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Functions with operator valued arguments acting on eigenstates

  1. Jan 25, 2012 #1
    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) )?
     
  2. jcsd
  3. Jan 25, 2012 #2

    Matterwave

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    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>.
     
  4. Jan 26, 2012 #3
    Thank you matterwave for the clear explanation
     
  5. Jan 26, 2012 #4
    Thus, if the operator has eigenvalues that exceed the convergence radius of the expansion, problems will arise.
     
  6. Jan 28, 2012 #5
    Yes, makes sense, thanks for pointing it out torquil.
     
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