Solutions to Problem Set 5 are now posted to Section IV of this website.
With the length of classes during the academic quarter shortened by 10%, there will not be sufficient time to treat time reversal symmetry in quantum mechanics in this course. A handout entitled Implications of Time-Reversal Symmetry in Quantum Mechanics, which is the lecture I would have given on this topic, along with a very nice treatment of the mathematics of antilinear and antiunitary operators, have been posted to Section V of this website. I also posted a link to a very nice review of time reversal in classical and quantum mechanics in Section X of this website.
In case we run out of time during the final lecture on Thursday, I am posting a revised version of the handout that provides an alternative proof of the Projection Theorem to Section V of this website. Make sure that you have the latest version (which is five pages long).
A handout, entitled
This handout, entitled The Addition Theorem of Spherical Harmonics, which provides a proof of the addition theorem that makes use of the theory of angular momentum operators in quantum mechanic, has been posted to Section V of this website.
Some of the important properties of the Wigner d matrix are presented along with their derivations in a handout entitled, Properties of the Wigner d-matrices, which has been posted to Section V of this website.
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 and Tuesdays 2--3 pm | |
haber@scipp.ucsc.edu | ||
web page | http://scipp.ucsc.edu/~haber/index.html | |
Lectures: Tuesdays and Thursdays, 11:40 am --1:15 pm, ISB 231
Modern Quantum Mechanics, 2nd Edition,
by J.J. Sakurai and Jim Napolitano
[Errata can be found here in
PDF
format.]
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
50% Homework (5 problem sets)
20% Midterm Exam (24 hour take-home exam; due Wednesday February 21, 2018 at 6 pm)
30% Final Exam (Tuesday March 20, 2018, from 8--11 am)
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.
Problem sets and exams are available in either PDF or Postscript formats.
The problem set and exam solutions are available in either PDF or postscript formats.
1. An excellent survey of the mathematical tools employed in quantum mechanics is provided in Chapter 1 of Principles of Quantum Mechanics, 2nd Edition, by Ramamurti Shankar. For your convenience, I am providing a copy of this chapter here. [PDF]
2. 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].
3. 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]
4. 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].
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. On Problem Set 1, you verified an integral representation of the Heavyside step function. You may be wondering whether this integral representation could be derived directly (say by inverting the Fourier transform). The answer is it can be if you are careful in employing generalized functions (also known as distributions). In this handout entitled Integral representation of the Heavyside step function, I provide the some notes on the derivation. [PDF | Postscript].
7. 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 provides a fourth derivation of the Sokhotski-Plemelj formula. [PDF | Postscript].
8. Generalized functions (also called distributions) are ubiquitous in quantum mechanics. We have already been introduced to a number of examples. In these notes entitled, Examples of Generalized Functions, I provide more details on the manipulating the generalized functions that you have already encountered, as well as introducing a number of additional generalized functions that are related to the function 1/x. [PDF | Postscript].
9 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].
10. Solving the time-dependent Schrodinger equation is equivalent to obtaining an explicit expression for the time evolution operator, U(t,t_{0}). If the Hamiltonian operator $H(t)$ is time dependent and if Hamiltonian operators are different times do not commute, then the form of U(t,t_{0}) can only be expressed as an infinite series. In this handout, I show how to write the time evolution operator as a time-ordered exponential. [PDF | Postscript].
11. In quantum mechanics courses, it is traditional to evaluate a Gaussian integral with a purely imaginary argument by taking the known formula for the integral of exp(-ax^{2}) for real positive values of a and applying it to the case of a purely imaginary a. However, this requires justification. A rigorous derivation of of the correct formula for a one-dimensional Gaussian integral with a purely imaginary argument is provided in this handout. [PDF | Postscript].
12. The method of stationary phase is used to obtain an approximate evaluation of an integral whose integrand consists of a smooth function g(x) multiplied by an oscillatory exponential of the form exp[ikf(x)]. The dominant contribution to the integral arises from the region of x in which the phase of the oscillating exponential is stationary, i.e. where df/dx=0. The approximate form of the integral makes use of a result from the previous handout. This treatment is taken from an appendix to a book by Larry B. Stotts, Free Space Optical Systems Engineering (John Wiley & Sons, New York, 2017). [PDF]
13. 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].
14. This handout entitled Expansion of plane waves in spherical
harmonics demonstrates that a plane wave can be expressed as a sum
over simultaneous eigenstates of the free particle Hamiltonian and the
angular momentum operators L_{z} and
L^{2}. In particular, the plane wave can be
viewed as a linear combination of incoming and outgoing spherical
waves. The derivation of the so-called partial wave expansion of the
plane wave is provided in these notes.
[PDF | Postscript].
15. This handout, entitled Properties of the Wigner d-matrices
provides a list of six useful properties satisfied by
d^{j}_{m'm}(θ)
along with derivations of each property. A useful theorem that shows
how the components of the angular momentum operator along two
different directions are related is proved in an Appendix.
[PDF | Postscript].
16. This handout, entitled The Addition Theorem of Spherical
Harmonics, provides a proof of the addition theorem that makes use of the theory of angular momentum operators
in quantum mechanics.
[PDF | Postscript].
17. This handout, entitled
18. In this handout, I provide a another proof the Projection
Theorem that differs from the one presented in Sakurai and Napolitano.
This theorem 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 Theorem is a simple consequence of the Wigner-Eckart theorem.
[PDF | Postscript].
19. With the length of classes during the academic quarter shortened by
10%, there was not sufficient time to treat time reversal symmetry
in quantum mechanics in this course. This handout entitled Implications of
Time-Reversal Symmetry in Quantum Mechanics
is the lecture I would have given on this topic.
[PDF | Postscript].
20. The time reversal operator is an antiunitary operator, which involves a number of subtleties. 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, entitled Antilinear Operators. [PDF].
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.
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]
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. Finally, 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. Yi-Zen Chi has posted a set of lecture notes entitled Analytical Methods in Physics on the arXiv. Chapters 2--6 of these lecture notes contain material that is especially relevant for the graduate quantum mechanics course. [PDF]
6. 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 difficult 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:
7. If you would like to learn even 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 Operators (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).
8. 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 generalized 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. The other classic reference is I.M. Gel'fand and G.E. Shilov, Generalized Functions, Volume 1 (AMS Chelsea Publishing, Providence, RI, 1964).
9. A book on the delta function and its friends that provides many detailed examples is Ram P. Kanwal, Generalized Functions: Theory and Applications, 3rd edition (Birkhäuser, Boston, 2004). I have also found the book by D.S. Jones, The Theory of Generalised Functions (Cambridge University Press, Cambridge, UK, 2008) to be quite useful.
1. Here is a webpage that provides links to numerous textbooks on quantum mechanics. [HTML]
2. I neglected to include one additional quantum mechanics book of interest on the General Information sheet handed out in class. The book is Nonrelativistic Quantum Mechanics, 3rd edition, by Anton Z. Capri (World Scientific, Singapore, 2002). The main unique feature of this book is its discussion of Hilbert space and rigged Hilbert space. It is written from a physicist's point of view, but treats some of the mathematical aspects in more depth than in many other books on quantum mechanics. A preview can be found here: [HTML]. Links to two of the relevant chapters are provided below.
3. 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.
4. 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-Gordan, Wigner-Eckart, coupling of angular momenta, spherical tensors, etc. (as noted by an enthusiastic reviewer on Amazon).
5. The ultimate reference on the quantum theory of angular momentum, which contains almost everything you will ever need to know about this subject is Quantum Theory of Angular Momentum, by D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii (World Scientific Publiching Co., Singaore, 1988).
1. An excellent non-technical article on the foundation of quantum mechanics appeared last year in The New York Review of Books by Steven Weinberg entitled The Trouble with Quantum Mechanics.
2. The following 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. Quantum mechanics implies that pure states can evolve only into pure states. However, if one tosses a pure state into a black hole, the black hole will eventually evaporate completely while emitting Hawking radiation. The Hawking radiation is predicted to be thermal (i.e, with a blackbody spectrum), which corresponds to a mixed state. This is the essence of the information loss paradox of black hole physics. A review of this paradox can be found in a recent article (in particular, check out the first three sections) by David Wallace, Why black hole information loss is paradoxical, arXiv:1719.03783 [gr-qc]. [PDF]
4. What is the meaning of the wave function of the universe? One
place to start is a famous paper by James B. Hartle and Stephen
W. Hawking entitled Wave
function of the Universe, which is published in Phys. Rev. D
28, 2960 (1983).
[PDF]
Perhaps a more gentler treatment can be found in Michael Cooke, An
Introduction to Quantum Cosmology which appeared as a master's
thesis. However, this subject matter goes way beyond the scope of our
class. [PDF]
5. In class, I presented a proof that there is no degeneracy in one-dimensional bound states. But, the proof requires a careful discussion of the conditions that the potential must satisfy in order for the theorem to be true. The following two articles discuss the relevant conditions and provide counterexamples if these conditions are not satisfied.
6. 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]
7. 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 derivation 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]
8. On Problem Set 3, you evaluated the propagator of the one dimensional harmonic oscillator. The method employed in computing the propagator is just one of a number of possible methods of calculation. In the following article, you can find details of two other techniques for computing the harmonic oscillator propagator.
9. A very nice pedagogical treatment of the double Dirac delta potential, which was the subject of problem 3 of the Midterm exam, can be found in the following reference.
10. Consider the operator φ which multiplies the wave function ψ(r,φ) by φ. In the polar coordinate representation, L_{z} is represented by -iℏ ∂/∂φ. One can then derive the commutation relation [φ , L_{z}] = iℏ I. The latter would seem to imply the uncertainty relation, ΔφΔL_{z} ≥ ℏ/2 . But, this uncertainty relation cannot be correct since the maximum value of Δφ cannot be larger than π, whereas it is possible to construct a state with arbitrarily small ΔL_{z}. The resolution of this apparent paradox is related to the fact that the operator L_{z} is self-adjoint only when acting on a Hilbert space of periodic functions with period 2π. However, multiplication of a periodic function by φ yields a function that is no longer periodic. Thus, one cannot make use of the generalized Heisenberg uncertainty relation derived in class, which is only valid for self-adjoint operators. A very short and elegant explanation can be found on pp. 14--17 of Rudolf Peierls, Surprises in Theoretical Physics (Princeton University Press, Princeton, NJ, 1979). For a more comprehensive treatment, feel free to consult the following references:
11. 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]
1. At UC Berkeley during the 2017--2018 academic year, the graduate quantum mechanics course is being given by Professor Robert Littlejohn. He maintains a comprehensive webpage for this course. 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-Gordan coefficients. The classic table that provides all Clebsch-Gordan 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-Gordan 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 .