Simple Nature - Light and Matter

The correspondence principle for E > U example 19The correspondence principle demands that in the classical limith → 0, we recover the correct result for a particle encounteringa barrier U, for both E < U and E > U. The E < U casewas analyzed in self-check H on p. 870. In the remainder of thisexample, we analyze E > U, which turns out to be a little trickier.The particle has enough energy to get over the barrier, and theclassical result is that it continues forward at a different speed (areduced speed if U > 0, or an increased one if U < 0), then regainsits original speed as it emerges from the other side. Whathappens quantum-mechanically in this case? We would like toget a “tunneling” probability of 1 in the classical limit. The expressionderived on p. 870, however, doesn’t apply here, because itwas derived under the assumption that the wavefunction insidethe barrier was an exponential; in the classically allowed case,the barrier isn’t classically forbidden, and the wavefunction insideit is a sine wave.o / A particle encounters astep of height U < E in theinteraction energy. Both sides areclassically allowed. A reflectedwave exists, but is not shown inthe figure.We can simplify things a little by letting the width w of the barriergo to infinity. Classically, after all, there is no possibility that theparticle will turn around, no matter how wide the barrier. We thenhave the situation shown in figure o. The analysis is the sameas for any other wave being partially reflected at the boundarybetween two regions where its velocity differs, and the result isthe same as the one found on p. 367. The ratio of the amplitudeof the reflected wave to that of the incident wave is R = (v 2 −v 1 )/(v 2 +v 1 ). The probability of reflection is R 2 . (Counterintuitively,R 2 is nonzero even if U < 0, i.e., v 2 > v 1 .)This seems to violate the correspondence principle. There is nom or h anywhere in the result, so we seem to have the result that,even classically, the marble in figure p can be reflected!p / The marble has zero probabilityof being reflected from theedge of the table. (This examplehas U < 0, not U > 0 as infigures o and q).The solution to this paradox is that the step in figure o was takento be completely abrupt — an idealized mathematical discontinuity.Suppose we make the transition a little more gradual, as infigure q. As shown in problem 17 on p. 380, this reduces the amplitudewith which a wave is reflected. By smoothing out the stepmore and more, we continue to reduce the probability of reflection,until finally we arrive at a barrier shaped like a smooth ramp.More detailed calculations show that this results in zero reflectionin the limit where the width of the ramp is large compared to thewavelength.q / Making the step more gradualreduces the probability ofreflection.Section 13.3 Matter As a Wave 871