- #1

fluidistic

Gold Member

- 3,857

- 188

## Homework Statement

I'll try to recreate from my memory the problem we've been assigned on a test more than one month ago. They gave the solution but I either misunderstood or miscopied it.

An electron with kinetic energy 5 eV goes from a region with potential [itex]V_0=6 eV[/itex] (let's call this region I) to a region with potential 0 (let's call this region II). Calculate the coefficient of transmission.

## Homework Equations

The professor said we didn't need to have the explicit formula for the transmission. Rather we should write the expression of a plane wave (I guess she meant standing wave) and use the formula of probability current with j that follows.

[tex]j= \frac{1}{2im} \left ( \Psi ^* \frac{\partial \Psi }{\partial x} - \Psi \frac{\partial \Psi ^* }{\partial x} \right )[/tex]

With the [itex]\Psi _I[/itex] of region I, this gives [itex]j _{\text {incident} }+ j_{\text {reflected} }[/itex] and for region II this gives [itex]j_ \text {transmitted} [/itex].

Here is my problem. The solution she gave was like [itex]\Psi _I (x)=Ae^{ik_1x}+Be^{ik_2x}[/itex] and [itex]\Psi _{II}(x)=Ce^{i k_2 x}[/itex] and that we should get [itex]0.14[/itex] for the coefficient of transmission.

## The Attempt at a Solution

So I tried to get [itex]\Psi _I (x)[/itex] but I don't get the same function at all. I get [itex]\Psi _I (x) =Ae^{k_1 x}+Be^{-k_1 x}[/itex] where [itex]k_1 =\sqrt { \frac{2m (v_0 -E)}{\hbar ^2 } }[/itex].

And even more than that, I'm almost sure that B must be worh 0, otherwise psi diverges when x tends to - infinity.

Am I right on this?!