in solving the time-dependent Schrödinger's equation for the delta potential, one obtain a set of non-normalizable solutions.(adsbygoogle = window.adsbygoogle || []).push({});

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 Schrodinger's equation), how does one prove that after a very long time (as time approaches infinity),

the integral:

[tex]\lim_{t\rightarrow +\infty}\int_{-\infty}^0\left|\Psi(x,t)\right|^2dx[/tex]

is the reflection coefficient?

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# Why the reflection and transmission?

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