602 BELL SYSTEM TECHNICAL JOURNAL 



So far as the region to the right of the hill {x > Xa) is concerned, a 

 solution having all of these qualities is the following: 



<l, = A le-^^-V^ + A^e+'^-<^, k = J^^ , 



(22) 



in which the two terms stand for wave-trains I and II, and Ai and A2 

 are adjustable constants. So far as the region to the left of the hill 

 (x < Xi) is concerned, a solution having all of the required qualities is 

 the following: 



^ = Cie-^'^^'^. (23) 



It stands for wave-train III and Ci is an adjustable constant. As I 

 have already said, our "intuition" based on notions of what particles 

 should do, expects Ci to vanish and ^2 to become equal to Ai when E 

 is less than neVm, but the solution of (21) does not consent to these 

 limitations. Our intuition also expects that when E is less than neVm 

 nothing will happen in the region comprised within the hill (xi<x<X2), 

 but here again the solution of (21) does not conform with it. For 

 in this region comprised within the hill, the solution must take the form : 



^ = 5^g-^x^„.r,„-£ ^ 526+^^^^"^^'"-^ (24) 



which looks at first glance like (22) but is essentially different, since 

 the exponents are real and not imaginary, and the terms do not repre- 

 sent progressive waves. The five coefficients — Ai, A2, Bi, B2, Ci — 

 must now be mutually adjusted so that at the sides of the hill (x = Xi 

 and X — X2) the expressions (22) and (23) and (24) flow smoothly each 

 into the next, with no discontinuity either of ordinate or of slope. 

 This imposes four conditions on the five coefficients, and therefore 

 fixes the relative values of all of them — in other words, determines 

 them completely except for a common arbitrary factor which corre- 

 sponds to the intensity of the incident beam, and is irrelevant to the 

 course of the argument. 



In particular, this requirement of continuity imposed by the funda- 

 mental principles of quantum mechanics upon the acceptable solution 

 of (21) fixes the ratio of the amplitudes d and Ai of "transmitted" 

 and "incident" wave-train. From Gurney and Condon I quote an 

 approximate formula -^ for the ratio of the squares of these amplitudes, 

 denoting them on the left by the customary symbols: 



^7^2T^' = ^^^^ ^'^P- ^- (4^«//0V2m(iVeF,„-£)], 



V^^ Jinc. i^2S) 



4>{E) = \6{ElneV^){\ - E/neVm). 

 ^^ Exact formula given by E. U. Condon, Reviews of Modern Physics, 3, 57 (1931). 



