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Coherent States (Quantum Fields)

  1. Dec 7, 2013 #1
    [itex]\Huge[/itex]1. The problem statement, all variables and given/known data

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

    Find a coherent state which satises these expectation values.


    2. Relevant equations

    [itex] \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)[/itex]

    Coherent State :

    [itex]a|\alpha \rangle =\alpha |\alpha \rangle[/itex]


    3. 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 :
    [itex]
    \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)[/itex]


    Thank you !
     
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  3. Dec 7, 2013 #2

    vanhees71

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    The point is to find a coherent state such that the expectation value of the electric-field components (operators) take the given (classical) field [itex]f[/itex].
     
  4. Dec 7, 2013 #3
    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 dont see whats so special here or what should I do ...

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

    Thank u ....
     
  5. Dec 7, 2013 #4

    vanhees71

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    No, it's not a known function. You just assume a function [itex]\vec{f}(t,\vec{x})[/itex] and look for a coherent state [itex]|\psi[/itex] of the electromagnetic field such that
    [tex]\langle \psi | \hat{\vec{E}}|\psi \rangle=\vec{f}(t,\vec{x}).[/tex]
     
  6. Dec 7, 2013 #5
    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.
     
    Last edited: Dec 7, 2013
  7. Dec 7, 2013 #6
    As u can see , my result is depended on r (vector) for an arbitrary coherent state |alpha>...
     
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