Graduate Course: Quantum Physics

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

Eigenvalue problems in Quantum Physics (2001-3)

Eigenvalue problems form an important part of any course on Quantum Physics. 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...

The main course book, Gasiorowicz (G), spends the whole of chapters 4 on 5 on this topic, and there is much that is useful there- we just can't do all of it explicitly. However, I will grade any problems you do from this book, and we might like to add some comments to particular aspects or problems which can be linked to this page. For reference the topics in the book are classified in the outline below. I have added some section and page number references to the alternate course book, Liboff (L). Note: all references are to Gasiorowicz 2nd Edition (1996) and Liboff 3rd Edition (1998). The corresponding page references for Gasiorowicz's 3rd edition (G3, 2003) and Liboff's 4th edition (L4, 2003) are given here.

Separation of variables (G p54-57, and/or L, section 3.4, p82-86)  

  • The product solution Y(x,t) = T(t).u(x) leads to the time independent Schroedinger equation for u(x).
  • The idea of a linear operator is introduced, with examples.
Particle in a box (G p58-73, and/or L, section 4.1, p 92-97)
This 1D example is used to demonstrate the expansion postulate and
  • Normalization (p61-63)
  • momentum eigenstates (p63-66)
  • normalization of current (p66-67)
  • degeneracy (p67-68)
  • parity, odd-even functions, [p,H] = 0, etc (p68-70)
  • and of course, PROBLEMS (p 70-73) One learns this stuff by doing them!!
1-dimensional potentials (G p74-113, and/or L, chapters 7-8, p 194-298)
There is an enormous amount of material here. I'd suggest a background read, questions in class, and then a concerted attack on a few problems which interest you. Some of these are on Problem set #2. Then we can share our questions and experience at a later date. 
  • Potential steps (p74-78) and wavelike reflection. Alternatives or supplements are L, section 7.6 (p230-235) the CUPS program, and Quantum Resources.
  • Potential wells (p78-79, or L, section 8.1, p290-298) and transmission resonance, also dealt with by Liboff in section 7.8 (p244-246), CUPS and Quantum Resources.
  • Potential barriers (p 79-82), leading to Tunneling (p 82-89, or L, section 7.7, p237-243) and the WKB method (p450-452 or L, section 7.10, p 253-270) including
  • Applications to cold field emission, STM, superconductors and nuclear matter.
  • Bound states in a potential well (p 89-93).
  • Delta-function potentials (p93-99) leading to the 
  • Kronig-Penney model of a lattice of d-function potentials (p 99-103), and finally
  • The Simple harmonic oscillator (p103-108)

Latest version of this document: 1st February 2002, amended 16 Feb, 16 Dec 03 and 4 Feb 04.