G. The Perturbation Method
The perturbation method is applied to a problem in which the Hamiltonian operator
can be written in the form
ĤĤ(0)+̂H′ (G-1)
such that̂H(0)gives a time-independent Schrödinger equation that can be solved:
Ĥ(0)Ψ(0)E(0)Ψ(0) (G-2)
TheĤ′term is called the perturbation. We assume that the perturbation term is spin-
independent, so that spin functions and integrations over spins can be omitted. We
now construct a new Hamiltonian operator by multiplying the perturbation term by a
fictitious parameter,λ:
̂H(λ)̂H(0)+λ̂H′ (G-3)
The actual Hamiltonian is recovered whenλ1.
G. 1 The Nondegenerate Case
In the nondegenerate case each energy eigenvalue corresponds to a single eigenfunc-
tion. We assume that the energy eigenvalues and energy eigenfunctions of state number
ncan be represented by a power series inλ, as in Eqs. (19.3-5) and (19.3-6). When
these power series are substituted into the time-independent Schrödinger equation we
obtain
(̂H(0)+λ̂H′)(Ψ(0)n +Ψ(1)nλ+Ψ(2)nλ^2 + ···)
(E(0)n +E(1)nλ+E(2)n λ^2 + ···)(Ψ(0)n +Ψ(1)nλ+Ψ(2)nλ^2 + ···) (G-4)
We multiply out the products in Eq. (G-4) and use the fact that if two power series
are equal to each other for all values of the independent variables, the corresponding
coefficients in the two series are then equal to each other. We equate the terms that are
independent ofλon the two sides of the equation, after which we equate the coefficients
of the terms proportional toλ, and so on. The terms independent ofλ(zero-order terms)
produce the same equation as Eq. (G-2):
Ĥ(0)Ψ(0)n E(0)n Ψn(0) (G-5)
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