Graduate Course: Quantum Physics

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

Eigenvalue problems in Quantum Physics

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, third edition (G3, 2003), spends chapters 3 and 4 on this topic, and there is much that is useful there, and in the second edition - we just can't do all of it explicitly. However, I will grade any problems you do from these books, 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 books are classified in the outline below. I have added some section and page number references to an alternate course book, Liboff, fourth edition (L4, 2003). The corresponding page references for Gasiorowicz's 2nd edition (G2, 1996) and Liboff's 3rd edition (L3, 1998) are given here.

Separation of variables (G3 p44-47, and/or L4, section 3.4, p80-84)  

  • 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 (G3, chapter 3 p47-65 and/or L4, section 4.1, p90-94)
This 1D example is used to demonstrate the expansion postulate and
  • Normalization (G3 p48-49; L4 p92-93)
  • momentum eigenstates (G3 p56-57, L4 p216-219)
  • normalization of current (G3 p57-59, L4 p216-219)
  • degeneracy (G3 p59-60, L4 p137-139, 219)
  • parity operator P, odd-even functions, [P,H] = 0, etc (G3 p60-62, L4 p176-180)
  • and of course, PROBLEMS (G3 p63-65, lots in L4) One learns this stuff by doing them!!
1-dimensional potentials (G3, chapter 4 p66-93, and/or L4, chapters 7-8, p187-289)
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 (G3 p66-69) and wavelike reflection. Alternatives or supplements are L4, section 7.6 (p222-241) the CUPS program, and Quantum Resources.
  • Potential wells (G3 p69-70, or L4, section 8.1, p290-298) and transmission resonance, also dealt with by Liboff in section 7.8 (p235-241), CUPS and Quantum Resources.
  • Potential barriers (G3 p71-73), leading to Tunneling (G3 p73-75, or L4, section 7.7, p237-243) and the WKB method (web supplement 4-A or L4, section 7.10, p245-267) including
  • Applications to cold field emission, STM, superconductors and nuclear matter (web supplement 4-B).
  • Bound states in a potential well (G3 p75-80, L4 p278-289).
  • Delta-function potentials (G3 p81-84) leading to the 
  • Kronig-Penney model of a lattice of d-function potentials (web supplement 4-C, L4 p289-303), and finally
  • The Simple harmonic oscillator (G3 p85-89), and MORE PROBLEMS (G3 p90-93)...

Latest version of this document: 30th January 2008.