### Derivation of 1st and 2nd Order Perturbation Equations

To keep track of powers of the perturbation in this derivation we will make the substitution where is assumed to be a small parameter in which we are making the series expansion of our energy eigenvalues and eigenstates. It is there to do the book-keeping correctly and can go away at the end of the derivations.

To solve the problem using a perturbation series, we will expand both our energy eigenvalues and eigenstates in powers of .

The full Schrödinger equation is

where the has been factored out on both sides. For this equation to hold as we vary , it must hold for each power of . We will investigate the first three terms.
The zero order term is just the solution to the unperturbed problem so there is no new information there. The other two terms contain linear combinations of the orthonormal functions . This means we can dot the equations into each of the to get information, much like getting the components of a vector individually. Since is treated separately in this analysis, we will dot the equation into and separately into all the other functions .

The first order equation dotted into yields

and dotted into yields

From these it is simple to derive the first order corrections

The second order equation projected on yields

We will not need the projection on but could proceed with it to get the second order correction to the wave function, if that were needed. Solving for the second order correction to the energy and substituting for , we have

The normalization factor played no role in the solutions to the Schrödinger equation since that equation is independent of normalization. We do need to go back and check whether the first order corrected wavefunction needs normalization.

The correction is of order and can be neglected at this level of approximation.

These results are rewritten with all the removed in section 22.1.

Jim Branson 2013-04-22