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## Main Question or Discussion Point

I begin with [tex]\int (\bar{\psi}(x) (\mathcal{H} \psi(x)) d^3x[/tex]

This is just

[tex]\int (\bar{\psi}(x) ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} \psi(x)) d^3x[/tex]

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

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

So that

[tex]\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[/tex]

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!

This is just

[tex]\int (\bar{\psi}(x) ({\frac{p^2}{2M} + \frac{1}{2}M \omega^2 (x)} \psi(x)) d^3x[/tex]

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

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

So that

[tex]\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[/tex]

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!

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