Consider a one-dimensional problem. We want to find the energies of the
bound states of a particle in a given potential well with U(x) finite at every
finite x.
The WKB approximation requires that

∫_{x1}^{x2} pdx = (n -
½)(h/2) or
∫_{cylce}pdx = ∫_{cylce}ħkdx
= (n - ½)h.

Here
n = 1, 2, ..., ∫_{cylce} denotes an
integral over one complete cycle of the classical motion, x_{1} and x_{2}
are the classical turning points, and k^{2} =
(2m/ħ^{2})(E - U(x)).

The WKB method for
bound states leads to the Wilson-Sommerfeld quantization rule except
that n is replaced by (n - ½). It leads to a quantization of the classical
action J = ∫_{cylce}pdq.

If we let n = 1, 2, ..., then n counts the number of anti-nodes of the wave
function in the well.

(We may also let n = 0, 1, 2, 3, ... , and write ∫_{x1}^{x2 }k dx
= (n + ½)π. Then n counts the number of zeros of the wave
function in the well.)

If our potential well has one or two vertical walls, the results of the WKB
approximation differ only in the number that is subtracted from n = 1, 2, 3, ...
, respectively.

For one vertical wall we have ∫_{x1}^{x2 }k dx = (n - ¼)π,

and for two vertical walls we have ∫_{x1}^{x2 }k dx = nπ.

Since the WKB approximation works best for large n in the
semi-classical regime, this distinction is more in appearance than in substance.

In regions where E > U(x) we have Φ(x) = Ak^{-1/2}exp(±i∫^{x}
k(x')dx')

and in regions where E < U(x) we have
Φ(x) = Aρ^{-1/2}exp(±∫^{x}
ρ(x')dx').

Let H = H_{0 }+ W.

Let {|Φ_{p}^{i}>} be an
orthonormal eigenbasis of H_{0}, H_{0}|Φ_{p}^{i}> = E_{0}^{p}|Φ_{p}^{i}>.

Here i denotes the degeneracy.

Let H|ψ_{p}> = E^{p}|ψ_{p}>.

Expand E^{p} and |ψ_{p}>.

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

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

Stationary perturbation theory yields (H_{0} - E_{0}^{p})|ψ_{p}^{0}>
= 0, |ψ_{p}^{0}> = ∑_{i}a_{p}^{i}|Φ_{p}^{i}
>.

If |Φ_{p}> is not degenerate then |ψ_{p}^{0}>
= |Φ_{p}>.

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

We have to diagonalize the matrix of W in the subspace spanned by the degenerate
states
{|Φ_{p}^{i}>}.

If E_{0}^{p}
is not degenerate then E_{1}^{p} = <Φ_{p}|W|Φ_{p}>.

This is the **first order energy correction for non-degenerate states**.

If E_{0}^{p} is not degenerate then

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

This is the **first order correction to a non-degenerate eigenvector**.

The **second order energy correction for non-degenerate states** is

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

Link: Derivation

Consider an arbitrary physical system with a time-independent Hamiltonian.
Let {E_{n}} be its eigenvalues and {|Φ_{n}>}
its eigenstates.

H|Φ_{n}> = E_{n}|Φ_{n}>.

Let E_{0} denote the energy of the ground state.

If |ψ>
is an arbitrary ket in the state space of the system, |ψ> = ∑_{n }a_{n}|Φ_{n}>.

Then <H> = <ψ|H|ψ>/<ψ|ψ> = ∑_{n}|a_{n}|^{2}E_{n}
/∑_{n}|a_{n}|^{2} ≥ E_{0}.

This is the basis for the **variational
method**. Choose a family of kets |ψ(α)>
which satisfy the boundary conditions and which depend on a number of free
parameters α. Calculate the mean value <H>(α). Minimize <H>(α)
with respect to α. The minimal value thus
obtained constitutes an approximation to the ground state energy E_{0}
of the system. The variational method yields an upper limit for the ground
state energy.