Expectation values with annihilation/creation operators

[QuarksAbove]

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

&lt;i(\hat{a} - \hat{a^{t}})&gt; = &lt;\psi|i\hat{a}-i\hat{a^{t}}|\psi&gt;<br />

<br /> &lt;\psi|i\hat{a}-i\hat{a^{t}}|\psi&gt; = &lt;\psi|i\hat{a}|\psi&gt; - &lt;\psi|i\hat{a^{t}}|\psi&gt;<br />

<br /> &lt;\psi|i\hat{a}|\psi&gt; - &lt;\psi|i\hat{a^{t}}|\psi&gt; = &lt;\psi|i\hat{a}|\psi&gt; - &lt;-i\hat{a}\psi|\psi&gt;<br />

<br /> i(\alpha e^{i\phi})^{-1}&lt;\psi|\psi&gt; + i(\alpha e^{i\phi})^{-1}&lt;\psi|\psi&gt;<br />

assuming \psi is normalized,

<br /> &lt;\psi|\psi&gt; = 1<br />

<br /> &lt;i(\hat{a} - \hat{a^{t}})&gt; = 2i(\alpha e^{i\phi})^{-1}<br />

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

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!

 

[fzero]

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