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 = H_{0 }+ H' = H_{0 }+ λW. H_{0}
is the unperturbed Hamiltonian whose eigenvalues E_{0}^{p} and
eigenstates |Φ^{p}_{i}> are known. Let {|Φ^{p}_{i}>}
denote an orthonormal eigenbasis of H_{0}, H_{0}|Φ^{p}_{i}>=
E_{0}^{p}|Φ^{p}_{i}>. Here i denotes the
degeneracy. Assume that the matrix elements of H' in the eigenbasis of H_{0}
are small compared to the matrix elements of H_{0}.

<Φ^{p}_{i}|H'|Φ^{p}_{i}> << <Φ^{p}_{i}|H_{0}|Φ^{p}_{i}>
= E_{0}^{p}.

We write H' = λW with λ << 1 and <Φ^{p}_{i}|H_{0}|Φ^{p}_{i}>
on the order of E_{0}^{p}.

We are looking for the eigenvalues E(λ) and the eigenstates |ψ(λ)> of H(λ) = H_{0}
+ λW.

H(λ)|ψ_{p}> = E^{p}(λ)|ψ_{p}> or, keeping the dependence
on λ in mind without specifically writing it down, H|ψ_{p}> = E^{p}|ψ_{p}>.

Since λW is small, we assume that E and |ψ> can be expanded as a power series in
λ.

E^{p} = E_{0}^{p} + λE_{1}^{p} + λ^{2}E_{2}^{p}
+ ...
,

|ψ_{p}> = |ψ_{p}^{0}> + λ|ψ_{p}^{1}> + λ^{2}|ψ_{p}^{2}>
+ ... .

We may then write

(H_{0} + λW)(|ψ_{p}^{0}> + λ|ψ_{p}^{1}>
+ λ^{2}|ψ_{p}^{2}> + ...)

= (E_{0}^{p} + λE_{1}^{p} + λ^{2}E_{2}^{p}
+ ...)(|ψ_{p}^{0}> + λ|ψ_{p}^{1}> + λ^{2}|ψ_{p}^{2}>
+ ...).

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.

- (H
_{0}- E_{0}^{p})|ψ_{p}^{0}> = 0 - (H
_{0}- E_{0}^{p})|ψ_{p}^{1}> = (E_{1}^{p}- W)|ψ_{p}^{0}> - (H
_{0}- E_{0}^{p})|ψ_{p}^{2}> = (E_{1}^{p}- W)|ψ_{p}^{1}> + E_{2}^{p}|ψ_{p}^{0}> - (H
_{0}- E_{0}^{p})|ψ_{p}^{3}> = (E_{1}^{p}- W)|ψ_{p}^{2}> + E_{2}^{p}|ψ_{p}^{1}> + E_{3}^{p}|ψ_{p}^{0}> - ...

(H_{0} - E_{0}^{p})|ψ_{p}^{0}> = 0
implies that |ψ_{p}^{0}> is a linear combination of unperturbed
eigenfuctionns |Φ^{p}_{i}> with the corresponding eigenvalue E_{0}.
We choose <ψ_{p}^{0}|ψ_{p}^{0}> = 1. |ψ_{p}^{s}>is
not uniquely defined. We can add an arbitrary multiple of |ψ_{p}^{0}> to
each |ψ_{p}^{s}> without affecting the left hand side of the
above equations. Most often this multiple is chosen so that <ψ_{p}^{0}|ψ_{p}^{s}>
= 0. The perturbed ket is then **not** normalized.

We then have

<ψ_{p}^{0}|(H_{0} - E_{0}^{p})|ψ_{p}^{s}>
= <ψ_{p}^{0}|(E_{1}^{p} - W)|ψ_{p}^{s-1}>
+ <ψ_{p}^{0}|E_{2}^{p}|ψ_{p}^{s-2}>
+ ... + <ψ_{p}^{0}|E_{s}^{p}|ψ_{p}^{0}>,

or

0 = -<ψ_{p}^{0}|W|ψ_{p}^{s-1}> + E_{s}^{p},

or

E_{s}^{p} = <ψ_{p}^{0}|W|ψ_{p}^{s-1}>.

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

**First-order energy corrections**

Consider a particular non-degenerate eigenvalue E_{0}^{n} of
H_{0}. H_{0}|Φ_{n}> = E_{0}^{n}|Φ_{n}>.
The other eigenvalues of H may or may not be degenerate. We have |ψ_{n}^{0}>
= |Φ_{n}> and E_{1}^{n} = <Φ_{n}|W|Φ_{n}>.

The first-order energy correction therefore is λE_{1}^{n} = <Φ_{n}|H'|Φ_{n}>.

We have E^{n} = E_{0}^{n} + <Φ_{n}|H'|Φ_{n}>
+ O(λ^{2}).

**First-order eigenvector corrections**

(H_{0} - E_{0}^{p})|ψ_{p}^{1}> = (E_{1}^{p}
- W)|ψ_{p}^{0}> = (E_{1}^{p} - W)|Φ_{p}>.
|ψ_{p}^{0}> is an eigenstate of the unperturbed Hamiltonian. We
may expand

|ψ_{p}^{1}> = ∑_{p'≠p,i }b_{p'}^{i}|Φ_{p'}^{i}>
in terms of the basis vectors |Φ_{p'}^{i}>.

In the expansion b_{p}
= 0 because <ψ_{p}^{0}|ψ_{p}^{i}> = 0.

∑_{p'≠p,i}(H_{0} - E_{0}^{p})_{ }b_{p'}^{i}|Φ_{p'}^{i}> = ∑_{p'≠p,i}(E_{0}^{p'} - E_{0}^{p})_{ }b_{p'}^{i}|Φ_{p'}^{i}>
= (E_{1}^{p} - W)|Φ_{p}>.

Multiply from the left by <Φ_{p''}^{i}| we obtain

(E_{0}^{p'} - E_{0}^{p})_{ }b_{p''}^{i}
= -<Φ_{p''}^{i}|W|Φ_{p}>,

or

b_{p'}^{i} = <Φ_{p'}^{i}|W|Φ_{p}>/(E_{0}^{p
}- E_{0}^{p'}), p' ≠ p.

Therefore

|ψ_{p}> = |Φ_{p}> + ∑_{p'≠p,i }<Φ_{p'}^{i}|H'|Φ_{p}>/(E_{0}^{p
}- E_{0}^{p'})]|Φ_{p'}^{i}> + O(λ^{2}).

**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.

E_{2}^{p} = <ψ_{p}^{0}|W|ψ_{p}^{1}>
= ∑_{p'≠p,i }b_{p'}^{i}<Φ_{p}|W|Φ_{p'}^{i}>
= ∑_{p'≠p,i }<Φ_{p'}^{i}|W|Φ_{p}><Φ_{p}|W|Φ_{p'}^{i}>/(E_{0}^{p
}- E_{0}^{p'}).

E_{2}^{p} = ∑_{p'≠p,i }|<Φ_{p'}^{i}|W|Φ_{p}>|^{2}/(E_{0}^{p
}- E_{0}^{p'}).

E^{p} =
E_{0}^{p} + <Φ_{n}|H'|Φ_{n}> + ∑_{p'≠p,i }|<Φ_{p'}^{i}|W|Φ_{p}>|^{2}/(E_{0}^{p
}- E_{0}^{p'}) + O(λ^{3})

= E_{0}^{p} + λE_{1}^{p} + λ^{2}E_{2}^{p}
+ ... .

H = H_{0 }+ H' = H_{0 }+ λW. In practice, after having derived
the perturbation expansion, we often set λ = 1 and let H' = W be small.

**First-order energy corrections**

Let H = H_{0} + H' = H_{0 }+ λW. Let {|Φ^{p}_{i}>}
denote an orthonormal eigenbasis of H_{0}, H_{0}|Φ_{p}^{i}>
= E_{0}^{p}|Φ_{p}^{i}>. Here i denotes the
degeneracy. Consider a particular degenerate eigenvalue E_{0}^{p}
of H_{0}. Assume that this eigenvalue is g-fold degenerate, i =
1,2,...,g. To find E_{1}^{p} we use

(H_{0} - E_{0}^{p})|ψ_{p}^{1}>
= (E_{1}^{p} - W)|ψ_{p}^{0}>, |ψ_{p}^{0}> = ∑_{i}a_{p}^{i}|Φ_{p}^{i}
>.

Multiplying from the left by <Φ_{p}^{j}| we obtain

0 = ∑_{i}a_{p}^{i}<Φ_{p}^{j}|(E_{1}^{p}
- W)|Φ_{p}^{i}> = E_{1}^{p}a_{p}^{j}
- ∑_{i}a_{p}^{i}<Φ_{p}^{j}|W|Φ_{p}^{i}
>,

or

∑_{i}<Φ_{p}^{j}|W
- E_{1}^{p}δ_{ij}|Φ_{p}^{i}>a_{p}^{i} =
0.

This is an eigenvalue equation for the operator W in the subspace E(0,p) of
vectors with the eigenvalue of H_{0} equal to E_{0}^{p}.
We find g eigenvalues E_{1}^{p,i} which may or may not be
degenerate. We then find the corresponding g eigenvectors. To solve for the
eigenvalues we set det(W - E_{1}^{p}) = 0 in the subspace
E(0,p). If we find a non-degenerate eigenvalue E_{1}^{p,i},
then the corresponding eigenvector is uniquely defined. If the eigenvalue E_{1}^{p,i}
is degenerate then the corresponding eigenvector is still not uniquely defined.
The degeneracy may or may not be removed in higher order.