130_notes.dvi

and dotted intoφkyields

〈φk|λH 1 |φn〉+E

(0)

k λc

(1)

nk=E

(0)

n λc

(1)

nk.

From these it is simple to derivethe first order corrections

λEn(1)=〈φn|λH 1 |φn〉

λc

(1)

nk=

〈φk|λH 1 |φn〉

E(0)n −Ek(0)

The second order equation projected onφnyields

∑

k 6 =n

λc

(1)

nk〈φn|λH^1 |φk〉=λ

(^2) E(2)
N.
We will not need the projection onφkbut could proceed with it to get the second order correction to
the wave function, if that were needed. Solving for thesecond order correction to the energy
and substituting forc(1)nk, we have
λ^2 En(2)=

∑

k 6 =n

|〈φk|λH 1 |φn〉|^2

En(0)−E(0)k

Thenormalization factorN(λ) played no role in the solutions to the Schr ̈odinger equation since
that equation is independent of normalization. We do need to go backand check whether the first
order corrected wavefunction needs normalization.

1

N(λ)^2 =〈φn+

∑

k 6 =n

λc(1)nkφk|φn+

∑

k 6 =n

λc(1)nkφk〉= 1 +

∑

k 6 =n

λ^2 |c(1)nk|^2

N(λ)≈ 1 −^12

∑

k 6 =n

λ^2 |c(1)nk|^2

The correction is of orderλ^2 andcan be neglectedat this level of approximation.

These results are rewritten with all theλremoved in section 22.1.

22.4.2 Derivation of 1st Order Degenerate Perturbation Equations

To deal with the problem of degenerate states, we will allow an arbitrary linear combination of
those states at zeroth order. In the following equation, the sum overiis the sum over all the states
degenerate withφnand the sum over k runs over all the other states.

ψn=N(λ)





∑

i∈N

αiφ(i)+

∑

k6∈N

λc(1)nkφk+...





whereN is the set of zeroth order states which are (nearly) degenerate withφn. We will only go
to first order in this derivation and we will useλas in the previous derivation to keep track of the
order in the perturbation.