Graduate Course: Quantum Physics

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

Operator/Matrix methods in Quantum Physics

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, such problems used to appear with remarkable regularity on the physics written comprehensive exam . This is no longer relevant, since the course is now primarily aimed at Materials, Chemistry and Nanoscience students (Quantum Physics hasn't changed, though).

The course book, Gasiorowicz, third edition (G3, 2003), unlike some other books, does not start by using operator methods or matrices but introduces them gradually. The scattering matrix first appears in the problems for chapter 4. Operator methods are introduced in chapter 5, and used in chapter 6 to discuss the simple harmonic oscillator, whose formulation in terms of matrix operators is set out at the beginning of chapter 9. Angular momentum is discussed in chapter 7, followed by the equivalent matrix formulation in chapter 9. Then, spin, which requires a matrix treatment, is introduced in chapter 10. The corresponding chapter references for the second edition (G2, 1996) are given here.

We have followed this path very quickly, and now have to recap to solve some problems. We may also need to absorb some of the web material, previously covered in appendices in the second edition, 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), this year on 03/31/08. Also, at this point, it may be relevant to use my QMMS course, in particular lecture 1, since this starts with the matrix formulation.

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 (it was in the past anyway, I've taken in down now: too complicated to reinstall!)

In a lighter vein, enjoy the fruits of my surfing for matrices, using either Alta Vista ("quantum physics" + matrix)- with "matrices" and "chaos" as alternates. I had another go in 1999 with Alta Vista, and repeated the exercise several times using the excellent Google search engine:

  • A course on Quantum Groups by Shahn Majid is available, originally in Cambridge, England. More recently 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..
  • 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. Some impressive scientists are orgainzing a "school" on the topic in 2004 at the Isaac Newton Institute in Cambridge. It is amazing what young persons (and some not so young) 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!!
  • Now of course, we also have the movie, so if you type in matrix+quantum you have only yourself to blame.
  • 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. This site has been asleep for more than two years, lost in hyperspace, and the entry about matrices in quantum whatever has long since disappeared, but don't you just love those titles?!

Latest version of this document: 27th February 2008.