## Time-independent approximations

### The WKB approximation for bound states (1D)

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
x1x2 pdx = (n - ½)(h/2) or ∫cylcepdx = ∫cylceħkdx = (n - ½)h.
Here n = 1, 2, ..., ∫cylce denotes an integral over one complete cycle of the classical motion, x1 and x2 are the classical turning points, and k2 = (2m/ħ2)(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 = ∫cylcepdq.
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 ∫x1x2 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 ∫x1x2 k dx =  (n - ¼)π,
and for two vertical walls we have ∫x1x2 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-1/2exp(±i∫x k(x')dx')
and in regions where E < U(x) we have Φ(x) = Aρ-1/2exp(±∫x ρ(x')dx').

### Stationary perturbation theory

Let H = H0 + W.
Let {|Φpi>} be an orthonormal eigenbasis of H0, H0pi> = E0ppi>.
Here i denotes the degeneracy.
Let H|ψp> = Epp>.
Expand Ep and |ψp>.
Ep = E0p + E1p + E2p + ... ,
p> = |ψp0> + |ψp1> + |ψp2> +... .
Stationary perturbation theory yields (H0 - E0p)|ψp0> = 0, |ψp0> = ∑iapipi >.
If |Φp> is not degenerate then |ψp0> = |Φp>.

E1p and api are found from ∑ipj|W - E1pδijpi>api = 0.
We have to diagonalize the matrix of W in the subspace spanned by the degenerate states {|Φpi>}.

If E0p is not degenerate then  E1p = <Φp|W|Φp>.
This is the first order energy correction for non-degenerate states.

If E0p is not degenerate then
p1> = ∑p'≠p,i bp'ip'i> , where bp'i = <Φp'i|W|Φp>/(E0p - E0p').
This is the first order correction to a non-degenerate eigenvector

The second order energy correction for non-degenerate states is
E2p = ∑p'≠p,i |<Φp'i|W|Φp>|2/(E0p - E0p').