Quantum Mechanics Spring 2006

Tuesdays 13-16, Fridays 9-12 Lecture room II.

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

Course Content:

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 computing.

Course materials:

D. Griffith: 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 has been ordered 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 probably 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. I will try to put a page indicating the mapping of new problems to old, 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 computing, 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.

Other recommended books: "The quantum world", J. C. Polkinghorne (focusses on philosophical issues), "The Feynman Lectures of Physics", Volume III, by Feynman, Leighton and Sands (particularly the first 12 chapters).

Course plan

The parts above the double horizontal line should be more or less stable. Below that things are still incomplete/subject to change.

7/2 Introduction to the course and to the ideas of quantum mechanics: Particles as waves, and a quick look at the Schrødinger equation; a brief overview of the mathematics of quantum mechanics, overview of the philosophical issues

10/2 States, wavefunctions, the Schrødinger equation and interpretation. Read Griffiths 1.1 - 1.4. Do problems 1.3, 1.4, 1.5

14/2 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, 2.1, 2.2 [1st edition problems 1.12, 1.14, 1.10, 2.1, 2.2 respectively]

17/2 Harmonic oscillator--ladders and polynomials. Read Griffiths 2.3. Do problems 2.5, 2.7, 2.8, 2.10 [1st edition problems 1.6 * , NEW, NEW, 13 * - NEW means this problem was not in the old editiion. * means the problem has been modified slightly.]

Note changes in the next three lectures, and in the problems! Square brackets indicate the problem number in the first edition. If there are none, then it's the same.

21/2 Harmonic oscillator, analytic method. Free particle and wave packets. Read Griffiths 2.3(second half)-2.4. Do problems 2.10 [13*], 2.12 [37], 2.15, 2.42[38]

24/2 Delta-function potential. Read Griffiths 2.5. Do problems 2.18[19*], 2.22, 2.24, 2.26[25]

28/2 Finite square well, and review of one-dimensional quantum mechanics. Read Griffiths 2.6. Do problems 2.27[26], 2.29[28].

Remember a "*" means the cooresponding problem in the old book is different in some way.

3/3 From vector spaces to 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 3.3[21*], 3.4[NEW], 3.5[NEW], 3.6[NEW]

7/3 Operators, observables and eigenfunctions; generalized uncertainty principle. Read Griffiths 3.3-3.6 (but not 122-123). Do problems 3.12[51], 3.14[39], 3.17[43], 3.27[NEW]

10/3 Quantum mechanics in three dimensions: The separation of variables. Read Griffiths 4.1. Do problems 4.2, 4.3, 4.7[*]

14/3 Hydrogen atom wavefunctions. Read Griffiths 4.2. Do problems 4.8, 4.9[*]

17/3 Theory of angular momentum. Read Griffiths 4.3. Do problems 4.10, 4.13, 4.14[NEW]

21/3 Half-integer angular momentum: Spin, electron in magnetic field. Read Griffiths 4.4-4.4.2. Do problems 4.18[19], 4.19[20], 4.24[25]

24/3 Addition of angular momenta. Read Griffiths 4.4.3 (but not 187-188). Do problems 4.27[28], 4.28[29], 4.32[33]. In this lecture we will also go through the first written assignment.

28/3 Review three-dimensional quantum mechanics. Do exam problems June 1994 Opgave 1 and June 1998, Opgave 1.

31/3 Identical particles, atoms. Read Griffiths 5.1-5.2. Do problems 5.1, 5.4[3], 5.6[5]

4/4. Solids. Read Griffiths 5.3. Do problems 5.8[7], 5.9[8], 5.16[13]

7/4 Time-independent perturbation theory. Read Griffiths 6.1-6.2. Do problems 6.1*, 6.2, 6.3

11/4 NO LECTURE. Suggested exam problems to practice on: June 1995 Opgave 2, June 1997, Opgave 1 and 2

18/4. Hydrogen fine structure. Read Griffiths 6.3. Do problems 6.8, 6.9, 6.14[13]

21/4. Weak- and strong- field Zeeman effect, hyperfine structure. Variational principle. Read Griffiths 6.4-6.4.2, 6.5, and 7.1. Problems 6.8, 6.9, 6.14[13] are re-assigned for this lecture.

25/4. 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[*]

28/4. Time-dependent perturbation theory: two-level systems. Read Griffiths 9.1. Do problems 7.6, 7.7, 7.15[7.13]

2/5. Time-dependent perturbation theory: radiation emission and absorption. Read Griffiths 9.2-9.3.2. Do problems 9.1, 9.2, 9.5

5/5. Quantum computing I: From spin-states to qubits to quantum gates. Re-read Griffiths 4.4.1 (spin-1/2) and read 12.1-12.3 (EPR paradox, Bell's theroem, no-cloning theorem). Also, pages will be handed-out/sent by email which are photocopied from the book "Quantum Computing" by Joachim Stolze and Dieter Suter (Wiley, 2004). You may be interested in skimming through an article available online at http://www.arxiv.org/abs/quant-ph?papernum=9708022, called Quantum Computing by Andrew Steane. It is a fairly comprehensive article on the whole subject dating from 1997. Do problems 9.7, 9.10[9], 9.15[14], 9.16[15]

9/5. No lecture (as mentioned on Friday 28/4, I have a meeting I cannot get out of that afternoon)

16/5. Quantum computing II: Algorithms, Deutsche and Shor.

Read the first half of chapter 8 of Stolze and Suter (pdf has been distributed by email). I have made a set of short problems (I'm pretty sure these are all zero star problems), available as a pdf.

19/5 Start of review period. From now on we will concentrate on exam problems. I sorted these into different categories on a separate page. Today (Friday 19/5) is the last scheduled lecture so we will start as usual at 9.15. After that it will be different (to be discussed). The program for today is to start thinking about one-dimensional quantum mechanics again. I will summarize that briefly, and then we will have a look at two exam problems. One I suggested before, which is June 1997, Opgave 1 and one which is I think pretty straightforward, Jun 1999 Opg 3. Another thing I want to do is have a little discussion about how the course has been and what could be done better.

23/5. Since we did not actually go over the two problems on Friday, they are assigned for Tuesday, along with June 2000, Opgave 2. Students will be there at 13.00, I will show up at 14.00. As well as the problems I will review 3D quantum mechanics.

Practicing

I have made a list of short "practice problems". These are for practicing the mathematical manipulations rather than quantum mechanics. I have also made a separate page containing the schedule for our remaining sessions.

NPB 290506