Question about the Heisenberg Picture

[Beer-monster]

Homework Statement

I've seen this example for using the Heisenberg equation of motion to solve the Simple Hamonic Oscillator.

http://en.wikipedia.org/wiki/Heisenberg_picture#Commutator_relations"

However, if you were only interested in one variable, let's say position, on how the the position operator varies in time and its resulting expectation values: Would is it necessary to find and solve the equation of motion for P as well?

Also once one has the relevant equation of potion:

\frac{dX(t)}{dt} = \frac{p(0)}{m}

Why is it so often solved by differentiating the equation a second time and solving, rather than just integrating?

Homework Equations

The Attempt at a Solution

 

[vela]

The equations for the SHO are
\begin{align*}
\dot{x}(t) &= \frac{p(t)}{m} \\
\dot{p}(t) &= -m\omega^2x(t)
\end{align*}Note that you have functions of time on both sides of the equations. You don't have p(0) on the right-hand side of the x equation. You differentiate one equation and substitute in the second one to get rid of one of the functions. The resulting second-order equation is straightforward to solve.

 

[Beer-monster]

Thanks for the reply, Vela.

I was thinking about it earlier as I walked the dog, but hadn't got around to sitting down with it. I did wonder if assuming that the answer in the commutator was p(0) was unfounded.

Glad to have someone confirm it and that'd I'm not a complete idiot, just a bit slow :D