- #1
Emspak
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Homework Statement
The task is to calculate the transmission probability when E<V0 given that the step potential barrier is of infinite length.
Homework Equations
So far it seems simple enough. For region I, where x<0, (we're assuming, as usual, that the particles/ waves are coming from the left) I had
[itex] \phi''_I = \frac{2mE}{\hbar^2}\phi[/itex] and [itex] \phi''_{II} = \frac{2m(E-V_0)}{\hbar^2}\phi[/itex]
and we will let [itex]\frac{2mE}{\hbar^2} = k_1^2[/itex] and [itex]\frac{2m(E-V_0)}{\hbar^2} = k_2^2[/itex]
The Attempt at a Solution
SO I go through the whole process of solving the DQ.
[itex] \phi_I = Ae^{ik_1x} + Be^{-ik_1x}[/itex]
[itex] \phi_{II} = Ce^{ik_2x} + De^{-ik_2x}[/itex]
[itex] \phi_I' = ik_1Ae^{ik_1x} -ik_1Be^{-ik_1x}[/itex]
[itex] \phi_{II}' = ik_2Ce^{ik_2x} -ik_2De^{-ik_2x}[/itex]
The two first derivatives are also equal at x=0, which sets another boundary condition.
D=0 here because the particles are all moving from the left to right and there isn't anything coming from the right.
That gets me
[itex]ik_1A -ik_1B = ik_2C \rightarrow k_1(-A + B) = k_2C \rightarrow \frac{C}{A} = \frac{2k_1}{k_1+k_2}[/itex]
and the transmission probability is
[itex]\frac{|C|^2}{|A|^2} = \frac{|2k_1|^2}{|k_1+k_2|^2}[/itex]
So far so good, I think. But then I noticed something. When I plugged in my k1 and k2 this happened:
[tex]\frac{|2k_1|^2}{|k_1+k_2|^2}=\frac{ \frac{8mE}{\hbar^2}}{\frac{2mE}{\hbar^2} +2 \left(\frac{\sqrt{2mE}}{\hbar} \right) \left(\frac{\sqrt{2m(E-V_0)}}{\hbar}\right)+\frac{2m(E-V_0)}{\hbar^2}}=\frac{4E}{2E+2\sqrt{2E}\sqrt{(E-V_0)}-V_0}[/tex]
Anyhow I was just curious if I did this right because what was interesting was that you get an imaginary component in there but I could make it into a complex conjugate. (At least the first two terms). And I was interested in whether I should do that to get a real coefficient. I suppose it's just algebra, but I wanted to make sure I did this right from the get go.
thanks in advance.