Quantum Mechanics Spring 2007

Tuesdays 13-16, Fridays 9.15-12 Lecture room II.

Instructor: Nicholas (Nick) Bailey, nbailey[at]ruc.dk, tel. 2254

First lecture: Tuesday February 6, 2007.

New (8/4) Quizzes for different chapters: chapter 1 quiz, chapter 2 quiz, chapter 3 quiz, chapter 4 quiz (which you saw already) chapter 5 quiz, chapter 6 quiz, chapter 7 quiz, chapter 9 quiz,

The content can be summarized as non-relativistic quantum mechanics of mostly a single particle (some of the material deals with many-particle systems). Topics: representation of particles by wavefunction, Schrødinger equation in one dimension for various potentials. Mathematical formalism. Schrødinger equation in three dimensions---Hydrogen atom and angular momentum. Systems of many identical particles, helium atom and solids. Time-independent perturbation theory. Variational principle. Time-dependent perturbation theory. Introduction to quantum cryptopgraphy. I have written about the content of the course in a little more detail here.

D. Griffiths: Introduction to Quantum Mechanics, second edition, Prentice Hall 2005. We will use chapters. 1-7 and 9, the afterword, and the appendix, but omitting pp 122-123, 187-188, 281-282. This will be available in the Campus bookshop.

Opgavesamling til dybdekursus i fysik, IMFUFA Tekst 429 (2004) and Eksamenopgavesæt sommer 2005.

Note: if you happen to have a copy already of the first edition of Griffiths, then you can use it. Most of the text, apart from chapter three, has not changed, though the page numbers are different. Below I use the section numbers to make it easier to follow in the first edition. Some of the problems are the same, but their numbering is different in most cases. If there are people using the first edition, I will always indicate the corresponding problem numbers in the first edition, but for those problems which are completely new, you will have to look at somebody else's book. At the end of the course we will cover some material on quantum cryptography, for which I will provide copies of appropriate articles.

Regarding problems: It is important to realize that much of the material of the course is contained in the problems. This means first of all that it is very important to do them. Secondly it means that your notes and solutions to the problems constitute an official part of the syllabus, and are thus allowed material in the exam.

- "The quantum world", J. C. Polkinghorne (focuses on philosophical issues)
- "The Feynman Lectures of Physics", Volume III, by Feynman, Leighton and Sands (particularly the first 12 chapters; good for two-state systems and their many interesting real-world applications).
- "The Principles of Quantum Mechanics", by Paul Dirac (1930). Written by one of the founders, the first chapters are still useful. It is here I first saw the idea that a quantum system may be defined as one where the act of measurement will necessarily disturb the system.
- Wikipedia article on Quantum Mechanics. While no one would ever cite Wikipedia in an article, 70% of physicists in a recent survey said they used Wikipedia to get a basic introduction for a topic outside their field (this would include me too) (Physics World, January 2007). There is a lot of stuff on QM, starting with the main article. I really encourage you to surf around a bit, to get a feeling of all the applications and related topics.
- Java-based physics simulations from the University of Colorado: Some are about quantum-mechanics.
- Some physics-blogs (or to be more correct, blogs written by physicists), which may have some discussions on quantum phenomena (apparently there's been a lot of argument about string theory recently "The String Wars"). They include Cosmic variance, Uncertain Principles, Not Even Wrong. Mixed States aggregates many physics blogs.
- More quantum simulations from Groningen in the Netherlands.
- Quantiki, a wiki for quantum information.
- Qwiki, a quantum physics wiki.

The parts above the double horizontal line should be more or less stable. Below that things are still incomplete/subject to change. Square brackets indicate the problem number in the first edition. If there are none, then it's the same. "NEW" means this problem was not in the old editiion. "*" means the problem has been modified slightly.

6/2 Summary of classical wave mechanics and a rough "derivation" of the Schrødinger equation. Review of the basics of linear algebra. A problem sheet will be given out, dealing with some mathematics and physics that is more or less assumed to be already known, but may be a little rusty.

9/2 The concept of the "state" of a system in classical mechanics and in QM. Statistical interpretation of the wavefunction, normalization, probability distributions and how to calculate expectation values. More reviewing of linear algebra. Read Griffiths 1.1 - 1.4. Do problems 1.3, 1.4, 1.5 [1.6,1.7,1.8]

13/2 Momentum, uncertainty principle, and time independent Schrødinger equation. Infinite square well. Read Griffiths sections 1.5 - 1.6, 2.1 - 2.2. Do problems 1.7, 1.9, 1.15 [1.12, 1.14, 1.10]

16/2 Harmonic oscillator--ladder operator method. Read Griffiths 2.3 (first-half: "algrebraic method"). Do problems 2.1, 2.2, 2.5[2.6*]. To hand-in: 1.3, 1.4, 1.9

20/2 Harmonic oscillator, analytic method. Free particle and wave packets. Read Griffiths 2.3(second half)-2.4. Do problems 2.7[NEW], 2.8[NEW], 2.10[13*]

At this point we have covered enough in the course for some of the physics simulations from the University of Colorado to be understandable and hopefully helpful to you. Click on "Quantum Phenomena" on the left panel. The "Quantum Bound States" simulation is about square wells while the "Fourier: Making Waves" one illustrates how fourier series work. I have trouble making the Java-based simulations work on Linux, but they worked under windows.

23/2 Delta-function potential. Read Griffiths 2.5. Do problems 2.12 [37], 2.15, 2.18[19*]. To hand-in: 2.5, 2.7, 2.8

27/2 Finite square well, and review of one-dimensional quantum mechanics. Read Griffiths 2.6. Do problems 2.24, 2.27[26]. If you have time, take a look at the University of Colorado simulation called "Quantum Tunnelling". It is very much about what we've just been looking at, 1D scattering problems.

At this point you might like to start looking at past exam problems. I have sorted all of them according to which system they involve here. Have a look at the ones under "1D", and try not to be intimidated by them. Usually they're not difficult in terms of calculation (easier than many in the book) but often they are stated in a non-standard way, which requires you to think a little. In fact, the weirder the problem seems at first, the more likely it is to not involve anything really tricky.

2/3 Hilbert spaces and Hermitian operators. Read Griffiths 3.1-3.2 (skim through the appendix to check that the material on linear algebra is familiar to you). Do problems 2.27, 2.29[28], To hand in: 2.12, 2.15.

6/3 Operators, observables and eigenfunctions; generalized uncertainty principle. Read Griffiths 3.3-3.6 (but not 122-123). Problems: 3.3[21*], 3.4[NEW], 3.5[NEW], 3.6[NEW]

9/3 Quantum mechanics in three dimensions: The separation of variables. Read Griffiths 4.1. Do problems 3.12[51], 3.17[43], 3.27[NEW]. To hand in: 2.29, 3.3, 3.5

13/3 Hydrogen atom wavefunctions. Read Griffiths 4.2. Do problems 3.13, 4.2, 4.3

16/3 Theory of angular momentum. Read Griffiths 4.3. Do problems 4.8, 4.9[*], 4.10. To hand in: 3.17, 3.27, 4.2

20/3 Angular momentum eigenfunctions. Half-integer angular momentum: Spin, electron in magnetic field. Read Griffiths 4.4-4.4.2. Do problems 4.13, 4.14[NEW],4.18[19]

23/3 NO LECTURE! Work on exam problems: Juni 1995 Opgave 1, June 1999 Opgave 1 (eave out the last part of 1.3) and June 1999 Opgave 3. You may treat this as a written assignment, which I will mark if you hand it in to me before the end of the day on Friday 23/3.

27/3 Addition of angular momenta. Read Griffiths 4.4.3 (but not 187-188). Do problems 4.24[25], 4.27[28], 4.28[29]. To hand in: 4.8, 4.13, 4.14

30/3 Review three-dimensional quantum mechanics. Do Griffiths problem 4.32[33] and exam problem June 1994, Opgave 1. To hand in: 4.24, 4.27

3/4 Identical particles, Helium atom. Read Griffiths 5.1-5.2 (but NOT 5.2.2, the Periodic Table)). Do exam problem June 1998, Opgave 1 and Griffiths problem 5.1

10/4. Solids. Read Griffiths 5.3. Do problems 5.4[3], 5.6[5], 5.8[7]. To hand in: June 1994, Opgave 1 and June 1998, Opgave 1.

13/4 Time-independent perturbation theory. Read Griffiths 6.1-6.2. Note: I think section 6.2.1, "Two-fold degeneracy" is a bit confusingly written, and it's not absolutely necessary, since that case is included in the general case covered in secion 6.2.2. (I will only do the general case in the lecture) So if 6.2.1 seems too difficult try skipping to 6.2.2, although you'll need to refer back to 6.2.2 for notation and a couple of other things. Do problems 5.9[8], 5.11. To hand in: 5.6

17/4. Hydrogen fine structure. Read Griffiths 6.3. Do problems 6.1*, 6.2, 6.3

20/4. Weak- and strong- field Zeeman effect, hyperfine structure. Variational principle. Read Griffiths 6.4-6.4.2, 6.5, and 7.1. Do Griffiths problem 6.14[13], June 1996, opgave 1 and opgave 2.

24/4. Applying the variational principle: He ground state and H molecular ion. Read Griffiths 7.2-7.3. Do problems 6.18[16], 7.1, 7.3[*]. To hand in: June 1996, opgave 1 and 2

27/4. Time-dependent perturbation theory: two-level systems. Read Griffiths 9.1. Do problems 7.7, 7.15[7.13], June 1997, opgave 2.

1/5. Time-dependent perturbation theory: radiation emission and absorption. Read Griffiths 9.2-9.3.2. Do problems 9.1, Jan 1978, opgave 1. To hand in: June 1997, opgave 2.

8/5. Quantum Cryptopgraphy I: Spin-states to EPR pairs/entanglement and qubit operations. Re-read Griffiths 4.4.1, 4.4.3 (but only the case of two spin-1/2, not the more general stuff in the last two pages); read 12.1, 12.3, 12.4 (optional: 12.2). Do Griffiths problem 9.18 [Hint: Look at problem 9.15, but don't do it], Exam June 2002, opgave 2.

11/5. Quantum Cryptography II: Algorithms for secure transmission. Have a look at the paper Quantum Cryptography by Hughes et al. (sections 6,7,8 can be left out). Another nice introduction to quantum information processing in general in the introduction Quantum Computing by Andrew Steane (up to page 11)

Between now and the exam will meet on Tuesday afternoons, starting May 15, we will meet to go through exam problems. Students will meet at 13.00 to work on them together (hopefully after trying tried them already) and I will join you at 14.00

15/5 June 1990 (Note: "udartning" means degeneracy). On this day we will also discuss the running of the course and possible improvements

29/5 To work on: Jan 1995, opgave 2, and june 1995, opgave 2. Please note: as I just wrote in an email, I need to leave at 14.50, so we should probably change the time. At least move it forward one hour (so you guys meet at 12.00 and I come at 13.00) or meet in the morning, or another day.

11/6. I think this is the day we agreed upon yesterday---the Monday after next. Please correct me if I am wrong. Re-do 1998 Question I, and do Questions II and III. Email me if you have questions about anything!

*NPB 010607 *