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diff --git a/source/know/concept/amplitude-rate-equations/index.md b/source/know/concept/amplitude-rate-equations/index.md new file mode 100644 index 0000000..0ca3248 --- /dev/null +++ b/source/know/concept/amplitude-rate-equations/index.md @@ -0,0 +1,94 @@ +--- +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's superposition coefficients through time. +They are known as the precursors for +[time-dependent perturbation theory](/know/concept/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](/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. |