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

---
 .../time-dependent-perturbation-theory/index.md    | 73 ++++++----------------
 1 file changed, 20 insertions(+), 53 deletions(-)

(limited to 'source/know/concept/time-dependent-perturbation-theory/index.md')

diff --git a/source/know/concept/time-dependent-perturbation-theory/index.md b/source/know/concept/time-dependent-perturbation-theory/index.md
index d39e321..b4b35e1 100644
--- a/source/know/concept/time-dependent-perturbation-theory/index.md
+++ b/source/know/concept/time-dependent-perturbation-theory/index.md
@@ -14,84 +14,47 @@ with time-varying perturbations to the Schrödinger equation.
 This is in contrast to [time-independent perturbation theory](/know/concept/time-independent-perturbation-theory/),
 where the perturbation is stationary.
 
-Let $$\hat{H}_0$$ be the base time-independent
-Hamiltonian, and $$\hat{H}_1$$ be a time-varying perturbation, with
-"bookkeeping" parameter $$\lambda$$:
+Let $$\hat{H}_0$$ be the base time-independent Hamiltonian,
+and $$\hat{H}_1$$ be a time-varying perturbation,
+with "bookkeeping" parameter $$\lambda$$:
 
 $$\begin{aligned}
     \hat{H}(t) = \hat{H}_0 + \lambda \hat{H}_1(t)
 \end{aligned}$$
 
 We assume that the unperturbed time-independent problem
-$$\hat{H}_0 \Ket{n} = E_n \Ket{n}$$ has already been solved, such that the
-full solution is:
-
-$$\begin{aligned}
-    \Ket{\Psi_0(t)} = \sum_{n} c_n \Ket{n} \exp(- i E_n t / \hbar)
-\end{aligned}$$
-
-Since these $$\Ket{n}$$ form a complete basis, the perturbed wave function
-can be written in the same form, but with time-dependent coefficients $$c_n(t)$$:
+$$\hat{H}_0 \Ket{n} = E_n \Ket{n}$$ has already been solved,
+such that the general solution for the full $$\hat{H}$$ can be written as:
 
 $$\begin{aligned}
     \Ket{\Psi(t)} = \sum_{n} c_n(t) \Ket{n} \exp(- i E_n t / \hbar)
 \end{aligned}$$
 
-We insert this ansatz in the time-dependent Schrödinger equation, and
-reduce it using the known unperturbed time-independent problem:
+These time-dependent coefficients are then governed by
+the [amplitude rate equations](/know/concept/amplitude-rate-equations/):
 
 $$\begin{aligned}
-    0
-    &= \hat{H}_0 \Ket{\Psi(t)} + \lambda \hat{H}_1 \Ket{\Psi(t)} - i \hbar \dv{}{t}\Ket{\Psi(t)}
-    \\
-    &= \sum_{n}
-    \Big( c_n \hat{H}_0 \Ket{n} + \lambda c_n \hat{H}_1 \Ket{n} - c_n E_n \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) \exp(- i E_n t / \hbar)
-    \\
-    &= \sum_{n} \Big( \lambda c_n \hat{H}_1 \Ket{n} - i \hbar \dv{c_n}{t} \Ket{n} \Big) \exp(- i E_n t / \hbar)
+    i \hbar \dv{c_m}{t}
+    = \sum_{n} c_n(t) \matrixel{m}{\lambda \hat{H}_1(t)}{n} \exp(i \omega_{mn} t)
 \end{aligned}$$
 
-We then take the inner product with an arbitrary stationary basis state $$\Ket{m}$$:
-
-$$\begin{aligned}
-    0
-    &= \sum_{n} \Big( \lambda c_n \matrixel{m}{\hat{H}_1}{n} - i \hbar \dv{c_n}{t} \Inprod{m}{n} \Big) \exp(- i E_n t / \hbar)
-\end{aligned}$$
-
-Thanks to orthonormality, this removes the latter term from the summation:
-
-$$\begin{aligned}
-    i \hbar \dv{c_m}{t} \exp(- i E_m t / \hbar)
-    &= \lambda \sum_{n} c_n \matrixel{m}{\hat{H}_1}{n} \exp(- i E_n t / \hbar)
-\end{aligned}$$
-
-We divide by the left-hand exponential and define
-$$\omega_{mn} \equiv (E_m - E_n) / \hbar$$ to get:
-
-$$\begin{aligned}
-    \boxed{
-        i \hbar \dv{c_m}{t}
-        = \lambda \sum_{n} c_n(t) \matrixel{m}{\hat{H}_1(t)}{n} \exp(i \omega_{mn} t)
-    }
-\end{aligned}$$
-
-So far, we have not invoked any approximation,
-so we can analytically find $$c_n(t)$$ for some simple systems.
-Furthermore, it is useful to write this equation in integral form instead:
+So far, we have not made any approximations at all.
+We rewrite this in integral form:
 
 $$\begin{aligned}
     c_m(t)
     = c_m(0) - \lambda \frac{i}{\hbar} \sum_{n} \int_0^t c_n(\tau) \matrixel{m}{\hat{H}_1(\tau)}{n} \exp(i \omega_{mn} \tau) \dd{\tau}
 \end{aligned}$$
 
-If this cannot be solved exactly, we must approximate it. We expand
-$$c_m(t)$$ in the usual way, with the initial condition $$c_m^{(j)}(0) = 0$$
-for $$j > 0$$:
+If this cannot be solved exactly, we must approximate it.
+We expand $$c_m(t)$$ as a power series,
+with the initial condition $$c_m^{(j)}(0) = 0$$ for $$j > 0$$:
 
 $$\begin{aligned}
     c_m(t) = c_m^{(0)} + \lambda c_m^{(1)}(t) + \lambda^2 c_m^{(2)}(t) + ...
 \end{aligned}$$
 
-We then insert this into the integral and collect the non-zero orders of $$\lambda$$:
+We then insert this into the integral and collect the nonzero orders of $$\lambda$$:
 
 $$\begin{aligned}
     c_m^{(1)}(t)
@@ -108,7 +71,11 @@ $$\begin{aligned}
 
 And so forth. The pattern here is clear: we can calculate the $$(j\!+\!1)$$th
 correction using only our previous result for the $$j$$th correction.
-We cannot go any further than this without considering a specific perturbation $$\hat{H}_1(t)$$.
+The only purpose of $$\lambda$$ was to help us collect its orders;
+in the end we simply set $$\lambda = 1$$ or absorb it into $$\hat{H}_1$$.
+Now we have the essence of time-dependent perturbation theory,
+we cannot go any further without considering a specific $$\hat{H}_1$$.
+
 
 
 ## Sinusoidal perturbation
-- 
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