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Diffstat (limited to 'source/know/concept/hellmann-feynman-theorem')
| -rw-r--r-- | source/know/concept/hellmann-feynman-theorem/index.md | 18 |
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/): |