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Well I'm reading Mukhanov & Winitzki's Introduction to quantum effects in gravity, and I got to the exercise 2.8 that ask to derive Heisenberg's equation of motion

\begin{equation} \frac{d\hat{A}}{dt} = - \frac{i}{\hbar} [\hat{A},\hat{H}] + \frac{\partial \hat{A}}{\partial t} \end{equation}

And there is a couple of identities earlier, the classic ones in this part of quantum mechanics i.e. $[\hat{q},f] = i\hbar\ \partial f / \partial p$ and $[\hat{p},f] = -i\hbar\ \partial f / \partial q$, where $f = f(\hat{p}, \hat{q})$. And also Hamilton's equations.

So I tried to solve that exercise only with commutators and the identities that the book has given to you at that point, without the tipical aproach that mix Schrödinger's picture (like some people do it here or here), and I couldn't get to anywhere.

So I as wondering if there's a way to solve it without requiring Schrödinger's picture.

Harish Chandra Rajpoot
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vxxm
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    You need to be specific in what you are to derive it from. Just saying "given earlier" is no help to anyone who does not have the book. – mike stone May 28 '21 at 00:01
  • Do you want me to give an extensive list of all the identities that the book gives? As I wrote, the main things that can be used are those commutators (and identities asociated with commutators), Hamilton's equations and that. Or at least that's the way im asking – vxxm May 28 '21 at 00:49
  • At some point you need to get dynamics $\phi(t) = e^{iHt}\phi(0)e^{-iHt}$ or something equivalent. – mike stone May 28 '21 at 14:26
  • I see i see, thanks – vxxm Jun 03 '21 at 07:39

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