**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.

**Maths**

Let the ground-state have quantum number n=1 and actual eigenfunction/value ψ

_{1}, E

_{1}.

We have: E

_{1}= <ψ

_{1}Hψ

_{1}>/<ψ

_{1}ψ

_{1}>

(the denominator to make sure the equation is correctly normalised).

Since we don't know ψ

_{1}, we approximate it by φ

_{1}, giving

E'

_{1}= <φ

_{1}Hφ

_{1}>/<φ

_{1}φ

_{1}>.

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}= <φ

_{1}Hφ

_{1}>/<φ

_{1}φ

_{1}> - 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.

**Maths**

Note that E'

_{n}= <ψ'

_{n}H'ψ'

_{n}> where ψ' is an eigenfunction of H'.

Let E'

_{n}= approx <ψ

_{n}H'ψ

_{n}> where ψ is an eigenfunction of H,

= <ψ

_{n}(H + δH)ψ

_{n}>

= E

_{n}+ <ψ

_{n}δHψ

_{n}>.

We can work out E

_{n}which is just the eigenvalue of the unperturbed Hamiltonian H. The expected value <ψ

_{n}δHψ

_{n}> 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}.