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Diffstat (limited to 'source/know/concept/ehrenfests-theorem')
| -rw-r--r-- | source/know/concept/ehrenfests-theorem/index.md | 28 |
1 files changed, 14 insertions, 14 deletions
diff --git a/source/know/concept/ehrenfests-theorem/index.md b/source/know/concept/ehrenfests-theorem/index.md index 4d96989..fba0192 100644 --- a/source/know/concept/ehrenfests-theorem/index.md +++ b/source/know/concept/ehrenfests-theorem/index.md @@ -9,10 +9,10 @@ layout: "concept" --- In quantum mechanics, **Ehrenfest's theorem** gives a general expression for the -time evolution of an observable's expectation value $\expval{\hat{L}}$. +time evolution of an observable's expectation value $$\expval{\hat{L}}$$. The time-dependent Schrödinger equation is as follows, -where prime denotes differentiation with respect to time $t$: +where prime denotes differentiation with respect to time $$t$$: $$\begin{aligned} \Ket{\psi'} = \frac{1}{i \hbar} \hat{H} \Ket{\psi} @@ -20,8 +20,8 @@ $$\begin{aligned} \Bra{\psi'} = - \frac{1}{i \hbar} \Bra{\psi} \hat{H} \end{aligned}$$ -Given an observable operator $\hat{L}$ and a state $\Ket{\psi}$, -the time-derivative of the expectation value $\expval{\hat{L}}$ is as follows +Given an observable operator $$\hat{L}$$ and a state $$\Ket{\psi}$$, +the time-derivative of the expectation value $$\expval{\hat{L}}$$ is as follows (due to the product rule of differentiation): $$\begin{aligned} @@ -48,12 +48,12 @@ the last term often vanishes. As a interesting side note, in the [Heisenberg picture](/know/concept/heisenberg-picture/), this relation proves itself, -when one simply wraps all terms in $\Bra{\psi}$ and $\Ket{\psi}$. +when one simply wraps all terms in $$\Bra{\psi}$$ and $$\Ket{\psi}$$. -Two observables of particular interest are the position $\hat{X}$ and momentum $\hat{P}$. -Applying the above theorem to $\hat{X}$ yields the following, -which we reduce using the fact that $\hat{X}$ commutes -with the potential $V(\hat{X})$, +Two observables of particular interest are the position $$\hat{X}$$ and momentum $$\hat{P}$$. +Applying the above theorem to $$\hat{X}$$ yields the following, +which we reduce using the fact that $$\hat{X}$$ commutes +with the potential $$V(\hat{X})$$, because one is a function of the other: $$\begin{aligned} @@ -76,7 +76,7 @@ $$\begin{gathered} } \end{gathered}$$ -Next, applying the general formula to the expected momentum $\expval{\hat{P}}$ +Next, applying the general formula to the expected momentum $$\expval{\hat{P}}$$ gives us: $$\begin{aligned} @@ -86,8 +86,8 @@ $$\begin{aligned} = \frac{1}{i \hbar} \Expval{[\hat{P}, V(\hat{X})]} \end{aligned}$$ -To find the commutator, we go to the $\hat{X}$-basis and use a test -function $f(x)$: +To find the commutator, we go to the $$\hat{X}$$-basis and use a test +function $$f(x)$$: $$\begin{aligned} \Comm{- i \hbar \dv{}{x}}{V(x)} \: f(x) @@ -113,8 +113,8 @@ $$\begin{gathered} \end{gathered}$$ There is an important consequence of Ehrenfest's original theorems -for the symbolic derivatives of the Hamiltonian $\hat{H}$ -with respect to $\hat{X}$ and $\hat{P}$: +for the symbolic derivatives of the Hamiltonian $$\hat{H}$$ +with respect to $$\hat{X}$$ and $$\hat{P}$$: $$\begin{gathered} \boxed{ |