## Stationary perturbation theory

Stationary perturbation theory is concerned with finding the changes in the discrete energy levels and the changes in the corresponding energy eigenfunctions of a system, when the Hamiltonian of a system is changed by a small amount.  Let H = H0 + H' = H0 + λW.  H0 is the unperturbed Hamiltonian whose eigenvalues E0p and eigenstates |Φpi> are known.  Let {|Φpi>} denote an orthonormal eigenbasis of H0, H0pi>= E0ppi>.  Here i denotes the degeneracy.  Assume that the matrix elements of H' in the eigenbasis of H0 are small compared to the matrix elements of H0.
pi|H'|Φpi>  <<  <Φpi|H0pi> = E0p.
We write H' = λW with λ << 1 and <Φpi|H0pi> on the order of E0p.
We are looking for the eigenvalues E(λ) and the eigenstates |ψ(λ)> of H(λ) = H0 + λW.
H(λ)|ψp> = Ep(λ)|ψp> or, keeping the dependence on λ in mind without specifically writing it down, H|ψp> = Epp>.
Since λW is small, we assume that E and |ψ> can be expanded as a power series in λ.

Ep = E0p + λE1p + λ2E2p + ... ,
p> = |ψp0> + λ|ψp1> + λ2p2> + ... .
We may then write
(H0 + λW)(|ψp0> + λ|ψp1> + λ2p2> + ...)
= (E0p + λE1p + λ2E2p + ...)(|ψp0> + λ|ψp1> + λ2p2> + ...).

This equation is must be valid over a continuous range of λ.  Therefore we equate coefficients of equal powers of λ on both sides to obtain a series of equations that represent successively higher orders of the perturbation.

• (H0 - E0p)|ψp0> = 0
• (H0 - E0p)|ψp1> = (E1p - W)|ψp0>
• (H0 - E0p)|ψp2> = (E1p - W)|ψp1> + E2pp0>
• (H0 - E0p)|ψp3> = (E1p - W)|ψp2> + E2pp1> + E3pp0>
• ...

(H0 - E0p)|ψp0> = 0 implies that |ψp0> is a linear combination of unperturbed eigenfuctionns |Φpi> with the corresponding eigenvalue E0.  We choose <ψp0p0> = 1.  |ψps>is not uniquely defined.  We can add an arbitrary multiple of |ψp0> to each |ψps> without affecting the left hand side of the above equations.  Most often this multiple is chosen so that <ψp0ps> = 0.  The perturbed ket is then not normalized.
We then have
p0|(H0 - E0p)|ψps> = <ψp0|(E1p - W)|ψps-1> + <ψp0|E2pps-2> + ... + <ψp0|Espp0>,
or
0 = -<ψp0|W|ψps-1> + Esp,
or
Esp = <ψp0|W|ψps-1>.

To calculate the energy to sth order, we only need to know the state vector to order s - 1.

### Perturbation theory for non-degenerate levels

First-order energy corrections

Consider a particular non-degenerate eigenvalue E0n of H0.  H0n> = E0nn>.  The other eigenvalues of H may or may not be degenerate.  We have |ψn0> = |Φn> and E1n = <Φn|W|Φn>.
The first-order energy correction therefore is λE1n = <Φn|H'|Φn>.
We have En = E0n + <Φn|H'|Φn> + O(λ2).

First-order eigenvector corrections

(H0 - E0p)|ψp1> = (E1p - W)|ψp0> = (E1p - W)|Φp>.  |ψp0> is an eigenstate of the unperturbed Hamiltonian.  We may expand
p1> = ∑p'≠p,i bp'ip'i> in terms of the basis vectors |Φp'i>.
In the expansion bp = 0 because <ψp0pi> = 0.
p'≠p,i(H0 - E0p) bp'ip'i> = ∑p'≠p,i(E0p' - E0p) bp'ip'i> = (E1p - W)|Φp>.

Multiply from the left by <Φp''i| we obtain
(E0p' - E0p) bp''i = -<Φp''i|W|Φp>,
or
bp'i = <Φp'i|W|Φp>/(E0p - E0p'),  p' ≠ p.
Therefore

p> = |Φp> + ∑p'≠p,i p'i|H'|Φp>/(E0p - E0p')]|Φp'i> + O(λ2).

Second-order energy corrections

Since we have found the expression for the state vector to first order, we can now find the expression for the energy to second order.
E2p = <ψp0|W|ψp1> = ∑p'≠p,i bp'ip|W|Φp'i> = ∑p'≠p,i p'i|W|Φp><Φp|W|Φp'i>/(E0p - E0p').

E2p = ∑p'≠p,i |<Φp'i|W|Φp>|2/(E0p - E0p').

Ep = E0p + <Φn|H'|Φn> + ∑p'≠p,i |<Φp'i|W|Φp>|2/(E0p - E0p') + O(λ3)
= E0p + λE1p + λ2E2p + ... .
H = H0 + H' = H0 + λW.  In practice, after having derived the perturbation expansion, we often set λ = 1 and let H' = W be small.

### Perturbation theory for degenerate levels

First-order energy corrections

Let H = H0 + H' = H0 + λW.  Let {|Φpi>} denote an orthonormal eigenbasis of H0, H0pi> = E0ppi>.  Here i denotes the degeneracy.  Consider a particular degenerate eigenvalue E0p of H0.  Assume that this eigenvalue is g-fold degenerate, i = 1,2,...,g.  To find E1p we use

(H0 - E0p)|ψp1> =  (E1p - W)|ψp0>, |ψp0> = ∑iapipi >.

Multiplying from the left by <Φpj| we obtain
0 = ∑iapipj|(E1p - W)|Φpi> = E1papj - ∑iapipj|W|Φpi >,
or
ipj|W - E1pδijpi>api = 0.

This is an eigenvalue equation for the operator W in the subspace E(0,p) of vectors with the eigenvalue of H0 equal to E0p.  We find g eigenvalues E1p,i which may or may not be degenerate.  We then find the corresponding g eigenvectors.  To solve for the eigenvalues we set det(W - E1p) = 0 in the subspace E(0,p).  If we find a non-degenerate eigenvalue E1p,i, then the corresponding eigenvector is uniquely defined.  If the eigenvalue E1p,i is degenerate then the corresponding eigenvector is still not uniquely defined.  The degeneracy may or may not be removed in higher order.