Schrodinger's Equation, Potential Energy Barrier U>E

[Oijl]

Homework Statement

Consider particles incident on a potential energy step with E<U.
(That is, a particle with total energy E travels along one dimension where U=0, then crosses, at point x=0 into a region where U>E.) (The particle is incident on the potential energy step from the negative x direction.)

Starting with the wave functions,

x < 0: Ψ0 = A’e^(ikx) + B’e^(-ikx), k = 2mE/h_bar2
x >= 0: Ψ1 = Ce^(k1x) + De^(-k1x), k1 = 2m(U-E)/h_bar2

Apply the boundary conditions for Ψ and dΨ/dx and show that the full wave intensity is reflected at the step [i.e., |A'|^2 = |B'|^2].

Homework Equations

Ψ0(x=0) = Ψ1(x=0)
dΨ0(x=0)/dx = Ψ1(x=0)/dx

The Attempt at a Solution

I set C=0, or else the wave function Ψ1 may become infinity.

The boundary conditions are stated above. They become
A' + B' = D
and
ikA' - ikB' = -k1D

How, from this, do I find that
|A'|^2 = |B'|^2
?

Thanks.

 

[thebigstar25]

you have two equations and three unknowns..

my hint for your:

write A and B in terms of D .. and i noted something in your solution (you may not need it, but i will say it anyway) when you wrote k = 2mE/h_bar2, and k1 = 2m(U-E)/h_bar2, they should actually be k^2 and k1^2 .. finally after you find A and B in terms of D find AA* (which is |A|^2) and BB* (which is |B|^2) you need to write them in that form since you will have to find a complex conjugate of both of them ..

good luck with this .. and tell us what you get .. :)

 

[Oijl]

Yay, I got it! I didn't have enough faith in algebra, was my problem. I often don't.

Thanks!

 

[thebigstar25]

:) u r welcome .. next time don't give up so fast..