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I learn from the text that the exponential of an operator could be expanded with a series such that

[itex]e^{\hat{A}} = \sum_{n=0}^{\infty} \frac{\hat{A}^n}{n!}[/itex]

So if the eigenvalue of the operator [itex]\hat{A}[/itex] is given as [itex]a_i[/itex]

[itex]e^{\hat{A}}|\psi\rangle[/itex]

will be a matrix with diagonal elements given as [tex]\exp(a_i)[/tex], is that right?

So I am wondering what happen if [itex]\hat{A}[/itex] is now written as a power form, i.e. [itex]\hat{A}^n[/itex], can we conclude that

So if the eigenvalue of the operator [itex]\hat{A}[/itex] is given as [itex]a_i[/itex]

[itex]e^{\hat{A}^n}|\psi\rangle[/itex]

gives a matrix with diagonal elements as [itex]\exp(a_i^n)[/itex] ?

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# Exponential of operator

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