter's notes can be
found here:
[PDF | Postscript]

7. In these notes, the calculation of the ground state energy of the helium atom using the variational principle, is given. These will fill in some of the steps omitted by Shankar. Details of all the steps in the calculation are provided (along with a number of integration tricks) here: [PDF | Postscript]

8. These notes review the Schrodinger equation for a charged particle in an external electromagnetic field. In order to obtain the relevant equation, we first examine the classical Hamiltonian of a charged particle in an electromagnetic field. We then use this result to obtain the Schrodinger equation using the principle of minimal substitution. As an example, the special case of a uniform magnetic field is exhibited. Finally, we demonstrate the origin of the coupling of the spin operator to the external magnetic field in the case of a charged spin-1/2 particle. [PDF | Postscript]

9. The book by Stephen Gasiorowicz, * Quantum Physics*, 3rd Edition
(John Wiley & Sons, Inc., Hoboken, NJ, 2003) includes free
supplemental material on the Wiley
website. Here, I provide Gasiorowicz's Supplement 8A, which provides
some clever techniques for evaluating the expectation values
<1/r^{n}> (n=1,2,3) with respect to the hydrogen atom radial wave functions.
These are needed in computation of the fine structure of hydrogen.
[PDF].

10. In quantum mechanics, one often deals with two-state systems. To solve such a system requires one to diagonalize a general 2x2 hermitian matrix. Having done so once, one can then apply these results in many different circumstances. The derivation of the eigenvalues and the diagonalizing unitary matrix can be found in this handout. [PDF | Postscript]

11. A handout entitled *The
Riemann-Lebesgue Lemma* is a very important result of
Fourier analysis. It has many applications in mathematical physics.
In this course, we employ the Riemann-Lebesgue lemma in our derivation
of the optical theorem. A simple proof of the Riemann-Lebesgue lemma
is given in this handout.
[PDF
| Postscript].

12. A handout entitled *The Optical Theorem*
provides two derivations of the celebrated optical theorem of
scattering theory. The first proof employs a interesting mathematical
identity that interprets
lim_{r→∞}e^{ik·x} as a distribution.
The second proof makes use of the abstract scattering theory
formalism.
[PDF
| Postscript].

1. A superb resource for both the elementary functions and the
special functions of mathematical physics is the
*Handbook
of Mathematical Functions* by Milton Abramowitz and Irene A. Stegun,
which is freely available on-line. The home page for this
resource can be found
here. There, you
will find links to a
frames
interface of the book. Another scan of the book can be found
here.
A third independent link to the book can be found
here.

2. The *NIST
Handbook of Mathematical Functions* (published
by Cambridge University Press), together with its Web
counterpart, the NIST Digital Library
of Mathematical Functions
(DLMF), is the culmination of a project that was conceived in 1996 at
the National Institute of Standards and Technology (NIST). The project
had two equally important goals: to develop an authoritative
replacement for the highly successful Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables, published in
1964 by the National Bureau of Standards (M. Abramowitz and
I. A. Stegun, editors); and to disseminate essentially the same
information from a public Web site operated by NIST. The new Handbook
and DLMF are the work of many hands: editors, associate editors,
authors, validators, and numerous technical experts.
The NIST
Handbook covers
the properties of mathematical functions, from elementary
trigonometric functions to the multitude of special functions.
All of the mathematical information contained in the Handbook is also
contained in the
DLMF,
along with additional features such as more
graphics, expanded tables, and higher members of some families of
formulas. A PDF copy of the handbook is provided here:
[PDF]

3. One of the classic references to special functions is a three
volume set entitled *Higher Transcendental Functions*
(edited by A. Erdelyi), which was
compiled in 1953 and is based in part on notes left by Harry Bateman.
This was the primary reference for a generation of physicists and
applied mathematicians, which is colloquially referred to as the
Bateman Manuscripts. This esteemed reference work
continues to be a valuable resources for students and professionals.
PDF versions of the three volumes are now available free of charge.
Check out the three volumes by clicking on the relevant links here:
[Volume 1 |
Volume 2 |
Volume 3].

4. Another very useful reference for both the elementary functions and the
special functions of mathematical physics is
*An Atlas
of Functions* (2nd edition) by Keith B. Oldham, Jan Myland and Jerome
Spanier, published by Springer Science in 2009.
This resource is freely available on-line to students
at the University of California at this
link.

5. Yet another excellent website for both the elementary functions and the special functions of mathematical physics is the Wolfram Functions site. This site was created with Mathematica and is developed and maintained by Wolfram Research with partial support from the National Science Foundation.

6. One of my favorite books on special functions is
*Special
Functions and Their Applications* by N.N. Lebedev
(Dover Publications, Inc., Mineola, NY, 1972). It
provides invaluable information on special functions, while being
extremely cheap to buy (and even cheaper to peruse on
Google
Books).

7. In your problem sets, you will encounter many difficult
integrals. Although you may be tempted to use Mathematica or Maple
(which sometimes is the easiest approach), I cannot overestimate the
value of a good table of integrals. Professionals always choose first
to consult *Table of Integrals, Series and Products*, 7th edition,
by I.S. Gradshteyn and Ryzhik, edited by Alan Jeffrey and
Daniel Zwillinger (Elsevier, Inc., Amsterdam,2007).
You can preview quite a few pages using
Google Books.

1. *Quantum Mechanics:
Fundamental Principles and Applications* by John F. Dawson
is a compilation of notes for a first year graduate course in
non-relativistic quantum mechanics which Professor Dawson
taught at the University of New Hampshire for a number of years.
[PDF]

2. *Lecture Notes in
Quantum Mechanics* by Doron Cohen is based on a course given
by Professor Cohen at Ben-Gurion University.
[PDF | Postscript]

3. The first four chapters of *
Advanced Modern Physics* by John Dirk Walecka is provided free of
charge by World
Scientific. Included are chapters on quantum mechanics, angular
momentum and scattering theory.
For further details, check out the links on the
World
Scientific website. Additional pages of this book can be
previewed on
the google
books website.
[PDF]

1. There are many classic books on angular momentum in quantum
mechanics. These include book by M.E. Rose; A.R. Edmonds;
D.M. Brink and D.R. Sachler; and L.C. Biedenharn and J.D. Louck.
However, for a modern introductory treatment of this subject,
I can
recommend *Angular
Momentum: An Illustrated Guide to Rotational Symmetries for
Physical Systems* by William J. Thompson published by
Wiley-VCH in 2004. Unfortunately, the University of California does not
provide free access to the Wiley Online Library. However, if you
happen to be at Stanford University, you can access the chapters
of this book from a Stanford computer. Otherwise, you can peruse
this book
here
courtesy of google books.

2. Another modern book on angular momentum in quantum mechanics,
entitled *Angular Momentum Techniques in Quantum Mechanics*, by
Varadarajan Devanathan (Kluwer Academic Publishers, New York, 2002),
provides excellent reference for formulae and identities for
rotations, Clebsch-Gordon, Wigner-Eckart, coupling of angular momenta,
spherical tensors, etc. (as noted by an enthusiastic reviewer on
Amazon). In fact, this book can be viewed online at
Scribd,
although some patience is required.

3. A very nice treatment of the semi-classical approximation to
quantum mechanics can be found in a book entitled *Qualitative
Methods in Quantum Theory*, by A.B. Migdal (W.A. Benjamin, Inc.,
Reading, MA, 1977), In this book, the connection formulae of the
WKB approximation are derived by a very clever technique that involves
analytic continuation into the complex plane. This book can be partially
previewed online courtesy of
at google books.

4. For the more traditional approach to the WKB approximation and the
derivation of the connection formulae, check out Chapter 14 of
*Quantum Mechanics:
Fundamental Principles and Applications* by John F. Dawson,
previously referenced in Section VI of this website.
Another good source for this material can be found in Chapter 7 of
Eugen Merzbacher, *Quantum Mechanics*, Third Edition
(John Wiley & Sons, Inc., New York, NY, 1998). In fact, the
Second Edition of this book can be viewed online at
Scribd,
although it takes a while to download.

1. For a nice review of time reversal, have a look at
*Time
reversal in classical and quantum mechanics* by J.M.
Domingos, International Journal of Theoretical Physics **18**,
213--230 (1979). [PDF]

2. For a very nice introduction to the technique of path integrals
and their applications in physics, I highly recommend
*Path
Integral Methods and Applications* by Richard MacKenzie.
[PDF | Postscript]

3. A slightly more sophisticated treatment of path integrals
in quantum mechanics can be
found in *Lecture
Notes on the Path Integral Approach to Quantum Mechanics*
by Matthias Blau.
[PDF |
gzipped Postscript]

4. For a pedagogical treatment of the WKB approximation to
three-dimensional spherically symmetric potentials, with applications
to the analysis of the energy levels of quark-antiquark (quarkonium)
bound states, check out C. Quigg and Jonathan L. Rosner,
*
Quantum mechanics with applications to quarkonium*, Physics
Reports **56** (1979) 167-235.
[PDF]

5. Following Chapter 11 of Baym, I showed in class that
to second order in perturbation theory, the wave function
renormalization constant was equal to the partial derivative of the perturbed
energy eigenvalue with respect to the unperturbed energy, keeping
fixed the matrix elements of the perturbation. In fact, this result is
true to all orders. For a general proof, see
*Normalization
of states in perturbation theories* by D.R. Bès,
G.G. Dussel, and H. M. Sofía, American Journal of
Physics **45** (1977) 191--192.
[PDF]

2. You should learn how to read a
table of Clebsch-Gordon
coefficients. The classic table that provides all Clebsch-Gordon
coefficients for the addition of angular momentum with all possible
values of *j* up to and including *j=*2 can be found in
PDF
format
courtesy of the Particle Data Group.
To obtain the Clebsch-Gordon coefficients from this table, it is
essential that you take note of the following instruction near the top
of the table that reads:
*
Note: A square-root sign is to be understood
over every coefficient, e.g., for -8/15 read
-√ 8/15
.*

3. In 1977, J.D. Jackson gave a colloquium in which he explained the connection between the fundamental intrinsic magnetic dipole moment and the hyperfine structure of the s-states of the hydrogen atom. Jackson provided a writeup of his colloquium as a CERN Yellow Report (CERN 77-17), which you can find here: [PDF]

haber@scipp.ucsc.edu

Last Updated: June 15, 2012