when calculate the energy from Hamiltonian. usually, we only know the exact form from a simple Hamiltonian but not the exact form of the Hamiltonian. Thus, we have to apply a perturbation to find the approximate energy.

the formula is :

if

is the exactly solvable Hamiltonian, energy and eigenstate for the n-h state, then the REAL energy for the total Hamiltonian

is

where

so far as I encounter, the exact state is not quite important.

Detail walk through.

Set the perturbation strength as

,

where is the switch for the perturbation.

Suppose the solution for n-th level is

,

with energy

.

Put in the equation,

expand and collect

we can see, for , this is the original, unperturbed solution.

For , this is

To get , we apply from the left, the first terms of left and right side will cancel out.

For higher order, one can calculate using the lower orders. for example,

The state can be constructed using eigen states,

put in the equation, using orthogonal relation,

For Coulomb Potential, here is some common expected values:

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