From 549b1e963183b6284aefbc52b9167009fefa551d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 20 Feb 2021 18:08:52 +0100
Subject: Fix "Time-independent perturbation theory"

---
 .../time-independent-perturbation-theory/source.md | 86 +++++++++++-----------
 1 file changed, 42 insertions(+), 44 deletions(-)

(limited to 'latex/know')

diff --git a/latex/know/concept/time-independent-perturbation-theory/source.md b/latex/know/concept/time-independent-perturbation-theory/source.md
index d076457..a3167cd 100644
--- a/latex/know/concept/time-independent-perturbation-theory/source.md
+++ b/latex/know/concept/time-independent-perturbation-theory/source.md
@@ -16,31 +16,31 @@ $$\begin{aligned}
 Where $\hat{H}_0$ is a Hamiltonian for which the time-independent
 Schrödinger equation has a known solution, and $\hat{H}_1$ is a small
 perturbing Hamiltonian. The eigenenergies $E_n$ and eigenstates
-$\ket{\psi_n}$ of the composite problem are expanded accordingly in the
+$\ket{\psi_n}$ of the composite problem are expanded in the
 perturbation "bookkeeping" parameter $\lambda$:
 
 $$\begin{aligned}
     \ket{\psi_n}
-    &= \ket{\psi_n^{(0)}} + \lambda \ket{\psi_n^{(1)}} + \lambda^2 \ket{\psi_n^{(2)}} + ...
+    &= \ket*{\psi_n^{(0)}} + \lambda \ket*{\psi_n^{(1)}} + \lambda^2 \ket*{\psi_n^{(2)}} + ...
     \\
     E_n
     &= E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + ...
 \end{aligned}$$
 
-Where $E_n^{(1)}$ and $\ket{\psi_n^{(1)}}$ are called the *first-order
+Where $E_n^{(1)}$ and $\ket*{\psi_n^{(1)}}$ are called the *first-order
 corrections*, and so on for higher orders. We insert this into the
 Schrödinger equation:
 
 $$\begin{aligned}
     \hat{H} \ket{\psi_n}
-    &= \hat{H}_0 \ket{\psi_n^{(0)}}
-    + \lambda \big( \hat{H}_1 \ket{\psi_n^{(0)}} + \hat{H}_0 \ket{\psi_n^{(1)}} \big) \\
-    &\qquad + \lambda^2 \big( \hat{H}_1 \ket{\psi_n^{(1)}} + \hat{H}_0 \ket{\psi_n^{(2)}} \big) + ...
+    &= \hat{H}_0 \ket*{\psi_n^{(0)}}
+    + \lambda \big( \hat{H}_1 \ket*{\psi_n^{(0)}} + \hat{H}_0 \ket*{\psi_n^{(1)}} \big) \\
+    &\qquad + \lambda^2 \big( \hat{H}_1 \ket*{\psi_n^{(1)}} + \hat{H}_0 \ket*{\psi_n^{(2)}} \big) + ...
     \\
     E_n \ket{\psi_n}
-    &= E_n^{(0)} \ket{\psi_n^{(0)}}
-    + \lambda \big( E_n^{(1)} \ket{\psi_n^{(0)}} + E_n^{(0)} \ket{\psi_n^{(1)}} \big) \\
-    &\qquad + \lambda^2 \big( E_n^{(2)} \ket{\psi_n^{(0)}} + E_n^{(1)} \ket{\psi_n^{(1)}} + E_n^{(0)} \ket{\psi_n^{(2)}} \big) + ...
+    &= E_n^{(0)} \ket*{\psi_n^{(0)}}
+    + \lambda \big( E_n^{(1)} \ket*{\psi_n^{(0)}} + E_n^{(0)} \ket*{\psi_n^{(1)}} \big) \\
+    &\qquad + \lambda^2 \big( E_n^{(2)} \ket*{\psi_n^{(0)}} + E_n^{(1)} \ket*{\psi_n^{(1)}} + E_n^{(0)} \ket*{\psi_n^{(2)}} \big) + ...
 \end{aligned}$$
 
 If we collect the terms according to the order of $\lambda$, we arrive
@@ -48,14 +48,14 @@ at the following endless series of equations, of which in practice only
 the first three are typically used:
 
 $$\begin{aligned}
-    \hat{H}_0 \ket{\psi_n^{(0)}}
-    &= E_n^{(0)} \ket{\psi_n^{(0)}}
+    \hat{H}_0 \ket*{\psi_n^{(0)}}
+    &= E_n^{(0)} \ket*{\psi_n^{(0)}}
     \\
-    \hat{H}_1 \ket{\psi_n^{(0)}} + \hat{H}_0 \ket{\psi_n^{(1)}}
-    &= E_n^{(1)} \ket{\psi_n^{(0)}} + E_n^{(0)} \ket{\psi_n^{(1)}}
+    \hat{H}_1 \ket*{\psi_n^{(0)}} + \hat{H}_0 \ket*{\psi_n^{(1)}}
+    &= E_n^{(1)} \ket*{\psi_n^{(0)}} + E_n^{(0)} \ket*{\psi_n^{(1)}}
     \\
-    \hat{H}_1 \ket{\psi_n^{(1)}} + \hat{H}_0 \ket{\psi_n^{(2)}}
-    &= E_n^{(2)} \ket{\psi_n^{(0)}} + E_n^{(1)} \ket{\psi_n^{(1)}} + E_n^{(0)} \ket{\psi_n^{(2)}}
+    \hat{H}_1 \ket*{\psi_n^{(1)}} + \hat{H}_0 \ket*{\psi_n^{(2)}}
+    &= E_n^{(2)} \ket*{\psi_n^{(0)}} + E_n^{(1)} \ket*{\psi_n^{(1)}} + E_n^{(0)} \ket*{\psi_n^{(2)}}
     \\
     ...
     &= ...
@@ -63,7 +63,7 @@ $$\begin{aligned}
 
 The first equation is the unperturbed problem, which we assume has
 already been solved, with eigenvalues $E_n^{(0)} = \varepsilon_n$ and
-eigenvectors $\ket{\psi_n^{(0)}} = \ket{n}$:
+eigenvectors $\ket*{\psi_n^{(0)}} = \ket{n}$:
 
 $$\begin{aligned}
     \hat{H}_0 \ket{n} = \varepsilon_n \ket{n}
@@ -79,28 +79,27 @@ $\varepsilon_n$ corresponds to one $\ket{n}$. At order $\lambda^1$, we
 rewrite the equation as follows:
 
 $$\begin{aligned}
-    (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket{\psi_n^{(1)}} = 0
+    (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket*{\psi_n^{(1)}} = 0
 \end{aligned}$$
 
 Since $\ket{n}$ form a complete basis, we can express
-$\ket{\psi_n^{(1)}}$ in terms of them:
+$\ket*{\psi_n^{(1)}}$ in terms of them:
 
 $$\begin{aligned}
-    \ket{\psi_n^{(1)}} = \sum_{m \neq n} c_m \ket{m}
+    \ket*{\psi_n^{(1)}} = \sum_{m \neq n} c_m \ket{m}
 \end{aligned}$$
 
 Importantly, $n$ has been removed from the summation to prevent dividing
-by zero later. This is allowed, because
-$\ket{\psi_n^{(1)}} - c_n \ket{n}$ also satisfies the $\lambda^1$-order
+by zero later. We are allowed to do this, because
+$\ket*{\psi_n^{(1)}} - c_n \ket{n}$ also satisfies the order-$\lambda^1$
 equation for any value of $c_n$, as demonstrated here:
 
 $$\begin{aligned}
-    (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket{\psi_n^{(1)}} - (\varepsilon_n - \varepsilon_n) c_n \ket{n} = 0
+    (\hat{H}_1 - E_n^{(1)}) \ket{n} + (\hat{H}_0 - \varepsilon_n) \ket*{\psi_n^{(1)}} - (\varepsilon_n - \varepsilon_n) c_n \ket{n} = 0
 \end{aligned}$$
 
-Where we used $\hat{H}_0 \ket{n} = \varepsilon_n \ket{n}$. Inserting the
-series form of $\ket{\psi_n^{(1)}}$ into the order-$\lambda^1$ equation
-gives us:
+Where we used $\hat{H}_0 \ket{n} = \varepsilon_n \ket{n}$.
+We insert the series form of $\ket*{\psi_n^{(1)}}$ into the $\lambda^1$-equation:
 
 $$\begin{aligned}
     (\hat{H}_1 - E_n^{(1)}) \ket{n} + \sum_{m \neq n} c_m (\varepsilon_m - \varepsilon_n) \ket{m} = 0
@@ -135,12 +134,12 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We isolate this result for $c_k$ and insert it into the series form of
-$\ket{\psi_n^{(1)}}$ to get the full first-order correction to the wave
+$\ket*{\psi_n^{(1)}}$ to get the full first-order correction to the wave
 function:
 
 $$\begin{aligned}
     \boxed{
-        \ket{\psi_n^{(1)}}
+        \ket*{\psi_n^{(1)}}
         = \sum_{m \neq n} \frac{\matrixel{m}{\hat{H}_1}{n}}{\varepsilon_n - \varepsilon_m} \ket{m}
     }
 \end{aligned}$$
@@ -166,9 +165,9 @@ $$\begin{aligned}
     = \matrixel{n}{\hat{H}_1}{\psi_n^{(1)}} - E_n^{(1)} \braket{n}{\psi_n^{(1)}}
 \end{aligned}$$
 
-We explicitly removed the $\ket{n}$-dependence of $\ket{\psi_n^{(1)}}$,
+We explicitly removed the $\ket{n}$-dependence of $\ket*{\psi_n^{(1)}}$,
 so the last term is zero. By simply inserting our result for
-$\ket{\psi_n^{(1)}}$, we thus arrive at:
+$\ket*{\psi_n^{(1)}}$, we thus arrive at:
 
 $$\begin{aligned}
     \boxed{
@@ -257,9 +256,8 @@ $$\begin{aligned}
 
 This is an eigenvalue problem for $E_n^{(1)}$, where $c_d$ are the
 components of the eigenvectors which represent the "good" states.
-Suppose that this eigenvalue problem has been solved, and that
-$\ket{n, g}$ are the resulting "good" states. Then, as long as
-$E_n^{(1)}$ is a non-degenerate eigenvalue of $M$:
+After solving this, let $\ket{n, g}$ be the resulting "good" states.
+Then, as long as $E_n^{(1)}$ is a non-degenerate eigenvalue of $M$:
 
 $$\begin{aligned}
     \boxed{
@@ -272,36 +270,36 @@ first-order wave function correction is also unchanged:
 
 $$\begin{aligned}
     \boxed{
-        \ket{\psi_{n,g}^{(1)}}
+        \ket*{\psi_{n,g}^{(1)}}
         = \sum_{m \neq (n, g)} \frac{\matrixel{m}{\hat{H}_1}{n, g}}{\varepsilon_n - \varepsilon_m} \ket{m}
     }
 \end{aligned}$$
 
 This works because the matrix $M$ is diagonal in the $\ket{n, g}$-basis,
 such that when $\ket{m}$ is any vector $\ket{n, \gamma}$ in the
-$\ket{n}$-eigenspace (except for $\ket{n,g}$ of course, which is
-explicitly excluded), then conveniently the corresponding numerator
+$\ket{n}$-eigenspace (except for $\ket{n,g}$, which is
+explicitly excluded), then the corresponding numerator
 $\matrixel{n, \gamma}{\hat{H}_1}{n, g} = M_{\gamma, g} = 0$, so the term
 does not contribute.
 
 If any of the eigenvalues $E_n^{(1)}$ of $M$ are degenerate, then there
-is still some information missing about the components $c_d$ of the
-"good" states, in which case we must find these states some other way.
+is still information missing about the components $c_d$ of the
+"good" states, in which case we must find them some other way.
 
-An alternative way of determining these "good" states is also of
-interest if there is no degeneracy in $M$, since such a shortcut would
-allow us use the formulae from non-degenerate perturbation theory
+Such an alternative way of determining these "good" states is also of
+interest even if there is no degeneracy in $M$, since such a shortcut would
+allow us to use the formulae from non-degenerate perturbation theory
 straight away.
 
-The method is to find a Hermitian operator $\hat{L}$ (usually using
-symmetry) which commutes with both $\hat{H}_0$ and $\hat{H}_1$:
+The trick is to find a Hermitian operator $\hat{L}$ (usually using
+symmetries of the system) which commutes with both $\hat{H}_0$ and $\hat{H}_1$:
 
 $$\begin{aligned}
-= [\hat{L}, \hat{H}_1] = 0
+    [\hat{L}, \hat{H}_0] = [\hat{L}, \hat{H}_1] = 0
 \end{aligned}$$
 
 So that it shares its eigenstates with $\hat{H}_0$ (and $\hat{H}_1$),
-meaning at least $D$ of the vectors of the $D$-dimensional
+meaning all the vectors of the $D$-dimensional
 $\ket{n}$-eigenspace are also eigenvectors of $\hat{L}$.
 
 The crucial part, however, is that $\hat{L}$ must be chosen such that
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