- #1
whatisreality
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Homework Statement
There is a stream of electrons with energy E, incident from x = -∞ on a potential step such that V(x) = ##V_{0}## for x<0 and 0 for x>0.
E>##V_{0}##>0.
Write the T.I.S.E for x<0 and x>0 and find the general solution for both.
Homework Equations
The Attempt at a Solution
My main problem is I don't know how to account for the potential dropping, as it does here - in other examples, the particles always seem to start in the region of 0 potential and hit an area with potential V. So if an adaptation has to be made, I'm not sure how to do it. Here's my attempt for x<0:
##-\frac{\hbar^2}{2m} \frac{\partial^2{\psi}}{\partial{x^2}} + V_{0}\psi = E\psi##, I think...If that's it, then the general solution is ##Ae^{-kx}+Be^{kx}##. I know more can be done to that, I think I read something about how this wavefunction would basically always be infinite so you set B = 0. Or maybe it was A...
And for x>0, I think
##-\frac{\hbar^2}{2m} \frac{\partial^2{\psi}}{\partial{x^2}} = E \psi##.
For this, the solution would be ##Ae^{ikx}+Be^{-ikx}##.
There are so many variations on this question, I feel like it's really important to understand how to adapt the Schrodinger equation! I'd hugely appreciate some general guidelines for different cases, for example E<V and E>V, stepping to lower or higher potential, that kind of thing. Because I don't really understand the changes made for each situation.
Of course, if these sorts of guidelines aren't really possible I'd still really appreciate help just on this specific question! :)