Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I have the following question about coherent states: It is known that the creation operator has no eigenket. However, the action of a creation operator [tex]a^{\dagger}[/tex] on a coherent ket [tex]|\alpha\rangle[/tex] can be written as

[tex]a^{\dagger}|\alpha\rangle = \left( \frac{\partial}{\partial \alpha} + \frac{\alpha^*}{2}\right)|\alpha\rangle.[/tex]

My question now concerns

[tex]e^{\lambda a^{\dagger}} |\alpha\rangle} = e^{\lambda a^*/2}e^{\lambda\partial_{\alpha}}|\alpha\rangle,[/tex]

which follows from the equation above. I wish to find an explicit expression for that one. It came to my mind that there may be an analogy with the displacement operator acting on position eigenstates

[tex]e^{\lambda \partial_x}|x\rangle = |x+\lambda\rangle.[/tex]

But does it hold? Or is there another way? Writing down the explicit form of a coherent state doesn't help me much because this way I can't get rid of the creation operator in the exponential.

Thank you very much for any thoughts!

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# Displacement operator for coherent states?

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