# Expectation values with annihilation/creation operators

## Homework Statement

Calculate $<i(\hat{a} - \hat{a^{t}})>$

## Homework Equations

$|\psi > = e^{-\alpha ^{2}/2} \sum \frac{(\alpha e^{i\phi })^n}{\sqrt{n!}} |n>$

$\hat{a}|n> = \sqrt{n}|n-1>$

I derived:
$\hat{a}|\psi> = (\alpha e^{i\phi})^{-1}|\psi>$

## The Attempt at a Solution

$<i(\hat{a} - \hat{a^{t}})> = <\psi|i\hat{a}-i\hat{a^{t}}|\psi>$

$<\psi|i\hat{a}-i\hat{a^{t}}|\psi> = <\psi|i\hat{a}|\psi> - <\psi|i\hat{a^{t}}|\psi>$

$<\psi|i\hat{a}|\psi> - <\psi|i\hat{a^{t}}|\psi> = <\psi|i\hat{a}|\psi> - <-i\hat{a}\psi|\psi>$

$i(\alpha e^{i\phi})^{-1}<\psi|\psi> + i(\alpha e^{i\phi})^{-1}<\psi|\psi>$

assuming $\psi$ is normalized,

$<\psi|\psi> = 1$

$<i(\hat{a} - \hat{a^{t}})> = 2i(\alpha e^{i\phi})^{-1}$

Now, I think I did this correctly.. What I don't understand is the significance of
$<i(\hat{a} - \hat{a^{t}})>$

Normally with expectation values, you can usually tell if your result is at least reasonable.. I don't understand what this expectation value is telling me, so I can't tell if my result is reasonable. =/

Any help would be much appreciated!!

It would probably help to write $\hat{a}, \hat{a}^\dagger$ in terms of the position and momentum operators.