CONTEMPORARY ADVANCES IN PHYSICS, XXVIII 603 



This expression does not vanish suddenly as soon as E drops below 

 neVm, but falls away continuously — and very rapidly, it must be 

 admitted, owing to the exponential factor — as E diminishes from neVm 

 on downwards. Its value depends on a, the breadth of the hill (Fig. 

 14) in such a way that the broader or thicker the hill of given height 

 the less the amplitude of the transmitted waves: the thicker the hill, 

 the more nearly it comes to fulfilling the classical quality of being a 

 perfect obstacle to particles having insufficient energy to climb it! 

 I rewrite (25) in the equivalent form, 



(^^*)t, 



(^^*)ine. 



- (47r//o r 



= 4>{E) exp. [- (47r//0 ■ ^2m{NeVm - E)dx, (26) 



the integral in the exponent being taken "through the hill" from Xi 

 to X2. This form is generalizable. Take the case of Fig. 14: the ratio 

 of the squared-amplitudes of transmitted and incident wave-trains is 

 given, according to Fowler and Nordheim, by an expression which is 

 of the type (26), except that 0(-E) is a somewhat different function 

 (it is A:i{EINeVm)0~ - ElNeVm)Ji-). The distance from Xi to X2, 

 over which the integration is carried, obviously depends on E in this 

 and every other case but the particular one of Fig. 14. Take finally 

 the general case of a rounded hump, such as appears in Fig. 12. 

 According to Gamow, a formula of type (26) is approximately — not 

 exactly — valid for every such case, ^(.E) being given by him as 

 simply the number 4 when the hill descends to the same level on 

 both sides as in Fig. 14; while in the general case where the potential- 

 curve approaches different asymptotes at — co and + oo — say zero 

 at the latter, Vr ait the former — the factor 4>{E) assumes the form 

 A[EI{E — neVr)'Ji''-. Now E was the kinetic energy of the particles 

 at infinity in the direction whence they come, and {E — neVr) will 

 be the kinetic energy of the particles at infinity in the direction whither 

 they are going. We have been denoting the first of these quantities by 

 K\ denote the second by Kr, and the corresponding velocities by v and 

 Vt. Then 0(£) can be written as ^vjvr. 



The question must now be answered: what is the actual relation 

 between the ratio (^^*)trans./(^^*)inc., and the probability that a 

 particle will traverse the hill? In associating waves with corpuscles, 

 it is the rule to postulate that the square of the amplitude of the 

 waves at any point is proportional to the number-per-unit-volume 

 of corpuscles in the vicinity of that point. If one prefers to think of a 

 single particle instead of a great multitude, one may say that the square 

 of the amplitude of the waves at any point is proportional to the proba- 



