Why the reflection and transmission?

[tim_lou]

in solving the time-dependent Schrödinger's equation for the delta potential, one obtain a set of non-normalizable solutions.

form the boundary condition and comparing the coefficients of the solution, one obtains the probability of transmission and reflection.

However, how can one be sure that such events occur in a mathematical standpoint? suppose one has a wave traveling from -infinity (a time-dependent localized wave packet that solves the time-dependent Schrödinger's equation), how does one prove that after a very long time (as time approaches infinity),

the integral:
$$\lim_{t\rightarrow +\infty}\int_{-\infty}^0\left|\Psi(x,t)\right|^2dx$$

is the reflection coefficient?

 

[tim_lou]

hmmm, I think I see it now, the probability current can give me some hand waving argument to this...

But is there a more rigorous way (mathematically), perhaps something involving the general form of a wave packet?

 

[Meir Achuz]

For a physicist, the probability that a particle moving to the right has not passed go in an infinite time defines R. A mathemetician would need an existence proof using epsilon and delta.