E=E 0 ±
√
γ^2 + ∆^2This is OK in both limits, ∆≫γ, andγ≫∆. It is also correct when the two corrections are of the
same order.
22.4 Derivations and Computations
22.4.1 Derivation of 1st and 2nd Order Perturbation Equations
To keep track of powers of the perturbation in this derivation we willmake the substitutionH 1 →
λH 1 whereλis assumed to be a small parameter in which we are making the series expansion of our
energy eigenvalues and eigenstates. It is there to do the book-keeping correctly and can go away at
the end of the derivations.
To solve the problem using aperturbation series, we will expand both our energy eigenvalues and
eigenstates in powers ofλ.
En = E(0)n +λEn(1)+λ^2 En(2)+...ψn = N(λ)
φn+∑
k 6 =ncnk(λ)φk
cnk(λ) = λc(1)nk+λ^2 c(2)nk+...Thefull Schr ̈odinger equationis
(H 0 +λH 1 )
φn+∑
k 6 =ncnk(λ)φk
= (En(0)+λE(1)n +λ^2 En(2)+...)
φn+∑
k 6 =ncnk(λ)φk
where theN(λ) has been factored out on both sides. For this equation to hold as we varyλ, it must
hold for each power ofλ. We will investigate the first three terms.
λ^0 H 0 φn=En(0)φn
λ^1 λH 1 φn+H 0 λ∑
k 6 =nc(1)nkφk=λEn(1)φn+λEn(0)∑
k 6 =nc(1)nkφkλ^2 H 0∑
k 6 =nλ^2 c(2)nkφk+λH 1∑
k 6 =nλc(1)nkφk=En(0)∑
k 6 =nλ^2 c(2)nkφk+λEn(1)∑
k 6 =nλc(1)nkφk+λ^2 En(2)φnThe zero order term is just the solution to the unperturbed problem so there is no new information
there. The other two terms contain linear combinations of the orthonormal functionsφi. This means
we can dot the equations into each of theφito get information, much like getting the components
of a vector individually. Sinceφnis treated separately in this analysis, we will dot the equation into
φnand separately into all the other functionsφk.
The first order equation dotted intoφnyields
〈φn|λH 1 |φn〉=λE(1)n