Amplitude rate equations
In quantum mechanics, the amplitude rate equations give
the evolution of a quantum state in a time-varying potential.
Although best known as the precursors of
time-dependent perturbation theory,
by themselves they are exact and widely applicable.
Let H^0 be the time-independent part of the total Hamiltonian,
and H^1 the time-varying part
(whose contribution need not be small),
Suppose that the time-independent problem
H^0∣n⟩=En∣n⟩ has already been solved,
such that its general solution is a superposition as follows:
Since these ∣n⟩ form a complete basis,
the full solution for H^0+H^1 can be written in the same form,
but now with time-dependent coefficients cn(t):
We put this ansatz into the full Schrödinger equation,
and use the known solution for H^0:
We then take the inner product with an arbitrary stationary basis state ∣m⟩:
Thanks to orthonormality, this moves the latter term outside the summation:
We divide by the left-hand exponential and define
ωmn≡(Em−En)/ℏ to arrive at
the desired set of amplitude rate equations,
one for each basis state ∣m⟩:
We have not made any approximations,
so it is possible to exactly solve for cn(t) in some simple systems.
This is worth pointing out, because these equations’ most famous uses
are for deriving time-dependent-perturbation theory
(by making a truncated power series approximation)
and Rabi oscillation
(by making the rotating wave approximation).
- D.J. Griffiths, D.F. Schroeter,
Introduction to quantum mechanics, 3rd edition,