Particle rest mass and the de Broglie wave packet

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  • 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 has been shown (i) that , by considering the de Broglie waves from alI possible observers, it is possible to form a nondispers ive de Broglie wave packet for a free par- ticle. When such a particle, of rest mass m 0, is t ravel l ing in the -{-x direct ion wi th velocity v relat ive to the observer, hav ing an associated de Broglie wave of angular f requency ~o and wave vector /Co, th is wave packet takes the form

    (1)

    where

    and

    (sin kr/kr) exp [i( eot - - kox) ] ,

    k = moC/h

    r L 1 - (v2 /c2)+y~+z2 .

    DE ]~ROGLI]~ has po inted out (2) that , if a D~ BROGLIE wave can be expressed in the form

    (2) a(r, t) exp [i(cot-- koX)] ,

    (3)

    where

    there is then a contr ibut ion to the sequence of the apparent rest mass o~ the part ic le of magn i tude

    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 = mo9

    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.

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