From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/dyson-equation/index.md | 169 ++++++++++++++++++++++++++++ 1 file changed, 169 insertions(+) create mode 100644 source/know/concept/dyson-equation/index.md (limited to 'source/know/concept/dyson-equation/index.md') diff --git a/source/know/concept/dyson-equation/index.md b/source/know/concept/dyson-equation/index.md new file mode 100644 index 0000000..82020ad --- /dev/null +++ b/source/know/concept/dyson-equation/index.md @@ -0,0 +1,169 @@ +--- +title: "Dyson equation" +date: 2021-11-01 +categories: +- Physics +- Quantum mechanics +layout: "concept" +--- + +Consider the time-dependent Schrödinger equation, +describing a wavefunction $\Psi_0(\vb{r}, t)$: + +$$\begin{aligned} + i \hbar \pdv{}{t}\Psi_0(\vb{r}, t) + = \hat{H}_0(\vb{r}) \: \Psi_0(\vb{r}, t) +\end{aligned}$$ + +By definition, this equation's +[fundamental solution](/know/concept/fundamental-solution/) +$G_0(\vb{r}, t; \vb{r}', t')$ satisfies the following: + +$$\begin{aligned} + \Big( i \hbar \pdv{}{t}- \hat{H}_0(\vb{r}) \Big) G_0(\vb{r}, t; \vb{r}', t') + = \delta(\vb{r} - \vb{r}') \: \delta(t - t') +\end{aligned}$$ + +From this, we define the inverse $\hat{G}{}_0^{-1}(\vb{r}, t)$ +as follows, so that $\hat{G}{}_0^{-1} G_0 = \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t')$: + +$$\begin{aligned} + \hat{G}{}_0^{-1}(\vb{r}, t) + &\equiv i \hbar \pdv{}{t}- \hat{H}_0(\vb{r}) +\end{aligned}$$ + +Note that $\hat{G}{}_0^{-1}$ is an operator, while $G_0$ is a function. +For the sake of consistency, we thus define +the operator $\hat{G}_0(\vb{r}, t)$ +as a multiplication by $G_0$ +and integration over $\vb{r}'$ and $t'$: + +$$\begin{aligned} + \hat{G}_0(\vb{r}, t) \: f + \equiv \iint_{-\infty}^\infty G_0(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \: \dd{\vb{r}}' \dd{t'} +\end{aligned}$$ + +For an arbitrary function $f(\vb{r}, t)$, +so that $\hat{G}{}_0^{-1} \hat{G}_0 = \hat{G}_0 \hat{G}{}_0^{-1} = 1$. +Moving on, the Schrödinger equation can be rewritten like so, +using $\hat{G}{}_0^{-1}$: + +$$\begin{aligned} + \hat{G}{}_0^{-1}(\vb{r}, t) \: \Psi_0(\vb{r}, t) + = 0 +\end{aligned}$$ + +Let us assume that $\hat{H}_0$ is simple, +such that $G_0$ and $\hat{G}{}_0^{-1}$ can be found without issues +by solving the defining equation above. + +Suppose we now add a more complicated and +possibly time-dependent term $\hat{H}_1(\vb{r}, t)$, +in which case the corresponding fundamental solution +$G(\vb{r}, \vb{r}', t, t')$ satisfies: + +$$\begin{aligned} + \delta(\vb{r} - \vb{r}') \: \delta(t - t') + &= \Big( i \hbar \pdv{}{t}- \hat{H}_0(\vb{r}) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t') + \\ + &= \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) G(\vb{r}, t; \vb{r}', t') +\end{aligned}$$ + +This equation is typically too complicated to solve, +so we would like an easier way to calculate this new $G$. +The perturbed wavefunction $\Psi(\vb{r}, t)$ +satisfies the Schrödinger equation: + +$$\begin{aligned} + \Big( \hat{G}{}_0^{-1}(\vb{r}, t) - \hat{H}_1(\vb{r}, t) \Big) \Psi(\vb{r}, t) + = 0 +\end{aligned}$$ + +We know that $\hat{G}{}_0^{-1} \Psi_0 = 0$, +which we put on the right, +and then we apply $\hat{G}_0$ in front: + +$$\begin{aligned} + \hat{G}_0^{-1} \Psi - \hat{H}_1 \Psi + = \hat{G}_0^{-1} \Psi_0 + \quad \implies \quad + \Psi - \hat{G}_0 \hat{H}_1 \Psi + &= \Psi_0 +\end{aligned}$$ + +This equation is recursive, +so we iteratively insert it into itself. +Note that the resulting equations are the same as those from +[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/): + +$$\begin{aligned} + \Psi + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi + \\ + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi + \\ + &= \Psi_0 + \hat{G}_0 \hat{H}_1 \Psi_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0 + + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \Psi_0 + \: ... + \\ + &= \Psi_0 + \big( \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G}_0 \hat{H}_1 \hat{G}_0 + \: ... \big) \hat{H}_1 \Psi_0 +\end{aligned}$$ + +The parenthesized expression clearly has the same recursive pattern, +so we denote it by $\hat{G}$ and write the so-called **Dyson equation**: + +$$\begin{aligned} + \boxed{ + \hat{G} + = \hat{G}_0 + \hat{G}_0 \hat{H}_1 \hat{G} + } +\end{aligned}$$ + +Such an iterative scheme is excellent for approximating $\hat{G}(\vb{r}, t)$. +Once a satisfactory accuracy is obtained, +the perturbed wavefunction $\Psi$ can be calculated from: + +$$\begin{aligned} + \boxed{ + \Psi + = \Psi_0 + \hat{G} \hat{H}_1 \Psi_0 + } +\end{aligned}$$ + +This relation is equivalent to the Schrödinger equation. +So now we have the operator $\hat{G}(\vb{r}, t)$, +but what about the fundamental solution function $G(\vb{r}, t; \vb{r}', t')$? +Let us take its definition, multiply it by an arbitrary $f(\vb{r}, t)$, +and integrate over $G$'s second argument pair: + +$$\begin{aligned} + \iint \big( \hat{G}{}_0^{-1} \!-\! \hat{H}_1 \big) G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} + = \iint \delta(\vb{r} \!-\! \vb{r}') \: \delta(t \!-\! t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} + = f +\end{aligned}$$ + +Where we have hidden the arguments $(\vb{r}, t)$ for brevity. +We now apply $\hat{G}_0(\vb{r}, t)$ to this equation +(which contains an integral over $t''$ independent of $t'$): + +$$\begin{aligned} + \hat{G}_0 f + &= \big( \hat{G}_0 \hat{G}{}_0^{-1} - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} + \\ + &= \big( 1 - \hat{G}_0 \hat{H}_1 \big) \iint_{-\infty}^\infty G(\vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}'} \dd{t'} +\end{aligned}$$ + +Here, the shape of Dyson's equation is clearly recognizable, +so we conclude that, as expected, the operator $\hat{G}$ +is defined as multiplication by the function $G$ followed by integration: + +$$\begin{aligned} + \hat{G}(\vb{r}, t) \: f(\vb{r}, t) + \equiv \iint_{-\infty}^\infty G(\vb{r}, t; \vb{r}', t') \: f(\vb{r}', t') \dd{\vb{r}}' \dd{t'} +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. -- cgit v1.2.3