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-rw-r--r--source/know/concept/hellmann-feynman-theorem/index.md31
1 files changed, 15 insertions, 16 deletions
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}$$