ELEMENTARY PARTICLE — SCHRODINGER 195 



intermediate region. This idea is not entirely wrong, but it is also 

 far from correct. One may remember the interference patterns re- 

 ferred to in section 2 in evidence of the wave nature of the electron. 

 They can be obtained w ith an arbitrarily faint bundle of cathode rays, 

 provided the exposure is prolonged. Thus a typical wave phenomenon 

 is produced here, irrespective of crowding. Another instance is this. 

 xV competent theoretical investigation of the collision of two particles, 

 whether of the same or of different kind, has to take account of their 

 wave nature. The results are duly applied to the collisions of cosmic- 

 ray particles w^ith atomic nuclei in the atmosphere, both being ex- 

 tremely rarefied in every sense of the word. But perhaps this is triv- 

 ial; it only means that even an isolated particle, which gives us the 

 illusion of transitory individuality, must yet not be likened to a classi- 

 cal particle. It remains subject to the uncertainty relation, of which 

 the only tolerable image is the guiding wave group. 



13. THE CONDITION FOR THE PARTICLE ASPECT 



The following is the quantitative condition for strings to develop 

 which counterfeit individuals and suggest the particle aspect: the 

 product of the momentum f and the average distance I between neigh- 

 boring particles must be fairly large compared with Planck's constant 

 A; thus 



Vl» h. 

 (The momentum j) — and not the velocity — is the thing we should 

 really have referred to when, in sections 4 and 5, we dealt with the 

 uncertainty relation ; p is simply the product of the mass and the ve- 

 locity, unless the latter is comparable with that of light.) 



A large I means a low density in ordinary space. What matters, 

 however, is the density in the manifold of states — or phase space, to 

 use the technical term. That is why the momentum p comes in. It is 

 gratifying to remember that those very obvious strings — visible tracks 

 in the cloud chamber or in the photographic emulsion, and simul- 

 taneous discharges of alined counters — are all produced by particles 

 with comparatively very large momentum. 



The above relation is familiar from the theory of gases, where it 

 expresses the condition which must be fulfilled in very good approxi- 

 mation in order that the old classical particle theory of gases should 

 apply in very good approximation. This theory has to be modified 

 according to quantum theory when the temperature is very low and at 

 the same time the density very high, so that the product pi is no longer 

 very large compared with h. This modification is called the theory 

 of degenerate gases, of which the most famous application is that by 

 A. Sommerfeld to the electrons inside a metal; we have mentioned 

 them before as an instance of extreme crowding. 



