(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

2. Relevant equations

The wave equations and their derivatives:

[tex]\psi_0(x) = A sin(k_0 x) + B cos( k_0 x)[/tex]

[tex]\psi_1(x) = C sin(k_1 x) + D cos( k_1 x)[/tex]

[tex]\psi_0'(x) = A k_0 cos(k_0 x) - B k_0 sin(k_0 x)[/tex]

[tex]\psi_1'(x) = C k_1 cos(k_1 x) - D k_1 sin(k_1 x)[/tex]

Plus values for k:

[tex]k_0 = \sqrt{\frac{2m}{\hbar^2} (7eV)}[/tex]

[tex]k_1 = \sqrt{\frac{2m}{\hbar^2} (1eV)}[/tex]

3. The attempt at a solution

First I solved the two wave functions evaluated at 0, which yields:

[tex]B = D[/tex]

Next, I solved the two derivatives evaluated at 0, which yields:

[tex]A k_0 = C k_1 \rightarrow C = \sqrt{7} A[/tex]

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.

[tex]\psi_0(x) = A' e^{i k_0 x} + B' e^{-i k_0 x}[/tex]

[tex]\psi_1(x) = C' e^{i k_1 x} + D' e^{-i k_1 x}[/tex]

[tex]\psi_0'(x) = i k_0 A' e^{i k_0 x} + i k_0 B' e^{-i k_0 x}[/tex]

[tex]\psi_1'(x) = i k_1 C' e^{i k_1 x} + i k_1 D' e^{-i k_1 x}[/tex]

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

[tex]\psi_0'(0) = \psi_1'(0) \rightarrow i k_0 A' + i k_0 B' = i k_1 C'[/tex]

[tex]\psi_0(0) = \psi_1(0) \rightarrow A' + B' = C'[/tex]

[tex]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')[/tex]

[tex]i k_0 = i k_1[/tex]

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

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# Penetration of an energy step when KE > U

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