Appendix A: Perturbation
theory
A
A.1 Mathematics of
perturbation theory 298
A.2 Interaction of classical
oscillators of similar
frequencies 299
Although degenerate perturbation theory is quite simple (cf. the series
solution of differential equations used to find the eigenenergies of the
hydrogen atom), it is often regarded as a ‘difficult’ and subtle topic
in introductory quantum mechanics courses. Degenerate perturbation
theory arises often in atomic physics, e.g. in the treatment of helium
(eqn 3.14). Equations 6.34 and 7.89 in the treatment of the Zeeman
effect on hyperfine structure and the a.c. Stark effect, respectively, also
have a similar mathematical form, as can be appreciated by studying
the general case in this appendix. Another aim of this appendix is
to underline the point made in Chapter 3 that degenerate perturbation
theory is not a mysterious quantum mechanical phenomenon (associated
with exchange symmetry) and that a similar behaviour occurs when two
classical systems interact with each other.A.1 Mathematics of perturbation theory
The Hamiltonian for a system of two levels of energiesE 1 andE 2 (where
E 2 >E 1 ) with a perturbation given byH′, as in eqn 3.10, can be written
in matrix form asH 0 +H′=
(
E 1 0
0 E 2
)
+
(
H 11 ′ H 12 ′
H 21 ′ H 22 ′
)
. (A.1)
The matrix elements of the perturbation areH′ 12 =〈ψ 1 |H′|ψ 2 〉,andsim-
ilarly for the others. The expectation values〈ψ 1 |H′|ψ 1 〉and〈ψ 2 |H′|ψ 2 〉
are the usual first-order perturbations. It is convenient to write the
energies in terms of a mean energyJand an energy interval 2that
takes into account the energy shift caused by the diagonal terms of the
perturbation matrix:E 1 +H 11 ′ =J−,
E 2 +H 22 ′ =J+.For simplicity, we assume that the off-diagonal terms are real, e.g. as for
the case of exchange integrals in helium and the other examples in this
book:^1(^1) This is normally the case when levels
are bound states. We shall find that
the eigenenergies depend onK^2 ,which
would generalise to|K|^2. H 12 ′ =H 21 ′ =K.