ISO BULL SYSTEM TECIIXICAL JOCRXAL 



length of the waves is equal to // divided by the monientuin of the 

 particles, we have the following value for the ratio between the 

 wave-speeds v' and v on the two sides of the surface: 



„y_x;.v.(|^)/(|*).,V,, (28) 



and the speed of the waves varies inversely as the momentum of the 

 corpuscles, which is just what is required in order that we may hold 

 both the theories simultaneously. 



But how about the theorem that corpuscle-speed is equal to group- 

 speed? Returning to the equations (25), we see that the introduction 

 of the potential energy has altered the relation between energy and 

 momentum; we now have: 



E = c^Jnioc' -f p- + eV. (29) 



But so long as we are comparing different electron-streams in the 

 same medium (vacuum, for instance), the potential energy is the 

 same for all and does not depend on the momentum; and differentiating 

 E with respect to p to obtain the value of the group-speed g, we get: 



.? - dE/dp = ^^ = c^m.nHX_-^ ^ 

 E - eV nioc-l^l - (3- 



and thus group-speed and corpuscle-speed are equal, as before. 



I will write down the expression of the phase-speed, although for 

 the physicist it is of minor importance, not being measurable — a fact 

 which exempts us, temporarily at least, from pondering over the 

 curious feature that it depends on the value of the potential energy 

 of the corpuscles, and therefore (for electrons) on the value accepted 

 for the electrostatic potential of the region where the wave-train is, 

 even though in practice it is generally assumed that electrostatic 

 potential may be measured from an arbitrary zero. The formula is 



v = E/p = -^^-h'^^i±U 



Wo«/Vl - /3- (30) 



= c-ju + U/p, 



and if we put the potential energy of the corpuscles equal to zero, we 



find the phase-speed varying inversely as the corpuscle-speed,^'- and 



greater than the speed of light. 



1- There is a paradox here which, as I can testify from personal experience, is a 

 dangerous source of confusion. The formula v = c'-jii sounds like an approximation 

 to tlie formula v = const' p which I have gi\'en as the retpiisite relation lictwecii 



