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zhillyz
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Homework Statement
A one-dimensional time-independent wave-function can be written as:
[itex] \Psi = 5e^{2ix} – e^{-2ix}, (x ≤ 0) [/itex]
[itex] \Psi = 4e^{3ix}, (x > 0)[/itex]
Show that this wave-function obeys the boundary conditions, find the probability that an incident particle represented by the wave-function is reflected, and sketch the potential energy function, indicating where the total energy of the particle lies.
The attempt at a solution
First part is easy enough, proving boundary condition are obeyed means both equations will equal each other at x=0, which would give, 5-1 = 4, which is correct.
I think the probability of the incident particle reflecting concentrates on the first equation which would be the superposition of the incident wave( [itex] 5e^{2ix}[/itex] ) and reflected wave ( [itex] -e^{-2ix}[/itex] ).
The probability would be;
[itex] R = \dfrac{|B|^{2}}{|A|^{2}} = \dfrac{1^{2}}{5^{2}} = \dfrac{1}{25}[/itex]
I think the potential energy function would be the wave function approaching a barrier with higher energy, with some transmitted part of lesser amplitude and a even smaller amplitude reflecting back. Where the total energy lies, I am not really sure?
If someone would be kind enough to confirm my attempt so far and help on the last bit.
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