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Derivation of 1st Order Degenerate Perturbation Equations

To deal with the problem of degenerate states, we will allow an arbitrary linear combination
of those states at zeroth order.
In the following equation, the sum over
is the sum over all the states degenerate with
and the sum over k runs over all the other states.

where
is the set of zeroth order states which are (nearly) degenerate with
.
We will only go to first order in this derivation and we will use
as in the previous
derivation to keep track of the order in the perturbation.
The full Schrödinger equation is.

If we keep the zeroth and first order terms, we have

Projecting this onto one of the degenerate states
, we get

By putting both terms together, our calculation gives us the full energy to first order, not just the
correction.
It is useful both for degenerate states and for nearly degenerate states.
The result may be simplified to

This is just the standard eigenvalue problem for the full Hamiltonian in the
**subspace of (nearly) degenerate states.**

Jim Branson
2013-04-22