# Question about the Heisenberg Picture

1. Oct 15, 2011

### Beer-monster

1. The problem statement, all variables and given/known data

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, lets say position, on how the the position operator varies in time and its resulting expectation values: Would is it neccessary 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?

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: Apr 26, 2017
2. Oct 15, 2011

### vela

Staff Emeritus
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.

3. Oct 15, 2011