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fluidistic
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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?!