Harmonic osccillator: solution for A in Y'AY

[flux!]

Homework Statement

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$^{2}$($\tau$)

Homework 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,

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.

 

[anathema]

Thank you for your post. The value of A in this equation is not a single variable, but rather a combination of operators and variables. In this case, A is equal to the sum of the second and third term on the right-hand side of the equation. This can be written as:

A = \frac{p^{2}(\tau)}{2m}+\frac{m\omega^{2}}{2}q^{2}(\tau)

I would recommend studying the properties and behavior of ladder operators in quantum mechanics to better understand how they are used in this equation. I hope this helps. Good luck with your studies!