Potential Step and Amplitudes

[erok81]

Homework Statement

A beam of particles of energy E and incident upon a potential step of U₀=5/4E is described by the wave function;

$$\psi(x)_{inc}=1e^{ikx}$$

Determine completely the reflected wave and the wave inside the step by enforcing the required continuity conditions to obtain their amplitudes.

Homework Equations

n/a

The Attempt at a Solution

In this case U₀ > E

My step has two three locations. (I) before the step, x=0 at the boundary, and (II) after the step.

In section (I) I'll have two plane waves, one incident and one reflected. These are given by:

ψ_I(x)=Ae^ikx+Be^-ikx

Section (II) I only have one wave function given by.

ψ_II(x)=Ce^-αx

At x=0 my two functions and their derivatives need to be continuous. So setting my plane waves and their derivatives equal at x=0 I get the two equations:

A+B=C
ik(A-B)=-αC

Where A is given to be one.

α and k are both given as √[2m(U₀-E)]/ℏ

Cancelling the α and k and setting the A as 1 I get

1+B=C
i-iB=-C

So anyway, I continue to solve for B and C and think I am done. I check the book and I am no where near close. The book shows:

$$\left(\frac{3}{5}- \frac{4}{5}i\right)e^{ikx},~\left(\frac{8}{5}- \frac{4}{5}i\right)e^{\alpha x}$$

Besides not getting the answer right, I don't see how I have an imaginary term after the step. Since it is a decaying potential not a plane wave.

Am I even approaching this correctly?

 

[nilesh_pat]

I know the book is right so if you could help me understand what I am missing it would be much appreciated. Thanks!