Penetration of an energy step when KE > U

[OEP]

Homework Statement

An electron with KE = 8eV is incident on the potential step of height 7eV. What's the probability the electron will be reflected from the step function?

Homework Equations

The wave equations and their derivatives:

$$\psi_0(x) = A sin(k_0 x) + B cos( k_0 x)$$
$$\psi_1(x) = C sin(k_1 x) + D cos( k_1 x)$$
$$\psi_0'(x) = A k_0 cos(k_0 x) - B k_0 sin(k_0 x)$$
$$\psi_1'(x) = C k_1 cos(k_1 x) - D k_1 sin(k_1 x)$$

Plus values for k:

$$k_0 = \sqrt{\frac{2m}{\hbar^2} (7eV)}$$
$$k_1 = \sqrt{\frac{2m}{\hbar^2} (1eV)}$$

The Attempt at a Solution

First I solved the two wave functions evaluated at 0, which yields:
$$B = D$$

Next, I solved the two derivatives evaluated at 0, which yields:
$$A k_0 = C k_1 \rightarrow C = \sqrt{7} A$$

From here I'm lost.

I also did the same thing with complex exponentials... but I think I ended up with a nonsensical k_0 = sqrt(7) k_0.

$$\psi_0(x) = A' e^{i k_0 x} + B' e^{-i k_0 x}$$
$$\psi_1(x) = C' e^{i k_1 x} + D' e^{-i k_1 x}$$

$$\psi_0'(x) = i k_0 A' e^{i k_0 x} + i k_0 B' e^{-i k_0 x}$$
$$\psi_1'(x) = i k_1 C' e^{i k_1 x} + i k_1 D' e^{-i k_1 x}$$

D' = 0, we are firing particles from right to left, not left to right.

$$\psi_0'(0) = \psi_1'(0) \rightarrow i k_0 A' + i k_0 B' = i k_1 C'$$
$$\psi_0(0) = \psi_1(0) \rightarrow A' + B' = C'$$

$$i k_0 A' + i k_0 B' = i k_1 C' \rightarrow i k_0 A' + i k_0 B' = i k_1 (A' + B')$$
$$i k_0 = i k_1$$

Which is nonsensical. Any ideas? This seems like one of those good ol' problems that you first learn to solve, but... our professor is just a bit zany and I can't follow him. :P

 

[lanedance]

I prefer the exponential notation as it is clear what is moving right & left

you have missed a negative in you differentiation

should get
A + B = C

and
k0(A-B) = k1C

then solve these for reflection amplitude is r = A/B, probability r^2

 

[OEP]

Thanks a lot! I guess it's being so over my head that I tend to overlook the smaller things... I have R being at 2.64 which I don't think is right for its square being a probability. We aren't looking for B / A are we?

 

[lanedance]

i know what you mean -

& yep, you're right B/A should do it

 

[OEP]

Thanks again. I'm off to trudge through the rain and turn this sucker in.