From aeacfca5aea5df7c107cf0c12e72ab5d496c96e1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 3 Jan 2023 19:48:17 +0100
Subject: More improvements to knowledge base

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+---
+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.
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