TITLE: How is it justified to compare the potential wall with a reflecting wall?
QUESTION [1 upvotes]: While solving the Schrödinger equation (in step potential questions) for the particle that would or would not cross the potential wall, we compare the solution of the differential equation with the traveling waves and while doing so we take the second coefficient=0 as there's no wall to reflect back in the other half. My question is that how is it correct to compare potential wall with reflecting surface and believe so strongly in it that we straight away declare the coefficient to be equal to 0.

REPLY [1 votes]: If one solves the equation withing the wall of a finite height, one obtains solution
$$
\psi(x) = Ae^{\kappa x} + Be^{-\kappa x}, \kappa=\frac{1}{\hbar}\sqrt{2m(V_0-E)}
$$
This is true, even if the wall is not sharp, when we look at the solution sufficiently deep inside the wall.

The first term diverges for $x\rightarrow\infty$, which means that the wave function cannot be normalized (Neither as probability density, $|\psi(x)|^2$, integrating to 1, nor even as a finite flux.) Thus, this term has to be omitted.
The other term decays inside the wall, so far away from the boundary, the electron dansity vanishes - this means that the electronw as reflected.
Another important point is that real wave function like this corresponds to zero flux, this again means that the electron was reflected.

As a reference I could recommend the chapter on the transimmison coefficient in Landau & Livshits' Quantum mechanics. Not only do they present rather general arguments, but the problems that follow the chapter contain exact solutions for a number of smooth potentials, e.g.,
$$
U(x)=\frac{U_0}{1+e^{-\alpha x}}.
$$
Another similar resource is the collection of problems by Flügge. The reason why such more realistic problems are rarely solved in basic QM courses, is that they require extensive use of special functions, which is often beyond the mathematical abilities of students and, more importantly, takes far away from the main message.
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