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Department of Physics
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Physics 215 Home Page---Winter 2017

This page contains copies of the class handouts, and other pertinent items of interest for the U.C. Santa Cruz Physics 215 graduate quantum mechanics class.


new!!! Solutions to the final exam have been posted to Section IV of this website.

The graded homework set 5 and the final exam are now available. Stop by my office any time during the spring quarter to retrieve your work.

Final grades have been posted. The final exam scores ranged from a low of 32 to a high of 91. The mean was 61. If you asked for a letter grade, the assigned grade was based on the cumulative course average, whcih was computed with the following weighting: problem sets: 50%; midterm exam: 20% and final exam: 30%. The corresponding numerical ranges were: A+ (93--100); A (86--93); A- (79--86); B+ (72--79); B (65--72); B- (58--65); C+ (54--58); C (50--54); D (46--50); F (0--46).

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

Table of Contents

[ I. General Information and Syllabus | II. Disability Statement to Students in Class | III. Problem Sets and Exams | IV. Solutions to Problem Sets and Exams | V. Other Class Handouts | VI. References on special functions of mathematical physics | VII. Free Textbooks on Quantum Mechanics | VIII. Other mathematical books of interest | IX. Other physics books of interest | X. Articles of interest | XI. 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, 9:50--11:35 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. A separate set of errata can be found here.]
Lectures on Quantum Mechanics, by Gordon Baym

Course Grading and Requirements

50% Homework (5 problem sets)
20% Midterm Exam (24 hour take-home exam; due Friday February 17, 2017 at 6 pm)
30% Final Exam (Tuesday March 21, 2017, from 12--3 pm)

Homework assignments are not optional. Homework assignments are due on Thursdays (with two weeks allotted for each homework set). 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 231. 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 any textbook of your choosing, your own handwritten notes, and any class handout (including solutions to the problem sets) during the final exam.

Course Syllabus

  1. Mathematical introduction [Shankar, Chapter 1]
  2. Postulates of quantum mechanics [Shankar, Chapter 4]
  3. Quantum dynamics [Shankar, Chapters 6 and 9; Baym, Chapters 3 and 5]
  4. The harmonic oscillator [Shankar, Chapter 7]
  5. Further aspects of quantum mechanics in one dimension [Shankar, Chapter 5; Baym Chapter 4]
  6. Scattering in one dimension [Shankar, Chapter 5; Baym, Chapters 4 and Chapter 11]
  7. Symmetries in quantum mechanics [Shankar, Chapter 11]
  8. Angular momentum in quantum mechanics [Shankar, Chapter 12; Baym, Chapter 6]
  9. Rotationally invariant problems in three dimensions [Shankar, Chapter 12; Baym, Chapter 6]
  10. The hydrogen atom [Shankar, Chapter 13; Baym, Chapter 7]
  11. Spin 1/2 [Shankar, Chapter 14; Baym, Chapter 14]
  12. Addition of Angular Momentum [Shankar, Chapter 15; Baym, Chapter 15]
  13. Irreducible Tensor Operators [Shankar, Chapter 15; Baym, Chapter 17]

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II. Disability Statement to Students in Class

UC Santa Cruz is committed to creating an academic environment that supports its diverse student body. If you are a student with a disability who requires accommodations to achieve equal access in this course, please submit your Accommodation Authorization Letter from the Disability Resource Center (DRC) to me privately during my office hours or by appointment, preferably within the first two weeks of the quarter. At this time, we would also like us to discuss ways we can ensure your full participation in the course. We encourage all students who may benefit from learning more about DRC services to contact DRC by phone at 831-459-2089 or by email at drc@ucsc.edu.

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III. Problem Sets and Exams

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

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IV. Solutions to Problem Sets and Exams

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

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V. Other Class Handouts

Class handouts are available in either PDF or postscript formats.

1. This handout discusses vector components (also called coordinates), matrix elements and changes of basis. An application to the matrix diagonalization problem is provided.     [PDF | Postscript].

2. This handout treats the diagonalization of hermitian and normal matrices by a unitary similarity transformation. It also provides details of the proof that two commuting hermitian matrices are simultaneously diagonalizable by a common unitary similarity transformation. The latter implies that two commuting hermitian matrices possess simultaneous eigenvectors.     [PDF | Postscript]

3. This handout demonstrates a number of basic properties of the characteristic polynomial of a matrix. These notes discuss how the coefficients of the characteristic polynomial are related to the eigenvalues, and provides a general formula for the coefficients in terms of traces of powers of the matrix. A proof is given of the Cayley-Hamilton theorem, which states that any matrix satisfies its own characteristic equation.     [PDF | Postscript].

4. The identity 1/(x ± iε)=P(1/x)∓iπ is called the Sokhotski-Plemelj formula. In a handout entitled The Sokhotski-Plemelj Formula, three different derivations of this formula are provided. In an appendix, the Fourier transform of a tempered distribution is discussed and then applied to the Sokhotski-Plemelj formula. Since the Fourier transform of a tempered distribution (and its inverse Fourier transform) are unique, this analysis proides a fourth derivation of the Sokhotski-Plemelj formula.   [PDF | Postscript].

5. A handout entitled The Riemann-Lebesgue Lemma describes a very important result of Fourier analysis. It has many applications in mathematical physics. A simple proof of the Riemann-Lebesgue lemma is given in this handout.   [PDF | Postscript].

6. In quantum mechanics, elements of the quantum mechanical Hilbert space are pure quantum states. Nevertheless, it is possible to construct an ensemble of quantum states that cannot be represented by a state vector of the Hilbert space. To describe such mixed quantum states, one introduces the density operator. This handout defines the density operator and its associated density matrix and provides proofs of some of its most important properties. An example of a density matrix of a mixed quantum state is derived and provides a model for how mixed quantum states can arise.     [PDF | Postscript].

7. This handout entitled Three Dimensional Rotation Matrices 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].

8. This handout, entitled Clebsch-Gordon coefficients and the tensor spherical harmonics, 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].

9. In this handout, I provide a complete derivation of the relation, djm'm(π) = (-1)j-m δm,-m, satisfied by Wigner's small-d function.     [PDF | Postscript].

10. 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 spring 2106 semester by Professor Robert Littlejohn. Links to his notes on time reversal in quantum mechanics are provided here:     [PDF | Postscript]

11. 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].

12. In this handout, I derive the Projection Theorem, which shows that the matrix elements of any vector operator are proportional to the corresponding matrix elements of the angular momentum operator. Moreover, a simple expression for the proportionality constant is given. The Projection is a simple consequence of the Wigner-Eckart theorem.     [PDF | Postscript].

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VI. 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. Additional mathematical information can be found on the related Wolfram MathWorld site.

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 working out quantum mechanics problems, you will sometimes encounter a difficult integral. 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 the Table of Integrals, Series and Products, 8th edition, by I.S. Gradshteyn and Ryzhik, edited by Daniel Zwillinger and Victor Moll (Academic Press, Elsevier, Inc., Amsterdam, 2015). This resource is freely available on-line to students at the University of California at this link.

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VII. 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. 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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VIII. Other mathematical books of interest

1. One of the important mathematical tools used in quantum mechanics is the theory of complex variables and analytic functions. There are many books that treat this topic. One very nice treatment can be found in Chapter 6 of Frederick W. Byron and Robert C. Fuller, Mathematics of Classical and Quantum Physics (Dover Publications, Inc., New York, 1992), originally published by the Addison-Wesley Publishing Company in 1970, but has now been reprinted in an inexpensive paperback edition by Dover Publications. Note that problem 25 on p.382 provides an approach to establishing eq.(1) given in problem 4(c) on Problem Set 1. For your convenience, I am providing a link to Chapter 6 here.   [PDF]

2. A somewhat terse but useful introduction to complex integration can be found in Chapter 2 of Debabrata Basu, Introduction to Classical and Modern Analysis and Their Application to Group Representation Theory (World Scientific, Singapore, 2011). The entire chapter is provided free of charge by Google Books. The treatment of analytic continuation at the beginning of Chapter 3 is especially noteworthy. In particular, the example provided in Section 3.1 is spectacular and deserves a place in any book on complex analysis!

3. Many fully solved problems in the theory of complex variables can be found in Schaum's outlines, Complex Variables (2nd edition) by Murray R. Spiegel, Seymour Lipschutz, John J. Schiller and Dennis Spellman (McGraw Hill, New York, 2009).   [PDF]

4. Fianlly, you can also peruse Chapter 1 of Alexander O. Gogolin, Lectures on Complex Integration, which surveys the basics of complex analysis and applies it to evaluating integrals of various types. This resource is available to University of California students.   [PDF]

5. Quantum mechanics relies on the mathematical theory of Hilbert spaces. You can learn about Hilbert spaces from any mathematical textbook on functional analysis. But, such books tend to be diificult for the average physics student. For physics graduate students, I can recommend three textbooks on mathematical physics, each of which have very readable chapter on the theory of Hilbert spaces:

Links to each one of these books can also be found at this webpage.

6. If you would like to learn more about Hilbert spaces, I cannot think of a better place to start than Guido Fano, Mathematical Methods of Quantum Mechanics (McGraw Hill, New York, 1971). Another possible reference, which is available online is Gerald Teschl, Mathematical Methods in Quantum Mechanics With Applications to Schrodinger Operatorors (American Mathematical Society, Providence, RI, 2009). Finally, for a briefer treatment, also aimed at physics students, have a look at Thomas F. Jordan, Linear Operators for Quantum Mechanics (Dover Publications, Inc., New York, 2006).

7. Delta functions are not functions. However, they are an example more a general mathematical object, called a generalized functions. The classic book that describes the mathematical properties of generalzed functions (and their relation to Fourier analysis) is M.J. Lighthill, Fourier Analysis and Generalised Functions (Cambridge University Press, Cambridge, UK, 1958). Remarkably, it is still in press, although it costs slightly more than the $2.25 that I paid for it when I was a student in the early 1970s.

8. Currently, my favorite book on the delta function and its friends is Ram P. Kanwal, Generalized Functions: Theory and Applications, 3rd edition (Birkhäuser, Boston, 2004).

IX. Other physics books of interest

1. If you want to learn more about supersymmetric quantum mechanics, a good place to start is Fred Cooper, Avinash Khare and Uday Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singaproe, 2001). This book is an outgrowth of a review article by the same authors published in Physics Reports, 251, 267 (1995).   [PDF]
For a more recent book on supersymmetric quantum mechanics, check out Asim Gangopadhyaya, Jeffry V. Mallow and Constantin Rasinariu, Supersymmetric Quantum Mechanics, An Introduction (World Scientific, Singapore, 2010).

2. 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.

3. 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.

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X. Articles of interest

1. An excellent non-technical article on the foundation of quantum mechanics recently appeared in The New York Review of Books by Steven Weinberg entitled The Trouble with Quantum Mechanics.

2. The followingd two articles may provide some insight on how the commutator [X,K]=iI is modified in a finite-dimensional approximation to the infinite dimensional Hilbert space of square-integrable functions, such that the trace paradox is avoided.

3. Shankar provides a proof that there is no degeneracy in one-dimensional bound states. But, he does not provide the all the conditions necessary to guarantee the veracity of this statement. The following two articles discuss the relevant conditions and provide counterexamples if these conditions are not satisfied.

4. There is an alternative solution of the quantum mechanics problem of the bound state energies of a particle in a one-dimensional square well that makes use of the Lambert W function. If you are interested in this alternative approach (or you are curious about the Lambert W function), have a look at Ken Roberts and S.R. Valluri, Tutorial: The quantum finite square well and the Lambert W function, Can. J. Phys. 95, 105 (2017).   [PDF]

5. The derivation of the propagator for the one dimensional harmonic oscillator given in Solution Set 3 is correct only for 0 < t < π/ω (if we interpret i -1/2 = e -iπ/4). For other time intervals, an additional phase factor arises, which requires a more subtle analysis. A simple deriation of this form factor an be found in Nora S. Thornber and Edwin F. Taylor, The evolution of oscillator wave functions, American Journal of Physics, 66, 1022 (1998).   [PDF]

6. 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]

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XI. Web pages of interest

1. At UC Berkeley, the graduate quantum mechanics course last year was given by Professor Robert Littlejohn. He maintains a comprehensive webpage for the course that was given in the spring 2016 academic semester. On this webpage, you can find an extensive set of notes that are quite well written. 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  .

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Last Updated: March 27, 2017