Understanding Second Quantization and Its Application in Quantum Mechanics

[QuantumClue]

I begin with $$\int (\bar{\psi}(x) (\mathcal{H} \psi(x)) d^3x$$

This is just

$$\int (\bar{\psi}(x) ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} \psi(x)) d^3x$$

If one identified that $$\bar{\psi}(x)$$ and $$\psi(x)$$ are creation and annihilation operators, I assume that I can simply restate my integral by replacing the appropriate expressions with the following:


$$\int (a^{\dagger}a ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} aa^{\dagger}) d^3x$$

So that

$$\int (\hbar \omega^{-1} \mathcal{H} - \frac{\hbar \omega}{2} ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} \hbar \omega^{-1} \mathcal{H} + \frac{\hbar \omega}{2}) d^3x$$

I am just asking if I have assumed to much. Am I allowed to do this, and if not, why not?

Thanks

edit

What am I doing wrong this time, the equations won't show? I love latex, but I hate it sometimes!

 

[QuantumClue]

I think I found the invalid move. Sorry about that folks! Latex will show soon I hope :)