0
$\begingroup$

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.

Improve this question
$\endgroup$
4
  • 4
    $\begingroup$ 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. $\endgroup$
    – mike stone
    Commented May 28, 2021 at 0:01
  • $\begingroup$ 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 $\endgroup$
    – vxxm
    Commented May 28, 2021 at 0:49
  • $\begingroup$ At some point you need to get dynamics $\phi(t) = e^{iHt}\phi(0)e^{-iHt}$ or something equivalent. $\endgroup$
    – mike stone
    Commented May 28, 2021 at 14:26
  • $\begingroup$ I see i see, thanks $\endgroup$
    – vxxm
    Commented Jun 3, 2021 at 7:39

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.