For , the solution is
There are two unknown coefficients and which will be determined by matching boundary conditions. We will not require normalization to one particle, since we have a beam with definite momentum, which cannot be so normalized. (A more physical problem to solve would use an incoming wave packet with a spread in momentum.)
Continuity of the wave function at
Continuity of the derivative of the wavefunction at
The coefficients are
We now have the full solution, given our assumption of particles incident from the left.
Classically, all of the particles would be transmitted, continuing on to infinity.
In Quantum Mechanics, some probability is reflected.
If we wish to compute the transmission probability, the easy way to do it is to say that its
We'll get the same answers for the reflection and transmission coefficients using the probability flux to solve the problem.
The transmission probability goes to 1 one (since there is no step). The transmission probability goes to 0 for (since the kinetic energy is zero).
Jim Branson 2013-04-22