I want to obtain the the solutions to the time independent Schrödinger Equation for a barrier potential. I have created three equations:
tise1 = -\[HBar]^2/2 m D[\[Psi]1[x], {x, 2}] == e0 \[Psi]1[x]
tise2 = -\[HBar]^2/2 m D[\[Psi]2[x], {x, 2}] == (e0 - v0) \[Psi]2[x]
tise3 = -\[HBar]^2/2 m D[\[Psi]3[x], {x, 2}] == e0 \[Psi]3[x]
and the boundary conditions
bc = {\[Psi]1[0] == \[Psi]2[0], \[Psi]2[a] == \[Psi]3[a],
D[\[Psi]1[x], x] == D[\[Psi]2[x], x] /. x -> 0,
D[\[Psi]2[x], x] == D[\[Psi]3[x], x] /. x -> a}; ic = {}
(I intended to use ic to remove "left moving" waves right of the potential (x>a), but I could no figure out how to do this)
Next I used
sol = DSolve[
Join[{tise1, tise2, tise3}, bc, ic], {\[Psi]1, \[Psi]2, \[Psi]3},
x]
But the results are overwhelming and as far as I can tell: wrong. I would like to be able to get the wave function as an piecewise concatenation of psi 1-3 to add the time development and to calculate transmission and reflection rates.
Hopefully this makes sense. Any help (or pointers to such) will be appreciated.
/Mogens