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Could someone explain how the following two definitions of the displacement operator are equal? The first is the standard one

1) [itex]e^{\alpha a^{\dagger}-\alpha*a}[/itex]

But what about this one? This is from a Fock state decomposition of a coherent state.

2) [itex]e^{\frac{-|\alpha^{2}|}{2}}e^{\alpha a^{\dagger}}[/itex]

What is the reasoning behind not using the second equation to define the operator that acts on a vacuum state to produce a coherent state? Or if it is fine to use definition 2), then can someone help me show that the two definitions are equivalent?

Thank you

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# Displacement Operator definition

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