Time-dependent perturbation theory
In quantum mechanics, time-dependent perturbation theory exists to deal
with time-varying perturbations to the Schrödinger equation.
This is in contrast to time-independent perturbation theory,
where the perturbation is stationary.
Let be the base time-independent Hamiltonian,
and be a time-varying perturbation,
with “bookkeeping” parameter :
We assume that the unperturbed time-independent problem
has already been solved,
such that the general solution for the full can be written as:
These time-dependent coefficients are then governed by
the amplitude rate equations:
So far, we have not made any approximations at all.
We rewrite this in integral form:
If this cannot be solved exactly, we must approximate it.
We expand as a power series,
with the initial condition for :
We then insert this into the integral and collect the nonzero orders of :
And so forth. The pattern here is clear: we can calculate the th
correction using only our previous result for the th correction.
The only purpose of was to help us collect its orders;
in the end we simply set or absorb it into .
Now we have the essence of time-dependent perturbation theory,
we cannot go any further without considering a specific .
Arguably the most important perturbation
is a sinusoidally-varying potential, which represents
e.g. incoming electromagnetic waves,
or an AC voltage being applied to the system.
In this case, has the following form:
We abbreviate ,
and take the first-order correction formula:
For simplicity, we let the system start in a known state ,
such that ,
and we assume that the driving frequency is close to resonance ,
such that the second term dominates the first, which can then be neglected.
We thus get:
Taking the norm squared yields the transition probability:
the probability that a particle that started in state
will be found in at time :
The result would be the same if .
However, if instead ,
the result is larger by a factor of ,
which can cause confusion when comparing literature.
In any case, the probability oscillates as a function of
with period ,
so after one period the particle is back in ,
and after the particle is in .
See Rabi oscillation
for a more accurate treatment of this “flopping” behaviour.
However, when regarded as a function of ,
the probability takes the form of
a sinc-function centred around ,
so it is highest for transitions with energy .
Also note that the sinc-distribution becomes narrower over time,
which roughly means that it takes some time
for the system to “notice” that
it is being driven periodically.
In other words, there is some “inertia” to it.
- D.J. Griffiths, D.F. Schroeter,
Introduction to quantum mechanics, 3rd edition,
- R. Shankar,
Principles of quantum mechanics, 2nd edition,