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 everywhere.  
The WKB approximation requires that
x1x2 pdx = (n + ½)(h/2) or ∫cylcepdx = ∫cylceħkdx = (n + ½)h.  
Here ∫cylce denotes an integral over one complete cycle of the classical motion and 
k2 = (2m/ħ2)(E - U(x)). 
The WKB method for bound states therefore 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 = 0, 1, 2, ..., then n counts the number of zero of the wave function in the well.  We may also let n = 1, 2, 3, ... , and write ∫x1x2 k dx =  (n - ½)π.  Then n counts the number of anti-nodes 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').


The variational method

Consider an arbitrary physical system with a time-independent Hamiltonian.  Let {En} be its eigenvalues and {|Φn>} its eigenstates.
H|Φn> = Enn>. 
Let E0 denote the energy of the ground state. 
If |ψ> is an arbitrary ket in the state space of the system, then <H> = <ψ|H|ψ>/<ψ|ψ> ≥ E0
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 E0 of the system.