Triangular Potential Well

[TMSxPhyFor]

Homework Statement

find Transmission and Reflection coefficients (QM) for the following triangular potential well:

U=U_{0}(1-\frac{x}{a}) : x\geq0 and U= 0 : x<0

and U_{0}>0 , a>0

Homework Equations

The Attempt at a Solution

Basically constructing the wave functions is very easy (from the left side of the plane we have flat de'broel wave, from the right side we have a solution of superposition of Airy's functions of the first and second kinds), but then the problems arise:

1- first problem that i can't understand, is when our wave function is super position (coefficients still maybe complex) purely real, the current will be anyway equal to zero (in case of right side with Airy functions), this mean that our particle will reflect with 100% probability, while the wave function still not zero at the right side (what means that the particle may be discovered there), how I should understand that? all of the books I checked keeps silent regarding this case.

2- the general solution has 3 coefficients (1 in the flat incoming & reflecting flat wave from the left and 2 for Airyi's right solution) and i have only two boundary conditions to combine the two solutions at x=0, and please note that in case of this problem divergence of Bi is not a threat here and the both functions are bounded, so there is no other boundary conditions that can be used, and we can't suppose that our wave function is asymptotically flat at x->\infinity becuase at that point interaction of the field with the particle is not zero as usually supposed (not so physical but anyway..)

I couldn't find a similar problem anywhere, except that a similar thing happens in studding semiconductors as is written here :http://www.iue.tuwien.ac.at/phd/gehring/node47.html#s:gundlach that mentioned some "GUNDLACH Method" that i couldn't find in other references to it, plus as i understood the suppose additional boundary condition not available for this problem.

 

[member 553083]

Finally, I can't understand what is the exact relation between the Transmission/Reflection coefficients and the wave functions, and how to combine the two solutions without additional boundary condition.I will be thankful for any help.