130_notes.dvi

c(1)nk =

〈φk|H 1 |φn〉

E(0)n −E(0)k

En(2) =

∑

k 6 =n

|〈φk|H 1 |φn〉|^2

E

(0)

n −E

(0)

k

audio

So thefirst order correction to the energyof thentheigenstate,En(1), is just the expectation
value of the perturbation in the unperturbed state. The first order admixture ofφk inψn,c(1)nk,
depends on a matrix element and the energy difference between states. Thesecond order correc-

tion to the energy,E(2)n , has a similar dependence. Note that the higher order correctionsmay
not be small if states are nearby in energy.

The application of the first order perturbation equations is quite simple in principal. The actual
calculation of the matrix elements depends greatly on the problem being solved.

• See Example 22.3.1:H.O. with anharmonic perturbation (ax^4 ).*

Sometimes the first order correction to the energy is zero. Then we will need to use the second order
termEn(2)to estimate the correction. This is true when we apply an electric fieldto a hydrogen
atom.

• See Example 22.3.2:Hydrogen Atom in a E-field, the Stark Effect.*

We will exercise the use of perturbation theory in section 23 when wecompute the fine structure,
and other effects in Hydrogen.

22.2 Degenerate State Perturbation Theory

The perturbation expansion has a problem for states very close in energy. The energy difference
in the denominators goes to zero and thecorrections are no longer small.The series does not
converge. We can very effectively solve this problem bytreating all the (nearly) degenerate
states like we didφnin the regular perturbation expansion. That is, the zeroth order state will be
allowed to be an arbitrary linear combination of the degenerate states and the eigenvalue problem
will be solved.

Assume that two or more states are (nearly) degenerate. DefineN to be the set of those nearly
degenerate states. Choose a set of basis state inNwhich are orthonormal

〈φ(j)|φ(i)〉=δji

whereiandjare in the setN. We will use the indicesiandjto label the states inN.

By looking at the zeroth and first order terms in the Schr ̈odinger equation and dotting it into one of
the degenerate statesφ(j), wederive(see section 22.4.2)the energy equation for first order (nearly)