# QM, Heisenberg's motion equations, harmonic oscillator

1. Apr 20, 2013

### fluidistic

1. The problem statement, all variables and given/known data
Hi guys, I don't really know how to solve the first part of a problem which goes like this:
Consider a 1 dimensional harmonic oscillator of mass m, Hooke's constant k and angular frequency $\omega = \sqrt{\frac{k}{m} }$.
Remembering the classical solutions, solve the Heisenberg's motion equations for the operators $\hat x$ and $\hat p$. Why does the quantum evolution match the classical one?

2. Relevant equations
Heisenberg's motion equation: $\frac{dA(t)}{dt}=\frac{i}{h}[\hat H, \hat A (t)]=\frac{i}{h}[\hat H, \hat A](t)$. (eq.1)
Where $\hat A(t)=U_t^*AU_t$.
Where $U_t=\exp \{ -i\hat H t/\hbar \}$. Since $\hat H$ is self-adjoint, it follows that $U_t$ is unitary. Thus $U_t^*=U_t^{-1}$. Also $U_t^*=U_{-t}$.
3. The attempt at a solution
From Classical mechanics, $H=\frac{p^2}{2m}+\frac{m\omega^2 q^2}{2}$.
This leads to $\dot p =-m\omega ^2 q$ and $\dot q=\frac{p}{m}$. I know I must solve them but before proceeding, let's see if I can find some similar equations to solve using Heisenberg's motion equations.
In QM, $\hat H=\frac{\hat p ^2}{2m}+\frac{m\omega ^2}{2} \hat x$.
Using eq.1, skipping arithmetical steps I reached that $\frac{d}{dt} \hat A (t) = \frac{i}{\hbar} \{ \frac{1}{2m} [\hat p ^2 , \hat A](t) +\frac{m \omega ^2}{2} [\hat x ^2 , A](t) \}$ where $\hat p=-i\hbar \frac{d}{dx}$ and $\hat x =x$.
Now, replacing $\hat A$ by $\hat x$, I get that $\frac{d\hat x}{dt}=\frac{i}{\hbar} \{ \frac{1}{2m}[\hat p^2,\hat x](t) \}=\frac{i}{2m\hbar}(-\hat x \hat p^2)(t)=-\frac{i}{2m\hbar}U_t^* (\hat x \hat p ^2)U_t$.
Here I am unsure of my step. I think that it's worth $-\frac{i}{2m\hbar}U_t ^* U_t \hat x \hat p^2=-\frac{i}{2m\hbar} \hat x \hat p ^2$.
So I get the differential equation $\frac{d\hat x}{dt}=-\frac{i}{2m\hbar}\hat x \hat p ^2$ which differs from the one of classical mechanics for $\dot q$. I don't know what I did wrong.

Similarly for $\frac{d \hat p}{dt}$, I found that it's worth $\frac{m \omega ^2 }{2}(\hat x \hat p -1)$ which again does not match the DE of CM.
I guess my approach is wrong?
I would like any pointer. Thank you.

2. Apr 21, 2013

### BruceW

I don't think this step is correct. try checking this again.