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diff --git a/source/know/concept/equation-of-motion-theory/index.md b/source/know/concept/equation-of-motion-theory/index.md new file mode 100644 index 0000000..81d4d46 --- /dev/null +++ b/source/know/concept/equation-of-motion-theory/index.md @@ -0,0 +1,197 @@ +--- +title: "Equation-of-motion theory" +date: 2021-11-08 +categories: +- Physics +- Quantum mechanics +layout: "concept" +--- + +In many-body quantum theory, **equation-of-motion theory** +is a method to calculate the time evolution of a system's properties +using [Green's functions](/know/concept/greens-functions/). + +Starting from the definition of +the retarded single-particle Green's function $G_{\nu \nu'}^R(t, t')$, +we simply take the $t$-derivative +(we could do the same with the advanced function $G_{\nu \nu'}^A$): + +$$\begin{aligned} + i \hbar \pdv{G^R_{\nu \nu'}(t, t')}{t} + &= \pdv{\Theta(t \!-\! t')}{t} \Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}} + + \Theta(t \!-\! t') \pdv{}{t}\Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}} + \\ + &= \delta(t \!-\! t') \Expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}} + + \Theta(t \!-\! t') \Expval{\Comm{\dv{\hat{c}_\nu(t)}{t}}{\hat{c}_{\nu'}^\dagger(t)}_{\mp}} +\end{aligned}$$ + +Where we have used that the derivative +of a [Heaviside step function](/know/concept/heaviside-step-function/) $\Theta$ +is a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$. +Also, from the [second quantization](/know/concept/second-quantization/), +$\expval{\comm{\hat{c}_\nu(t)}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}}$ +for $t = t'$ is zero when $\nu \neq \nu'$. + +Since we are in the [Heisenberg picture](/know/concept/heisenberg-picture/), +we know the equation of motion of $\hat{c}_\nu(t)$: + +$$\begin{aligned} + \dv{\hat{c}_\nu(t)}{t} + = \frac{i}{\hbar} \comm{\hat{H}_0(t)}{\hat{c}_\nu(t)} + \frac{i}{\hbar} \comm{\hat{H}_\mathrm{int}(t)}{\hat{c}_\nu(t)} +\end{aligned}$$ + +Where the single-particle part of the Hamiltonian $\hat{H}_0$ +and the interaction part $\hat{H}_\mathrm{int}$ +are assumed to be time-independent in the Schrödinger picture. +We thus get: + +$$\begin{aligned} + i \hbar \pdv{G^R_{\nu \nu'}}{t} + &= \delta_{\nu \nu'} \delta(t \!-\! t')+ \frac{i}{\hbar} \Theta(t \!-\! t') + \Expval{\Comm{\comm{\hat{H}_0}{\hat{c}_\nu} + \comm{\hat{H}_\mathrm{int}}{\hat{c}_\nu}}{\hat{c}_{\nu'}^\dagger}_{\mp}} +\end{aligned}$$ + +The most general form of $\hat{H}_0$, for any basis, +is as follows, where $u_{\nu' \nu''}$ are constants: + +$$\begin{aligned} + \hat{H}_0 + = \sum_{\nu' \nu''} u_{\nu' \nu''} \hat{c}_{\nu'}^\dagger \hat{c}_{\nu''} + \quad \implies \quad + \comm{\hat{H}_0}{\hat{c}_\nu} + = - \sum_{\nu''} u_{\nu \nu''} \hat{c}_{\nu''} +\end{aligned}$$ + +<div class="accordion"> +<input type="checkbox" id="proof-commH0"/> +<label for="proof-commH0">Proof</label> +<div class="hidden"> +<label for="proof-commH0">Proof.</label> +Using the commutator identity for $\comm{A B}{C}$, +we decompose it like so: + +$$\begin{aligned} + \comm{\hat{H}_0}{\hat{c}_\nu} + &= \sum_{\nu' \nu''} u_{\nu \nu''} \comm{\hat{c}_{\nu'}^\dagger \hat{c}_{\nu''}}{\hat{c}_\nu} + = \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \hat{c}_{\nu'}^\dagger \comm{\hat{c}_{\nu''}}{\hat{c}_\nu} + + \comm{\hat{c}_{\nu'}^\dagger}{\hat{c}_\nu} \hat{c}_{\nu''} \Big) +\end{aligned}$$ + +Bosons have well-known commutation relations, +so the result follows directly: + +$$\begin{aligned} + \comm{\hat{H}_0}{\hat{b}_\nu} + &= \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \hat{b}_{\nu'}^\dagger \comm{\hat{b}_{\nu''}}{\hat{b}_\nu} + + \comm{\hat{b}_{\nu'}^\dagger}{\hat{b}_\nu} \hat{b}_{\nu''} \Big) + = - \sum_{\nu''} u_{\nu \nu''} \hat{b}_{\nu''} +\end{aligned}$$ + +Fermions only have anticommutation relations, +so a bit more work is necessary: + +$$\begin{aligned} + \comm{\hat{H}_0}{\hat{f}_{\!\nu}} + &= \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \hat{f}_{\!\nu'}^\dagger \comm{\hat{f}_{\!\nu''}}{\hat{f}_{\!\nu}} + + \comm{\hat{f}_{\!\nu'}^\dagger}{\hat{f}_{\!\nu}} \hat{f}_{\!\nu''} \Big) + \\ + &= \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \hat{f}_{\!\nu'}^\dagger \acomm{\hat{f}_{\!\nu''}}{\hat{f}_{\!\nu}} + - 2 \hat{f}_{\!\nu'}^\dagger \hat{f}_{\!\nu} \hat{f}_{\!\nu''} + + \acomm{\hat{f}_{\!\nu'}^\dagger}{\hat{f}_{\!\nu}} \hat{f}_{\!\nu''} + - 2 \hat{f}_{\!\nu} \hat{f}_{\!\nu'}^\dagger \hat{f}_{\!\nu''} \Big) + \\ + &= \sum_{\nu' \nu''} u_{\nu' \nu''} \Big( \delta_{\nu \nu'} \hat{f}_{\!\nu''} + - 2 \acomm{\hat{f}_{\!\nu'}^\dagger}{\hat{f}_{\!\nu}} \hat{f}_{\!\nu''} \Big) + = - \sum_{\nu''} u_{\nu \nu''} \hat{f}_{\!\nu''} +\end{aligned}$$ +</div> +</div> + +Substituting this into $G_{\nu \nu'}^R$'s equation of motion, +we recognize another Green's function $G_{\nu'' \nu'}^R$: + +$$\begin{aligned} + i \hbar \pdv{G^R_{\nu \nu'}}{t} + &= \delta_{\nu \nu'} \delta(t \!-\! t') + \frac{i}{\hbar} \Theta(t \!-\! t') + \bigg( \Expval{\comm{\comm{\hat{H}_\mathrm{int}}{\hat{c}_\nu}}{\hat{c}_{\nu'}^\dagger}_{\mp}} + - \sum_{\nu''} u_{\nu \nu''} \Expval{\comm{\hat{c}_{\nu''}}{\hat{c}_{\nu'}^\dagger}_{\mp}} \bigg) + \\ + &= \delta_{\nu \nu'} \delta(t \!-\! t') + + \frac{i}{\hbar} \Theta(t \!-\! t') \Expval{\comm{\comm{\hat{H}_\mathrm{int}}{\hat{c}_\nu}}{\hat{c}_{\nu'}^\dagger}_{\mp}} + + \sum_{\nu''} u_{\nu \nu''} G_{\nu''\nu'}^R(t, t') +\end{aligned}$$ + +Rearranging this as follows yields the main result +of equation-of-motion theory: + +$$\begin{aligned} + \boxed{ + \sum_{\nu''} \Big( i \hbar \delta_{\nu \nu''} \pdv{}{t} - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(t, t') + = \delta_{\nu \nu'} \delta(t \!-\! t') + D_{\nu \nu'}^R(t, t') + } +\end{aligned}$$ + +Where $D_{\nu \nu'}^R$ represents a correction due to interactions $\hat{H}_\mathrm{int}$, +and also has the form of a retarded Green's function, +but with $\hat{c}_{\nu}$ replaced by $\comm{-\hat{H}_\mathrm{int}}{\hat{c}_\nu}$: + +$$\begin{aligned} + \boxed{ + D^R_{\nu'' \nu'}(t, t') + \equiv - \frac{i}{\hbar} \Theta(t \!-\! t') \Expval{\comm{\comm{-\hat{H}_\mathrm{int}(t)}{\hat{c}_\nu(t)}}{\hat{c}_{\nu'}^\dagger(t')}_{\mp}} + } +\end{aligned}$$ + +Unfortunately, calculating $D_{\nu \nu'}^R$ +might still not be doable due to $\hat{H}_\mathrm{int}$. +The key idea of equation-of-motion theory is to either approximate $D_{\nu \nu'}^R$ now, +or to differentiate it again $i \hbar \idv{D_{\nu \nu'}^R}{t}$, +and try again for the resulting corrections, +until a solvable equation is found. +There is no guarantee that that will ever happen; +if not, one of the corrections needs to be approximated. + +For non-interacting particles $\hat{H}_\mathrm{int} = 0$, +so clearly $D_{\nu \nu'}^R$ trivially vanishes then. +Let us assume that $\hat{H}_0$ is also time-independent, +such that $G_{\nu'' \nu'}^R$ only depends on the difference $t - t'$: + +$$\begin{aligned} + \sum_{\nu''} \Big( i \hbar \delta_{\nu \nu''} \pdv{}{t} - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(t - t') + = \delta_{\nu \nu'} \delta(t - t') +\end{aligned}$$ + +We take the [Fourier transform](/know/concept/fourier-transform/) +$(t \!-\! t') \to (\omega + i \eta)$, where $\eta \to 0^+$ ensures convergence: + +$$\begin{aligned} + \sum_{\nu''} \Big( \hbar \delta_{\nu \nu''} (\omega + i \eta) - u_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(\omega) + = \delta_{\nu \nu'} +\end{aligned}$$ + +If we assume a diagonal basis $u_{\nu \nu''} = \varepsilon_\nu \delta_{\nu \nu''}$, +this reduces to the following: + +$$\begin{aligned} + \delta_{\nu \nu'} + &= \sum_{\nu''} \Big( \hbar \delta_{\nu \nu''} (\omega + i \eta) - \varepsilon_\nu \delta_{\nu \nu''} \Big) G^R_{\nu'' \nu'}(\omega) + \\ + &= \Big( \hbar (\omega + i \eta) - \varepsilon_\nu \Big) G^R_{\nu \nu'}(\omega) +\end{aligned}$$ + +For a non-interacting, time-independent Hamiltonian, +we therefore arrive at: + +$$\begin{aligned} + \boxed{ + G^R_{\nu \nu'}(\omega) + = \frac{\delta_{\nu \nu'}}{\hbar (\omega + i \eta) - \varepsilon_\nu} + } +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. |