Categories: Physics, Quantum mechanics.

In quantum mechanics, the **amplitude rate equations** give
the evolution of a quantum state’s superposition coefficients through time.
They are known as the precursors for
time-dependent perturbation theory,
but by themselves they are exact and widely applicable.

Let $\hat{H}_0$ be a “simple” time-independent part of the full Hamiltonian, and $\hat{H}_1$ a time-varying other part, whose contribution need not be small:

$\begin{aligned} \hat{H}(t) = \hat{H}_0 + \hat{H}_1(t) \end{aligned}$We assume that the time-independent problem $\hat{H}_0 \Ket{n} = E_n \Ket{n}$ has already been solved, such that its general solution is a superposition as follows:

$\begin{aligned} \Ket{\Psi_0(t)} = \sum_{n} c_n \Ket{n} e^{- i E_n t / \hbar} \end{aligned}$Since these $\Ket{n}$ form a complete basis, the full solution for $\hat{H}_0 + \hat{H}_1$ can be written in the same form, but now with time-dependent coefficients $c_n(t)$:

$\begin{aligned} \Ket{\Psi(t)} = \sum_{n} c_n(t) \Ket{n} e^{- i E_n t / \hbar} \end{aligned}$We put this ansatz into the full Schrödinger equation, and use the known solution for $\hat{H}_0$:

$\begin{aligned} 0 &= \hat{H}_0 \Ket{\Psi(t)} + \hat{H}_1 \Ket{\Psi(t)} - i \hbar \dv{}{t}\Ket{\Psi(t)} \\ &= \sum_{n} \Big( c_n \hat{H}_0 \Ket{n} + c_n \hat{H}_1 \Ket{n} - c_n E_n \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) e^{- i E_n t / \hbar} \\ &= \sum_{n} \Big( c_n \hat{H}_1 \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) e^{- i E_n t / \hbar} \end{aligned}$We then take the inner product with an arbitrary stationary basis state $\Ket{m}$:

$\begin{aligned} 0 &= \sum_{n} \Big( c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \inprod{m}{n} \Big) e^{- i E_n t / \hbar} \end{aligned}$Thanks to orthonormality, this moves the latter term outside the summation:

$\begin{aligned} i \hbar \dv{c_m}{t} e^{- i E_m t / \hbar} &= \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} e^{- i E_n t / \hbar} \end{aligned}$We divide by the left-hand exponential and define $\omega_{mn} \equiv (E_m - E_n) / \hbar$ to arrive at the desired set of amplitude rate equations, one for each basis state $\ket{m}$:

$\begin{aligned} \boxed{ i \hbar \dv{c_m}{t} = \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} e^{i \omega_{mn} t} } \end{aligned}$We have not made any approximations, so it is possible to exactly solve for $c_n(t)$ 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, Cambridge.

© 2023 Marcus R.A. Newman,
CC BY-NC-SA 4.0.