The Scattering of Particles in One Dimension

We have seen earlier that the exponentials having positive arguments correspond to particles traveling from left to right, etc. We signify this with arrows in Fig. 2-17a. 

The ratio A A/B B is the relative fluxes of particles traveling toward the right or left in the region to the left of the step. If 1 = 151, the fluxes are equal, corresponding to total reflection of the beam from the step potential. 

It is not difficult to show (Problem 2-24) that A = B whenever i/qeft has real value and slope at any point, i.e., for any trigonometric wave, and so the potential of Fig. 2-17a gives total reflection if E U. (The fact that some particle density exists at X 0 due to barrier penetration does not affect this conclusion. The evaluation of 

Chapter 2 Quantum Mechanics of Some Simple Systems 

This can be seen to range from zero, when k = k,io one, when = 0. k = k when E = E—U,i-Q-, when U is negligible compared to E. So zero reflection (total transmission) is approached in the high-energy limit. Only when E = U does =0, so total reflection occurs only when E equals the barrier height (or, as we saw previously, is lower). A plot of the fraction of particles reflected versus E/U appears in Fig. 2-17c. The transmission, equal to 1.0-reflection, is plotted in Fig. 2-17d. 

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