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Physics 216 Home Page---Spring 2012

This page contains copies of the class handouts, and other pertinent items of interest for the U.C. Santa Cruz Physics 216 graduate quantum mechanics class.


new!!! The solutions to the last problem set and the solutions to the final exam are available for download. An access problem preventing the download has been rectified.

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!!!

Table of Contents

[ I. General Information and Syllabus | II. Problem Sets and Exams | III. Solutions to Problem Sets and Exams | IV. Other Class Handouts | References on special functions of mathematical physics | VI. Free Textbooks on Quantum Mechanics | VII. Other books of interest | VIII. Articles of interest | IX. Web pages of interest ]

I. General Information and Syllabus

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
e-mail haber@scipp.ucsc.edu
web page http://scipp.ucsc.edu/~haber/index.html

Class Hours

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

Required Textbooks

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.

Course Grading and Requirements

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.

Course Syllabus

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

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II. Problem Sets and Exams

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

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III. Solutions to Problem Sets and Exams

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

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IV. Other Class Handouts

Class handouts are available in either PDF or postscript formats.

1. This handout examines the properties and derives an explicit form for the most general 3x3 proper rotation matrix R(n,θ) corresponding to a rotation by θ about an axis n. Using these results one can easily determine the rotation axis and the angle of rotation given an arbitrary real 3x3 orthogonal matrix of determinant +1. An arbitrary rotation can also be parameterized by three Euler angles. The relations between the Euler angles and (n,θ) are derived in these notes.     [PDF | Postscript].

2. This handout defines the tensor spherical harmonics as the simultaneous eigenfunctions of J2, Jz , L2 and S2 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/rn> (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 limr→∞eik·x as a distribution. The second proof makes use of the abstract scattering theory formalism.   [PDF | Postscript].

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V. References on special functions of mathematical physics

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.

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VI. Free Textbooks on Quantum Mechanics

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]

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VII. Other books of interest

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.

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VIII. Articles of interest

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. Bs, G.G. Dussel, and H. M. Sofa, American Journal of Physics 45 (1977) 191--192.     [PDF]

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IX. Web pages of interest

1. At UC Berkeley, the graduate quantum mechanics course during the past two academics years has been given by Professor Robert Littlejohn. He maintains a comprehensive webpage for this year's course (given over the 2011--2012 academic year). On this webpage, you can find an extensive set of notes that are quite well written, along with many problems and solutions. Feel free to explore if you have a chance.

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]

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Last Updated: June 15, 2012