Consider the Schrodinger equation with the step-function potential(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

V(x)=\begin{cases}

0, & x<0 \\

U, & x>0

\end{cases}\qquad .

[/tex]

A pulse with E>U comes in from the left with unit amplitude and undergoes partial reflection. The reflection has an amplitude (ignoring phase) that is given by the usual kinematic expression for any wave, [itex]R=(v_2-v_1)/(v_2+v_1)[/itex], where v_{1}is the velocity for x<0 and v_{2}for x>0.

Since the right-hand side is classically allowed, there should be a classical limit as [itex]h\rightarrow0[/itex] in which R approaches zero. If you look at a sampling of presentations of this, what you seem to find over and over is a swindle in which [itex]E\gg U[/itex] is identified as the classical limit. This is just plain wrong.

What's really going on is that the classical limit is one in which the wavelength of the particle gets short, but for the idealized step function there is no scale with which we can compare in order to say what "short" means. When you change the discontinuous step into a ramp with width w, you get the correct classical limiting behavior in which R approaches 0 as the wavelength gets small compared to w (Branson 1979).

What I'm looking for now is a nicer pedagogical demonstration of this than I've been able to come up with so far.

Branson takes the brute-force approach of solving the Schrodinger equation exactly in terms of Airy functions. Although only the first page of Branson is available online, Vern 2006 lays out all the gory details for an almost identical situation. Applying it to the ramp potential and then taking the classical limit seems like a mess.

I wrote up a very non-rigorous treatment here: http://www.lightandmatter.com/html_books/0sn/ch13/ch13.html#Section13.3 [Broken] (see the example titled "The correspondence principle for E>U"). It's actually probably about right for the majority of the students I teach, but I'm not completely happy with the treatment of the limiting process. I think it's clear that R approaches zero for a potential consisting of n tiny steps, as n approaches infinity, if each step has a width that's large compared to the width of the pulse. The step after that is one that I feel is too sloppy to satisfy me.

One approach is to try to apply approximations to the exact solution in terms of Airy functions. The exact solution of the Schrodinger equation for [itex]V=\alpha x[/itex] is of the form [itex]\Psi=c_1 Ai(u)+c_2 Bi(u)[/itex], where [itex]u=(-\alpha)^{-2/3}b(\alpha x-E)[/itex] and [itex]b=(2m/(\hbar)^2)^{1/3}[/itex]. In the classical limit, b is large, and if we're in the classically allowed region, then u is large and negative. In this limit, we have [itex]Ai(u) \approx \pi^{-1/2} (-u)^{-1/4}\sin(2/3 (-u)^{3/2}+\pi/4)[/itex], and Bi is approximated by the same expression but with a cosine. I think one can argue that since u is large, these functions are well approximated by a sine and a cosine, and if the sine and cosine form a basis for the set of solutions, that implies that the solution is well approximated by a free wave, so there's no reflection. This seems like it would work if the details of the limiting process were filled in carefully, and it would be somewhat less grotty than Branson's approach, but it still seems like it wouldn't work well for lower-division students who have never heard of Airy functions.

Is there an approach that has some decent level of rigor and yet will not come off as voodoo to college sophomores who aren't physics majors?

A related issue is the rigorous justification for the WKB procedure of integrating only over the classically forbidden region; actually if the classically allowed region has abrupt steps, this gives the wrong answer, because you get partial reflection at those steps.

D. Branson. 'The correspondence principle and scattering from potential steps', American Journal of Physics, Vol.47, 1101-1102, 1979. First page available at http://www.deepdyve.com/lp/american...-scattering-from-potential-steps-tKM85ATfDZ/1

Vern, "Airy wave packets as quantum solutions for recovering classical trajectories," BYU senior thesis, 2006, http://www.physics.byu.edu/faculty/vanhuele/Research/VernThesis.pdf

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# Correspondence principle for a step potential

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