Graduate Course: Quantum Physics

Time dependent Perturbation Theory #2

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 
Lecture notes by John A. Venables. Note that this lecture needs the Symbol font enabled on your browser.

1. Electromagnetism and Quantum Mechanics

2. The role of radiation

The qualitative arguments about radiation were covered earlier in lectures on transitions in atoms, particularly in relation to selection rules. These involved parity changes to accomodate the DL = + or -1 rule, etc, including the role of polarization (linear or circular). We also discussed the amplitudes in terms of radial wave functions. Quantitative arguments require the use of the correct A.p terms in the perturbation Hamiltonian, and time-dependent PT. These arguments are set out by Gasiorowicz (G3 chapter 17, or G2 chapter 21). Liboff has a useful section on the same material, and a summary table (handout) on the time dependencies expected for several types of perturbation.

This is also a good time to look at the thermodynamic arguments used by Einstein in establishing his A and B coefficients. This fascinating material is set out by G3 in an early Web-appendix (supplement 1A) and in G2 in chapter 22 (p367-369). This treatment is amazing in that the formula deduced (G3 eqn 1A-15, or G2 eqn 22-17) contains the factor (1 + n), where n is the number of photons. This factor does not come out naturally from the semi-classical treatment of radiation, but does from the QED treatment, where the radiation field is also quantized (as a type of simple harmonic oscillator), and thereby has "zero-point" energy, even when n = 0. This modification due to QED is noted by Gasiorowicz in G2 (p348), and treated more fully in G3 (p262 and web supplement 18A), even if, as he delicately puts it, certain features are "swept under the rug".

You can see that we are heavily into web supplements at this stage of the course, and you can be thankful that you haven't had to carry them to class throughout the semester. More immediate concerns might well include problem set 5, your project or revising for the Final Exam, but you can always study this stuff in your own time.

3. The transition rate between states, as given by Fermi's 'golden rule'


Return to Timetable 2, Module 5 or to course home page.

Latest version of this document: 10 April 2008, ex 2nd May 2004.