Graduate Course: Quantum Physics

Approximation Methods: Time dependent Perturbation Theory

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

This page is the first of three pages on the Time-dependent perturbation theory. You can go directly to 2: The role of radiation or 3: The transition rate between states, as given by Fermi's 'golden rule'.

1.1 Introduction

Perturbation theory (PT) comprises an important set of approximation methods in Quantum Physics. It is traditionally divided into: Time dependent PT is the subject of this web page. In particular, we aim to get a semi-quantitative understanding of In the books we are using, the topics are covered by G3 (chapters 15-18, plus several web supplements) and G2 (chapters 21 and 22, with Special Topic 4). Liboff has much of this material in Chapter 13, sections 5-9.

1.2 The incorporation of E&M into quantum theory

This topic is the marriage of particle dynamics (p, E, etc) and EM Fields, i.e. E, B, c, etc. Note that EM since Maxwell (1865) is relativistic (invariant under the Lorentz transformation), whereas QM pre-Dirac (1928) is not relavistically correct. Moreover, the true marriage of these two topics is Quantum Electrodynamics (QED). In this subject, pioneered by Feynman and others, we have quantized fields as well as particles, and this is beyond the version of QM we are currently studying. What we can readily do is sometimes referred to as 'semi-classical'. The full relativistic theory is typically studied only by specialists; everyone else runs out of time before they get there. If you want to tangle with QED, you can start with Baym's chapter 13, or Feynman's lectures: Quantum Electrodynamics (Benjamin, 1962), starting at lecture 1 (handout). This book also contains reprints of his Nobel prize-winning papers (Phys. Rev. 76 (1949) 749-759 & 769-789).

1.3 The choice of gauge

Many of you will have had the following material in an EM course, but let's aim to be self-contained here. The first stage is to introduce the Vector potential, A(r, t), and the Scalar potential f(r, t). From these one can derive the fields by vector differentiation, as in


        (1)

These equations are in Gaussian units as used by G2; consult G3 chapter 16 or EM books for the same equations in SI units. Since E and B are defined by derivatives of A, there is an indeterminacy of A itself; this is the choice of 'gauge' as indicated below:


        (2)

One therefore gets different differential equations for A and f, depending on the gauge adopted (G3 p247, G2 p216-7, or Baym pages 262-4). The Coulomb gauge is most convenient for the arguments developed here. Indeed, if one uses the retarded potential, f(t - r/c) in the Lorentz gauge, the next term compensates for retardation and gives results as for the Coulomb gauge. One can see that this discussion can get technical quite quickly...

1.4 The perturbation Hamiltonian

The Hamiltonian containing A and f potentials is given, for an electron, by G3 p249 or G2 p217, as


        (3)

Note however, that we need some care over signs. First, this equation (13-19) in G2 has a plus sign for the ef term, but this is corrected in the error list on the web, and in G3. Then, if you look in either Liboff or Baym, you will find (p +eA/c) and +ef for charged particles generally. This means that the form used by Gasiorowicz is specifically for an electron, with charge -|e|.

So (-)ef is just another potential, and (-)eA/c is a contribution to the momentum. At normal fields encountered on earth, this term in A is small, so that (eA/c)2 can be neglected. In that case the perturbation Hamiltonian H1 is given by


        (4)

We now incorporate the (-)ef term into the potential V(r), because, using the Coulomb gauge, we can show that f is constant unless the charge density is varying at the point in question, see G3 p247 or G2 p217. This means we can concentrate on the role of the vector potential A. Think carefully about the representation of p, and whether p and A commute. Then, using the Coulomb gauge you will find that the perturbation is


        (5)

There are two types of application. For static, or slowly varying fields, B is Bz, and G3 p250, G2 p219-220 shows that A = -r´B/2. The term A.p is therefore -r´B.p/2 = -p´r.B/2 = -L.B/2. This energy is just that due to orbital magnetism, with angular momentum L. Gasiorowicz (G3 p250 or G2 p220) then shows that this is only comparable with electronic energy levels at fields of order 109 Gauss (105 Tesla). So this is a justification for neglecting the quadratic term, since we can only achieve ~10 Tesla for steady fields and maybe up to 100 T in pulses.

However, this may not be true in some astrophysical situations. We have noted the normal Zeeman effect in a strong magnetic field, and wondered why it was less common than the anomalous Zeeman effect, where the field is weaker, and is not sufficient to break e.g. the L.S coupling of atomic states. Kevin Healey pointed out to me that this was because Zeeman discovered his normal effect in Hydrogen spectra from sunspots, where the magnetic field is ~100 T. Now on the surface of a neutron star, we have something else ...

In the second type of application we have dynamic, rapidly varying, fields. Here the EM field is fully described by A(r, t), but we can have several subcases depending on the precise t-dependence. The first case is a continuous wave, i.e. we are talking about radiation at angular frequency w.

2. The role of radiation

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: 10th April 2008.