Solving Reflection Coefficient for Step Potential: Why B ≠ 0?

[Thyestean]

Homework Statement

I'm working through a step potential and I am confused as to why one of the coefficients doesn't go to zero.

V(x) = 0 when x < 0;
V(x) = V_not when x > 0;

a. Calculate reflection coefficient when E < V_not
I can solve the reflection part, it is a step towards it that i am confused about.

Homework Equations

The Attempt at a Solution

Now i know the solutions solve to:

Aexp(ikx)+Bexp(-ikx) where k= sqrt(2mE)/hbar when x<0
Cexp(lx) where l=sqrt(-2m(E-V_not)/hbar when x>0

So my question is why doesn't B=0? Because when x->-infinity it goes to infinity so B has to be 0. The only reason i can think it wouldn't is because of tunneling. If this is the case how do I spot this. Is it only relevant in step potentials?

 

[vela]

e^-ikx is oscillatory. It doesn't blow up as x goes to -∞.

 

[Thyestean]

Ah thank you. Now that brings up another question for the same problem but now E>Vo.

The wave equations go to:
Aexp(ikx) + Bexp(-ikx) when x < 0. k=sqrt(2mE)/hbar
Cexp(ilx) + Dexp(-ilx) when x > 0. l=sqrt(2m(E-Vo))/hbar

Now in this case why does D=0.

 

[vela]

It's a boundary condition essentially. The idea here is you have an incident wave coming from the left. That corresponds to the A term. When it hits the potential step, you get a reflection, the B term, and a transmitted wave, the C term. The D term would correspond to a wave traveling to the right from x=+∞. You could certainly solve a problem with D not equal to 0, but it would be a different physical situation than the one you're interested in.

 

[paranormal]

The reason B ≠ 0 is not due to tunneling, but rather due to the boundary conditions at the step potential. In this case, the potential changes abruptly from 0 to V_not at x = 0. This means that the wave function and its derivative must also be continuous at this point. This leads to the condition that A + B = C, where C is the amplitude of the wave function on the right side of the step potential. This condition is necessary for the wave function to remain continuous and well-behaved at the step potential.

Furthermore, the reflection coefficient is not just determined by the value of B, but also by the value of A. The reflection coefficient R is given by |B/A|^2. So even if B ≠ 0, the value of R can still be small if A is large.

Tunneling is a phenomenon that occurs when a particle has a non-zero probability of crossing a potential barrier, even if its energy is less than the potential barrier. In this case, the wave function will have a non-zero value on the other side of the potential barrier, indicating that the particle has tunneled through the barrier. Tunneling is not limited to step potentials and can occur in various other situations as well.

In summary, the value of B ≠ 0 in the step potential due to the boundary conditions, and the presence of tunneling can affect the overall behavior of the wave function in this system. It is important to carefully consider all factors in order to accurately solve for the reflection coefficient in this case.