- #1
tim_lou
- 682
- 1
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 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?
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?