---
title: "Amplitude rate equations"
sort_title: "Amplitude rate equations"
date: 2023-01-03
categories:
- Physics
- Quantum mechanics
layout: "concept"
---

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](/know/concept/time-dependent-perturbation-theory/),
by themselves they are exact and widely applicable.

Let $$\hat{H}_0$$ be the time-independent part of the total Hamiltonian,
and $$\hat{H}_1$$ the time-varying part
(whose contribution need not be small),
so $$\hat{H}(t) = \hat{H}_0 + \hat{H}_1(t)$$.

Suppose 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](/know/concept/rabi-oscillation/)
(by making the [rotating wave approximation](/know/concept/rotating-wave-approximation/)).



## References
1.  D.J. Griffiths, D.F. Schroeter,
    *Introduction to quantum mechanics*, 3rd edition,
    Cambridge.