Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Operator/Matrix methods in Quantum Physics (2001-3)

Matrix methods are needed any course on Quantum Physics, both for formulating and for solving problems. Since this is a one-semester course with (at least in part) a review character, we can't spend nearly as much time on this topic as we would do in a two or three semester course sequence. Nonetheless, as I'm sure you are aware, such problems appear with remarkable regularity on the comprehensive exam... (Have I mentioned this before?)

The course book, Gasiorowicz (G), unlike some other books, does not start by using operator methods or matrices but introduces them gradually: all references are to the 2nd Edition (1996). The scattering matrix first appears in chapter 5, with some useful problems. Operator methods are used in chapter 7 to discuss the simple harmonic oscillator, whose formulation in terms of matrix operators is set out at the beginning of chapter 14. Angular momentum is discussed in chapter 11, followed by the equivalent matrix formulation in chapter 14. Then, spin is introduced which requires a matrix treatment.

We have followed this path very quickly, and now have to recap to solve some problems. We may also need to absorb Chapter 6 and Appendix B as we go. You may do this by attempting some problems, not necessarly for credit, part revision and consolidation in preparation for the mid-term exam, where I will concentrate on conceptual problems as much as possible. The more extended problems on this material will be on problem set#3, due (after Spring Break) on March 27th.

The first search of the web that we did in 1998 under (Matrix + quantum) gave 99 entries, potentially a useful number, but I was unprepared for the mixture of doctoral theses and alternative connectedness which I found.... Most of the sites I found are now well and truly dead, but see below for some which are still there.

You could also go to the library or get out your old math books, but this is more fun. (To any member of faculty who thinks we have lost our marbles, please note that we are also doing just that..). In '98 Jennifer Trelewicz and Hu Zhan produced a nice page about linear operator/matrix methods, and Jing Tao and Hu Zhan produced a very fancy page on matrices (and tricks) which could be useful/ educational for the class and may be developed further this year.

In a lighter vein, enjoy the fruits of my surfing for matrices, using either Alta Vista ("quantum physics" near "matrix")- with "matrices" and "chaos" as alternates. I had another go in 1999 with Alta Vista, which is now getting dominated by adverts, and repeated the exercise in January 2000 and 2002 using the excellent Google search engine:

  • A course on Quantum Groups by Shahn Majid was available last year in Cambridge, England. This year he has moved to Queen Mary College in London, and has put the lectures notes into a book. He has a nice explanation of what Quantum Groups are. Relax! we are not going to be as mathematical as this..
  • If you are turned on by quantum computation, try the Stanford qcomp collaboration, including Dilbert's view of the topic.
  • Suitable topics for a project later include density functional theory, but we are probably not quite ready for this paper yet.
  • A young mathematician got funding in 1998 from the Australian research grants council to study random matrices in relation to quantum chaos. It is amazing what young persons get up to these days...
  • Alternatively, you could vist the alternative site at mystic planet to get well and truly connected. Unfortunately this site crashes older computers, so be warned, the aliens really are out there!!
  • And, finally, for those who just like the idea that all of us on the planet are now somehow in contact with each other, try the Maths colloquia at the University of Tasmania. The entry about matrices in quantum whatever has long since disappeared, but don't you just love those titles?!

Latest version of this document: 16th February 2002, amended 16 Feb and 16 Dec 03.