# Harmonic osccillator: solution for A in Y'AY

1. Nov 7, 2011

### flux!

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

What is A in $\bar{\varphi}$A$\varphi$, if

$\bar{\varphi}$A$\varphi$ = $\frac{-ip(\tau)\dot{q}(\tau)}{\hbar}$+$\frac{p^{2}(\tau)}{2m}$+$\frac{m\omega^{2}}{2}$q$^{2}$($\tau$)

2. Relevant equations

provided that $\bar{\varphi}$ and $\varphi$ are ladder operators of the form:

$\varphi$ = $\left(\frac{m\omega}{2\hbar}\right)^{1/2}$$\left(q\left(\tau\right)+\frac{ip\left(\tau\right)}{m\omega}\right)$

$\bar{\varphi}$ = $\left(\frac{m\omega}{2\hbar}\right)^{1/2}$$\left(q\left(\tau\right)-\frac{ip\left(\tau\right)}{m\omega}\right)$

p is the momentum and q is the position in real space,

3. The attempt at a solution
A possible solution might be to extract all those variable to get A, but $\bar{\varphi}$ and $\varphi$ are operators, so i am in complete darkness here, i hope you have a possible solution to this, thank you.