##

General Time Dependent Perturbations

Assume that we solve the unperturbed energy eigenvalue problem exactly:
.
Now we add a perturbation that depends on time,
.
Our problem is now inherently time dependent so we go back to the
**time dependent Schrödinger equation**.

We will
**expand**
in terms of the eigenfunctions:
with
.
The time dependent Schrödinger equations is

Now dot
into this equation to get the time dependence of one coefficient.

Assume that at
, we are in an **initial state**
and hence all the other
are equal to zero:
.

Now we want to calculate transition rates.
To first order, all the
are small compared to
, so the sum can be neglected.

This is the **equation to use to compute transition probabilities for a general
time dependent perturbation**.
We will also use it as a basis to compute transition rates for the specific problem of harmonic potentials.
Again we are assuming
is small enough that
has not changed much.
This is not a limitation. We can deal with the decrease of the population of the initial state later.
Note that, if there is a large energy difference between the initial and final states,
a slowly varying perturbation can average to zero.
We will find that the perturbation will need frequency components compatible with
to cause transitions.

If the first order term is zero or higher accuracy is required, the second order term can be computed.
In second order, a transition can be made to an intermediate state
, then a transition to
.
We just put the first order
into the sum.

* Example:
Transitions of a 1D harmonic oscillator in a transient E field.*

Jim Branson
2013-04-22