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Diffstat (limited to 'source/know/concept/greens-functions')
| -rw-r--r-- | source/know/concept/greens-functions/index.md | 22 |
1 files changed, 8 insertions, 14 deletions
diff --git a/source/know/concept/greens-functions/index.md b/source/know/concept/greens-functions/index.md index ddba2cd..eda5671 100644 --- a/source/know/concept/greens-functions/index.md +++ b/source/know/concept/greens-functions/index.md @@ -21,6 +21,7 @@ but in general they are not the same, except in a special case, see below. + ## Single-particle functions If the two operators are single-particle creation/annihilation operators, @@ -146,11 +147,8 @@ $$\begin{gathered} G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t') \end{gathered}$$ -<div class="accordion"> -<input type="checkbox" id="proof-time-diff"/> -<label for="proof-time-diff">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-time-diff">Proof.</label> + +{% include proof/start.html id="proof-time-delta" -%} We will prove that the thermal expectation value $$\expval{\hat{A}(t) \hat{B}(t')}$$ only depends on $$t - t'$$ for arbitrary $$\hat{A}$$ and $$\hat{B}$$, @@ -189,8 +187,7 @@ because $$\hat{H}$$ is time-independent by assumption. Note that thermodynamic equilibrium is crucial: intuitively, if the system is not in equilibrium, then it evolves in some transient time-dependent way. -</div> -</div> +{% include proof/end.html id="proof-time-delta" %} If the Hamiltonian is both time-independent and non-interacting, then the time-dependence of $$\hat{c}_\nu$$ @@ -214,6 +211,7 @@ $$\begin{aligned} \end{aligned}$$ + ## As fundamental solutions In the absence of interactions, @@ -237,11 +235,8 @@ $$\begin{aligned} = \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r}) \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-commH0"/> -<label for="proof-commH0">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-commH0">Proof.</label> + +{% include proof/start.html id="proof-commutator" -%} In the second quantization, the Hamiltonian $$\hat{H}_0$$ is written like so: @@ -307,9 +302,8 @@ $$\begin{aligned} &= \frac{\hbar^2}{2 m} \sum_{\nu'} \hat{c}_{\nu'} \nabla^2 \psi_{\nu'}(\vb{r}) = \frac{\hbar^2}{2 m} \nabla^2 \hat{\Psi}(\vb{r}) \end{aligned}$$ +{% include proof/end.html id="proof-commutator" %} -</div> -</div> After substituting this into the equation of motion, we recognize $$G^R(\vb{r}, t; \vb{r}', t')$$ itself: |