Coherent States (Quantum Fields)

[Raz91]

$\Huge$

Homework Statement

Consider a state of the EM field which satisfies
$\left\langle \textbf{E}_x(\vec{r})\right\rangle =f(\vec{r})$

Find a coherent state which satises these expectation values.

Homework Equations

$\textbf{E}(\textbf{r})=\frac{i}{\sqrt{2 V}}\sum _{\textbf{k},\lambda } \sqrt{\omega _k}\left(e^{-i \textbf{k}\textbf{ r}} a^{\dagger }{}_{\textbf{k},\lambda } \hat{\epsilon }^*{}_{\textbf{k},\lambda }+e^{-i \textbf{k}\textbf{r}} a_{\textbf{k},\lambda } \hat{\epsilon }_{\textbf{k},\lambda }\right)$

Coherent State :

$a|\alpha \rangle =\alpha |\alpha \rangle$

The Attempt at a Solution

I tried to calculate this , but i just don't understand what am I suppose to prove here?
isn't it trivial that the expectation value will be a function of r (vector) ?

I've got this :
$\left\langle \textbf{E}_x(r)\right\rangle =\sum _{k,\lambda } \sqrt{\frac{2 \omega _k}{V}} \textbf{Im}\left(\alpha e^{-i k r} \epsilon _{x_{k,\lambda }}\right)$Thank you !

 

[vanhees71]

The point is to find a coherent state such that the expectation value of the electric-field components (operators) take the given (classical) field $f$.

 

[Raz91]

I still don't understand where is it given ?
it's just a "new name" for <Ex> , isn't it?

of course the expectation value won't be an operator... so I don't see what's so special here or what should I do ...

or f(r) is a known function in Electrodynamics that i should know ?

Thank u ...

 

[vanhees71]

No, it's not a known function. You just assume a function $\vec{f}(t,\vec{x})$ and look for a coherent state $|\psi$ of the electromagnetic field such that
$$\langle \psi | \hat{\vec{E}}|\psi \rangle=\vec{f}(t,\vec{x}).$$

 

[Raz91]

but according to the defination of the electric field , any coherent state will lead to such an expectation value because it's an eigen-state of the annihilation operator.

 

[Raz91]

As u can see , my result is depended on r (vector) for an arbitrary coherent state |alpha>...