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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