Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Probability Amplitude and Time Evolution Operator

  1. Feb 9, 2009 #1

    I am fairly new to quantum physics (I'm actually an engineer, not a physicist). I think I am getting a decent grasp on things, but I have a question.

    Suppose you have two time dependent states: [tex]\Psi_{1}[/tex] and [tex]\Psi_{2}[/tex].

    Also, suppose that we have a constant potential, represented in our Hamiltonian as V.

    Our Hamiltonian is thus represented as: [tex]\hat{H}=1/(2m)\partial^2/x^2+V(x)[/tex]

    Now suppose that we want to find the probability amplitude to go from state 1 to the 2nd state in a time interval t1 to t2.

    It is my understanding that the following operation is used:


    This is of course equal to


    However, how do I evaluate the following operation?


    Many thanks.
  2. jcsd
  3. Feb 10, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    The general idea of evaluating a function of an operator is to expand the function...

    exp(H) = 1 + H + H^2/2 + H^3/6 + ...

    If Psi is an eigenfunction of H with eigenvalue E, then it is not hard to show that exp(H) Psi = exp(E) Psi.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Probability Amplitude and Time Evolution Operator