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-rw-r--r--source/know/concept/hellmann-feynman-theorem/index.md18
1 files changed, 9 insertions, 9 deletions
diff --git a/source/know/concept/hellmann-feynman-theorem/index.md b/source/know/concept/hellmann-feynman-theorem/index.md
index 9ffff34..e18acc2 100644
--- a/source/know/concept/hellmann-feynman-theorem/index.md
+++ b/source/know/concept/hellmann-feynman-theorem/index.md
@@ -9,7 +9,7 @@ layout: "concept"
---
Consider the time-independent Schrödinger equation,
-where the Hamiltonian $\hat{H}$ depends on a general parameter $\lambda$,
+where the Hamiltonian $$\hat{H}$$ depends on a general parameter $$\lambda$$,
whose meaning or type we will not specify:
$$\begin{aligned}
@@ -17,7 +17,7 @@ $$\begin{aligned}
= 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}
@@ -26,7 +26,7 @@ $$\begin{aligned}
= \delta_{mn} E_n
\end{aligned}$$
-We differentiate this with respect to $\lambda$,
+We differentiate this with respect to $$\lambda$$,
which could be a scalar or a vector.
This yields:
@@ -43,9 +43,9 @@ $$\begin{aligned}
In order to simplify this,
we differentiate the orthogonality relation
-$\Inprod{\psi_m}{\psi_n} = \delta_{mn}$,
+$$\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{\nabla_\lambda \psi_m}{\psi_n} = - \Inprod{\psi_m}{\nabla_\lambda \psi_n}$$:
$$\begin{aligned}
0
@@ -54,7 +54,7 @@ $$\begin{aligned}
= \Inprod{\nabla_\lambda \psi_m}{\psi_n} + \Inprod{\psi_m}{\nabla_\lambda \psi_n}
\end{aligned}$$
-Using this result to replace $\Inprod{\nabla_\lambda \psi_m}{\psi_n}$
+Using this result to replace $$\Inprod{\nabla_\lambda \psi_m}{\psi_n}$$
in the previous equation leads to:
$$\begin{aligned}
@@ -62,9 +62,9 @@ $$\begin{aligned}
&= (E_m - E_n) \Inprod{\psi_m}{\nabla_\lambda \psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
\end{aligned}$$
-For $m = n$, we therefore arrive at the **Hellmann-Feynman theorem**,
+For $$m = n$$, we therefore arrive at the **Hellmann-Feynman theorem**,
which is useful when doing numerical calculations
-to minimize energies with respect to $\lambda$:
+to minimize energies with respect to $$\lambda$$:
$$\begin{aligned}
\boxed{
@@ -73,7 +73,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-While for $m \neq n$, we get the **Epstein generalization**
+While for $$m \neq n$$, we get the **Epstein generalization**
of the Hellmann-Feynman theorem, which is for example relevant for
the [Berry phase](/know/concept/berry-phase/):