This question concerns the outcome when operator valued functions act on an energy eigenstate. Given an eigenstate at t =0, say |Ej > , I have seen or inferred in some of the literature that the following applies :(adsbygoogle = window.adsbygoogle || []).push({});

exp(-iHt/h) |Ej > = exp(- iEj t/h) |Ej >

Where h = h-bar

Ej is energy eigenstate j

H is the Hamiltonian

I am unclear in particular why we can say that if we apply a function with an operator as the argument to this energy eigenstate, it will return the function with the eigenvalue as the argument times the energy eigenstate. (assuming I have inferred what is in the notes correctly)

Would this then be a general result for all functions or do only certain functions satisfy this relationship. (So for example would it be true for sin(H) instead of exp(-iHt/h) )?

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# Functions with operator valued arguments acting on eigenstates

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