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Suppose a particle has energy $E>V(+/-\infty)=0$, then the solutions to the Schrodinger equation outside of the potential will be $\psi(x)=Ae^{i k x}+Be^{-i k x}$. How can one show or explain that $|B|/|A|$ gives the probability that a particle scattering off the potential is reflected?

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  • $\begingroup$ You can either argue with probability currents (though this was always confusing to me). The nice way is to construct wave-packets with the scattering solutions of a small range of momenta and then calculate what is the probability of the particle to be scattered. $\endgroup$ – Fabian Aug 31 '12 at 5:32
  • $\begingroup$ Can you explain more? If I start with a gaussian wavepacket, how do I get it to travel? If I apply the schrodinger equation to find its time evolution, it just stays at the same point but flattens out, how can I input its initial speed? $\endgroup$ – Hobo Sep 2 '12 at 15:55
  • $\begingroup$ @Hobo: work with the momentum first. Try something like $\int dk A\,e^{-\alpha k^{2}}e^{-i\beta kx}$, with $A, \alpha,$ and $\beta$ constant. $\endgroup$ – Jerry Schirmer Sep 2 '12 at 16:18
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    $\begingroup$ See physics.stackexchange.com/questions/12611/… $\endgroup$ – jjcale Sep 2 '12 at 16:55
  • $\begingroup$ @jjcale: voted to close as duplicate, I can't see how to answer without replicating the answer to the previous question verbatim. $\endgroup$ – Ron Maimon Sep 2 '12 at 21:48

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