LETTEI~E AL b;UOVO CIMI~IN'~O VOL. 31, N. 2 9 Maggie 1981
Particle Rest Mass and the de Broglie Wave Packet.
L. -~IACKIN NON
Physics Departmen, U~iversity of Essex Wiveahoe Park, Colchester, Essex C04 3SQ, England
( r icevuto il 4 Marzo 1981)
i t h a s been shown (i) t h a t , b y cons ider ing t h e de Brogl ie waves f rom alI poss ible observers , i t is possible to form a nond i spe r s ive de Brogl ie wave p a c k e t for a free par- ticle. W h e n such a par t ic le , of res t mass m 0, is t r ave l l i ng in t he -{-x d i rec t ion w i t h ve loc i ty v r e l a t ive to t he observer , h a v i n g an assoc ia ted de Brogl ie w a v e of angu l a r f r e q u e n c y ~o a n d w a v e vec to r /Co, t h i s w a v e p a c k e t t akes t h e form
(1)
where
a n d
(sin kr/kr) exp [i( eot - - kox) ] ,
k = moC/h
r L 1 - ( v 2 / c 2 ) + y ~ + z 2 .
DE ]~ROGLI]~ h a s p o i n t e d ou t (2) t h a t , if a D~ BROGLIE wave can be expressed in t he fo rm
(2) a(r, t) exp [i(cot-- koX)] ,
(3)
where
t he re is t h e n a c o n t r i b u t i o n to t h e sequence of t h e a p p a r e n t r e s t mass o~ t h e pa r t i c l e of m a g n i t u d e
c ~ \ a ] '
[ ] . . . . V 2 . c 2 8t 2
tl) L. MACKINNON: Found. Phys., 8, 157 (1978). (') L. DE BRO(~LIE: Nonlinear ~Vave Mechanics, Chapter X (Amsterdam, 1960).
37
38 L. MACKIN:NON
I t is of interest to calculate this contribution to the apparent rest mass for the nondispersive wave packet where, from (1) and (2), a(r , t )= (sin kr/kr). ~Vhen this calculation is carried out, one finds
(4) 6too = mo�9
Therefore the entire rest mass of a free particle is related, in the manner put forward by D~ BROGLI~, to the amplitude term describing the wave packet.
Physically, the nondispersive wave packet has been formed by considering the extent of the space-time available to the particle if its behaviour is to be observer in- dependent. Thus the nonuniformity of a(r, t) has not been produced by de Broglie wave superposition of the type which arises from reflection off obstacles, etc.; it is therefore of a very different physical origin. What the result of eq. (4) does show, however, is that there is a very interesting overall consistency in the shape of the nondispersive wave packet, a consistency which one would not have immediately predicted.
