Functions with operator valued arguments acting on eigenstates

qtm912

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 :

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) )?

Matterwave

Functions of operators are defined from the power series for those functions. So, because H|E>=E|E>, and H^2|E>=H(E|E>)=E(H|E>)=E^2|E>, etc., as long as these power series converge, then you will have f(H)|E>=f(E)|E>.

qtm912

Thank you matterwave for the clear explanation

torquil

Thus, if the operator has eigenvalues that exceed the convergence radius of the expansion, problems will arise.

qtm912

Yes, makes sense, thanks for pointing it out torquil.