What is the Wavefunction for Transmission Resonances in a 1-D Potential Barrier?

[sachi]

We have a 1-d potential barrier,
V(x)=0 for x<0,x>a
V(x)=V1 for 0<=x<=a
we are considering the case where E>V1. we consider the case when transmission resonance occurs i.e when there is no reflected wave. this is when
[2m/(hbar^2) * (E-V1)]^0.5 = n*Pi/a
where n is an integer
we are then asked to describe what the wavefunction looks like in the region 0<=x<=a when transmission resonance occurs. we have already found the wavefunction in this region to be
psi = Aexp(Kx) + Bexp(-Kx)
where K = [2m/(hbar^2) * (V1-E)]^0.5
we know that K is imaginary as V1<E, which would suggest that we had sinusoidal wavefunctions. However, we also find that both A and B are themselves complex! I'm very confused about this situation. I think we essentially have complex valued sinusoids in this region. any suggestions would be greatly appreciated.

Sachi

 

[Physics Monkey]

The complex exponentials have a definite interpretation in terms of incoming and outgoing waves. There is nothing wrong in general with having complex coeffecients multiplying these plane waves. If you want to convert to sines and cosines then in general they will have complex coeffecients too, but this is also perfectly ok.