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

---
 .../know/concept/hellmann-feynman-theorem/index.md | 31 +++++++++++-----------
 1 file changed, 15 insertions(+), 16 deletions(-)

(limited to 'source/know/concept/hellmann-feynman-theorem')

diff --git a/source/know/concept/hellmann-feynman-theorem/index.md b/source/know/concept/hellmann-feynman-theorem/index.md
index e18acc2..c6bf720 100644
--- a/source/know/concept/hellmann-feynman-theorem/index.md
+++ b/source/know/concept/hellmann-feynman-theorem/index.md
@@ -9,20 +9,20 @@ layout: "concept"
 ---
 
 Consider the time-independent Schrödinger equation,
-where the Hamiltonian $$\hat{H}$$ depends on a general parameter $$\lambda$$,
-whose meaning or type we will not specify:
+where the Hamiltonian $$\hat{H}$$ depends on some parameter $$\lambda$$
+whose meaning we will not specify:
 
 $$\begin{aligned}
-    \hat{H}(\lambda) \Ket{\psi_n(\lambda)}
-    = E_n(\lambda) \Ket{\psi_n(\lambda)}
+    \hat{H}(\lambda) \ket{\psi_n(\lambda)}
+    = E_n(\lambda) \ket{\psi_n(\lambda)}
 \end{aligned}$$
 
-Assuming all eigenstates $$\Ket{\psi_n}$$ are normalized,
+Assuming all eigenstates $$\ket{\psi_n}$$ are normalized,
 this gives us the following basic relation:
 
 $$\begin{aligned}
     \matrixel{\psi_m}{\hat{H}}{\psi_n}
-    = E_n \Inprod{\psi_m}{\psi_n}
+    = E_n \inprod{\psi_m}{\psi_n}
     = \delta_{mn} E_n
 \end{aligned}$$
 
@@ -38,33 +38,32 @@ $$\begin{aligned}
     + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
     + \matrixel{\psi_m}{\hat{H}}{\nabla_\lambda \psi_n}
     \\
-    &= E_m \Inprod{\psi_m}{\nabla_\lambda \psi_n} + E_n \Inprod{\nabla_\lambda \psi_m}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
+    &= E_m \inprod{\psi_m}{\nabla_\lambda \psi_n} + E_n \inprod{\nabla_\lambda \psi_m}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
 \end{aligned}$$
 
 In order to simplify this,
 we differentiate the orthogonality relation
-$$\Inprod{\psi_m}{\psi_n} = \delta_{mn}$$,
-which ends up telling us that
-$$\Inprod{\nabla_\lambda \psi_m}{\psi_n} = - \Inprod{\psi_m}{\nabla_\lambda \psi_n}$$:
+$$\inprod{\psi_m}{\psi_n} = \delta_{mn}$$:
 
 $$\begin{aligned}
     0
     = \nabla_\lambda \delta_{mn}
-    = \nabla_\lambda \Inprod{\psi_m}{\psi_n}
-    = \Inprod{\nabla_\lambda \psi_m}{\psi_n} + \Inprod{\psi_m}{\nabla_\lambda \psi_n}
+    = \nabla_\lambda \inprod{\psi_m}{\psi_n}
+    = \inprod{\nabla_\lambda \psi_m}{\psi_n} + \inprod{\psi_m}{\nabla_\lambda \psi_n}
 \end{aligned}$$
 
-Using this result to replace $$\Inprod{\nabla_\lambda \psi_m}{\psi_n}$$
+Meaning that $$\inprod{\nabla_\lambda \psi_m}{\psi_n} = - \inprod{\psi_m}{\nabla_\lambda \psi_n}$$.
+Using this result to replace $$\inprod{\nabla_\lambda \psi_m}{\psi_n}$$
 in the previous equation leads to:
 
 $$\begin{aligned}
     \delta_{mn} \nabla_\lambda E_n
-    &= (E_m - E_n) \Inprod{\psi_m}{\nabla_\lambda \psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
+    &= (E_m - E_n) \inprod{\psi_m}{\nabla_\lambda \psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
 \end{aligned}$$
 
 For $$m = n$$, we therefore arrive at the **Hellmann-Feynman theorem**,
 which is useful when doing numerical calculations
-to minimize energies with respect to $$\lambda$$:
+that often involve minimizing energies with respect to $$\lambda$$:
 
 $$\begin{aligned}
     \boxed{
@@ -79,7 +78,7 @@ the [Berry phase](/know/concept/berry-phase/):
 
 $$\begin{aligned}
     \boxed{
-        (E_n - E_m) \Inprod{\psi_m}{\nabla_\lambda \psi_n}
+        (E_n - E_m) \inprod{\psi_m}{\nabla_\lambda \psi_n}
         = \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
     }
 \end{aligned}$$
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