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 H^0 be the base time-independent Hamiltonian,
and H^1 be a time-varying perturbation,
with “bookkeeping” parameter λ:
We assume that the unperturbed time-independent problem
H^0∣n⟩=En∣n⟩ has already been solved,
such that the general solution for the full H^ can be written as:
And so forth. The pattern here is clear: we can calculate the (j+1)th
correction using only our previous result for the jth correction.
The only purpose of λ was to help us collect its orders;
in the end we simply set λ=1 or absorb it into H^1.
Now we have the essence of time-dependent perturbation theory,
we cannot go any further without considering a specific H^1.
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, H^1 has the following form:
For simplicity, we let the system start in a known state ∣a⟩,
such that cn(0)=δna,
and we assume that the driving frequency is close to resonance ω≈ωma,
such that the second term dominates the first, which can then be neglected.
We thus get:
The result would be the same if H^1≡Vcos(ωt).
However, if instead H^1≡Vexp(−iωt),
the result is larger by a factor of 4,
which can cause confusion when comparing literature.
In any case, the probability oscillates as a function of t
with period T=2π/(ωma−ω),
so after one period the particle is back in ∣a⟩,
and after T/2 the particle is in ∣b⟩.
See Rabi oscillation
for a more accurate treatment of this “flopping” behaviour.
However, when regarded as a function of ω,
the probability takes the form of
a sinc-function centred around (ωma−ω),
so it is highest for transitions with energy ℏω=Em−Ea.
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.