Defining the state [itex]| \alpha > [/itex] such that:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]| \alpha > = Ce^{\alpha {\hat{a}}^{\dagger}} | 0 >\ ,\ C \in \mathbf{R};\ \alpha \in \mathbf{C};[/tex]

Now, [itex]| \alpha >[/itex] is an eigenstate of the lowering operator [itex]\hat{a}[/itex], isn't it?

In other words, the statement that [itex] \hat{a} | \alpha >\ =\ \alpha | \alpha > [/itex] is true, right?

(for the eigenvalue [itex]\alpha[/itex]?)

Is that the correct use of the Dirac notation?

How might one go about proving the above?

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# Harmonic Oscillator, Ladder Operators, and Dirac notation

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