1. The Variational Method
Purpose: to calculate the ground energy state (e.g. of an atom) when we don't know the correct eigenfunction.
Method: guess the eigenfunction and compute the eigenvalue (= the ground-state energy). If we guess a function with a free parameter, we may adjust this parameter for fine-tuning.
Let the ground-state have quantum number n=1 and actual eigenfunction/value ψ1, E1.
We have: E1 = <ψ1 Hψ1>/<ψ1ψ1>
(the denominator to make sure the equation is correctly normalised).
Since we don't know ψ1, we approximate it by φ1, giving
E'1 = <φ1 Hφ1>/<φ1 φ1>.
If φ contains a variable b, then E'1 will be a function of b, E'1(b), and we can differentiate to find the value of b (the 'best' eigenfunction φ(b)) which minimises E'1. This is our required approximation.
The only practical issue with this method is the labour involved in evaluating
E'1 = <φ1 Hφ1>/<φ1 φ1> - multiple integrals,
and the need to guess a 'good' eigenfunction which closely approximates ψ. Note that it's much harder to use this method to compute higher energy states, where n > 1.
2. Perturbation Methods.
Purpose: to calculate the energy state E' (e.g. of an atom) where the Hamiltonian H' is too complex to solve directly. (We do know the relevant eigenfunctions for the related unperturbed Hamiltonian H).
Method: Split the Hamiltonian function H' into a simple unperturbed part H, which we can solve, and a first-order 'perturbation' δH which we can also solve. So
H' = H + δH -- (to first order).
Accuracy may be improved by going to second or higher orders.
Note that E'n = <ψ'n H'ψ'n> where ψ' is an eigenfunction of H'.
Let E'n = approx <ψn H'ψn> where ψ is an eigenfunction of H,
= <ψn (H + δH)ψn>
= En + <ψn δHψn>.
We can work out En which is just the eigenvalue of the unperturbed Hamiltonian H. The expected value <ψn δHψn> of the first order perturbation δH, the first-order energy 'correction', is also intended to be easy to work out. So we hopefully have a good approximation to E'n.