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
∫_x1^x2 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 ∫_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^-1/2exp(±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 {|Φ_pⁱ>} be an orthonormal eigenbasis of H₀, H₀|Φ_pⁱ> = E₀^p|Φ_pⁱ>.
Here i denotes the degeneracy. 
Let H|ψ_p> = E^p|ψ_p>. 
Expand E^p and |ψ_p>.
E^p = E₀^p + E₁^p + E₂^p + ... ,
|ψ_p> = |ψ_p⁰> + |ψ_p¹> + |ψ_p²> +... .
Stationary perturbation theory yields (H₀ - E₀^p)|ψ_p⁰> = 0, |ψ_p⁰> = ∑_ia_pⁱ|Φ_pⁱ >.
If |Φ_p> is not degenerate then |ψ_p⁰> = |Φ_p>.

E₁^p and a_pⁱ are found from ∑_i<Φ_p^j|W - E₁^pδ_ij|Φ_pⁱ>a_pⁱ = 0.
We have to diagonalize the matrix of W in the subspace spanned by the degenerate states {|Φ_pⁱ>}. 

If E₀^p is not degenerate then  E₁^p = <Φ_p|W|Φ_p>.
This is the first order energy correction for non-degenerate states.

If E₀^p is not degenerate then 
|ψ_p¹> = ∑_p'≠p,i b_p'ⁱ|Φ_p'ⁱ> , where b_p'ⁱ = <Φ_p'ⁱ|W|Φ_p>/(E₀^p - E₀^p').
This is the first order correction to a non-degenerate eigenvector. 

The second order energy correction for non-degenerate states is 
E₂^p = ∑_p'≠p,i |<Φ_p'ⁱ|W|Φ_p>|²/(E₀^p - E₀^p').

Link:  Derivation

The variational method

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 E₀ 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|²E_n /∑_n|a_n|² ≥ 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.  The variational method yields an upper limit for the ground state energy.