- #1

erkokite

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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:

[tex]<\Psi_{2}|exp(-i\hat{H}(t2-t1))|\Psi_{1}>[/tex]

This is of course equal to

[tex]\int\Psi_{2}*exp(-i\hat{H}(t2-t1))\Psi_{1}dx[/tex]

However, how do I evaluate the following operation?

[tex]exp(\hat{H})\Psi_{1}[/tex]

Many thanks.