Correspondence principle for a step potential

[bcrowell]

Consider the Schrodinger equation with the step-function potential

[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 v₁ is the velocity for x<0 and v₂ 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 (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

 

[bcrowell]

I thought I was getting somewhere based on a presentation of the WKB approximation by Hayes (link below), but I couldn't quite get it to work. Hayes treats the classically allowed region by writing the wavefunction as [itex]\Psi=A e^{i\phi}[/itex]. The time-independent Schrodinger equation then becomes the following two equations:

[tex] A''/A-\phi'^2+p^2/\hbar^2=0 \quad \text{[1]}[/tex]

[tex] A^2\phi'=\text{const} \quad \text{[2]}[/tex]

In [1], the classical limit can be viewed as either a short-wavelength limit, which makes [itex]\phi'^2[/itex] approach infinity, or as the limit [itex]\hbar\rightarrow0[/itex], which makes [itex]p^2/\hbar^2[/itex] approach infinity. Therefore the first term in [1] becomes negligible, and we have [itex]\phi'=\pm p/\hbar[/itex].

Although Hayes doesn't say so, [2] is simply an expression for the probability current j. It has to be a constant because this is the time-independent Schrodinger equation, and the continuity equation for probability is [itex]\partial\rho/\partial t+\partial j/\partial x[/itex].

So although there are some clear links between my problem and the treatment of the classically allowed region in the WKB approximation, it seems that they don't quite link up nicely, because in the WKB approximation you can get away with using the time-independent Schrodinger equation, whereas for my problem you can't.

Hayes, Phys 4100 lecture notes, http://www.rpi.edu/dept/phys/courses/phys410/main.html

 

[DrClaude]

Have a look at Walter Appel's treatment of the problem:

http://press.princeton.edu/chapters/s8452.pdf

section 1.1.c

 

[bcrowell]

Have a look at Walter Appel's treatment of the problem:

http://press.princeton.edu/chapters/s8452.pdf

section 1.1.c

Aha, nice! He uses a different form for the potential, which happens to result in a simple expression for the reflected amplitude. Unfortunately the actual calculation is given in reference 58, and I can't tell what reference 58 refers to.

But maybe that's beside the point. To demonstrate that the equivalence principle holds, we really need to prove that as long as V(x) is in some sense sufficiently well behaved, you don't get any reflection from a classically allowed region in the classical limit [itex]\lambda\rightarrow0[/itex]. I don't know what "sufficiently well behaved" means ... maybe it's enough that it's continuously differentiable (C¹)?

Of course, under a sufficiently powerful microscope, all well-behaved functions look the same. If you want to approximate such a function as a series of ramps (piecewise linear), or as pieces of the form used by Appel, you can. In fact, approximating your function with a series of steps would be just as valid, and certainly simpler analytically. But it's not obvious to me how to translate this into a more rigorous argument. Your incoming wave packet is always going to have some finite width to it, and for instance if you break down your smooth function into smaller and smaller ramps, there will come a point at which the ramps are smaller than the wave packet.

 

[DrClaude]

Unfortunately the actual calculation is given in reference 58, and I can't tell what reference 58 refers to.

I don't have the book with me right now. I'll post the reference when I'll be back in my office.

 

[DrClaude]

Ref. 58 is Landau and Lifgarbagez, Quantum Mechanics.

 

[bcrowell]

Ref. 58 is Landau and Lifgarbagez, Quantum Mechanics.

Thanks!