17cS BULL SYSTEM TECIISICAL JOURNAL 



r'.ROrP-Sl'EED AXn rORI'lSCI.K-SPEED 



Thus far I have said thai if we wish to use wave-theory and corpuscle- 

 theory alternatively, we must make the momentum of the corpuscle 

 equal to the quotient of the constant h by the wave-length of the waves; 

 but I have said nothing about the energy of the corpuscle. 



Let us adopt the universal assumption — based on a multitude of 

 experiments, for instance those on the photoelectric effect — that the 

 energy £ of a corpuscle of light is equal to the product of its frequency 

 V b>- the same universal constant //; and let us extend it to the other 

 kinds of corpuscles which we may associate with other kinds of waves, 

 and vice versa. 



Then the complete description of the particles associated with waves 

 of wave-length X is as follows: 



p = ///X, E= hv^ hv/X. (21) 



Here, as before, v stands for the phase-speed of the waves (not the 



particles). 



Returning to the formula (20) for the group-speed, we now can write 



it thus: 



r^= V - \{dv:d\) = v\ - \dip\)ld\ , 



= -XHdp/dX) = - i\'lh)idE/d\). ^ ' 



Suppose next that the energy and the momentum of the corpuscles 

 in question are related to each other and to their speed in the well- 

 known fashion of ponderable bodies, to which it is known that electrons 

 conform. Thus for sufficiently low speeds, the relations are practically 

 those of the "classical" mechanics: 



p = Wo/^ E = i^Wo«-, whence E = p-,2n!Q. (23) 



Here nio stands for the constant mass, u for the speed of the corpuscles 

 (not the waves). 



The energy of the corpuscles is a function of the momentum only, 

 and continuing to develop the formula (22) for the group-speed, we 



find: 



o = {- }^jh)(d E;dp){dp;d\) = dE;dp .^^. 



— p^i — I3~lm() = II. 



The group-speed of the waves is equal to the speed of the corpuscles. 

 The same conclusion follows if we use the relativistic definitions for 

 the energy and the momentum of a particle, 



E = WocVVl - (3-, p = Wn^f/Vl - f3- (J3 = II 'c), 



E = c^niifc- -\- p- 

 as the reader mav test for himself. 



