ON ELECTROMAGNETIC- AND DE BROGLIE WAVES

BY A. KOMJÄTHY, BUDAPEST.

(Received Oct. 10, 1949.)

It has among other results been shown in this paper that theelectromagnetic waves are a special case of . De Broglie waves cha-racterized by the connected relations v ---*- C, m o —> 0, v resp. mo beingthe velocity resp. rest-mass of the particle belonging to the wave.The wave itself is unequivocally determined by one of the threeconditions u — k, u ` = k, u — u*, u resp. u ` being its phase-resp. group-speed, each leading to the common relation u = u* = c. In otherwords there exist no De Broglie waves apart from the electromaig--netic ones having constant phase -resp, group-speed, or allowing theequality of them.

Ít has been deduced in a. previous paper of the author' the-well known formula(1) 2_ h (1 —v2 lc2)"=

m0 vwhere k is the wave length of the De Broglie wave belonging tothe material particle having the rest-mass mo -and moving with thevelocity v.

From (1) we get immediately( he(2) V _ (mó C$ 22 +, h2) 1I

The proof was performed on the ground of the following threeassumptions

a) , * 0b) the exigencies of the special relativity theory are

f ulfillledc) the De Broglie wave of the , particle satisfies the , usual

wave equation of Optics.It wasn't necessary to assume neither the validity of the

relation E — - m2 CQ i = h P, nor suppose the velocity v of the

particle to be equal to, the groupspéed u*` of the De Broglie wave.It is well known the defining equation of the group-speed u*

to be as follows

(3) u-2 dT =u'

u being the phase-speed, A the wave length.

1 Hung. Acta Phys. Vol, I. No 3. 1948.

A. KOMJATHY: ELECTROMAGNETIC- AND DE BROGLIE WAVES 33

With regard to , (1), (3) is easily transformed into the follo-wing , equation

(4) u+ v(I _v'/c2)u' =u*in which u resp. u* have now to be regarded as functions of v.

Our first task is to establish the .appropiate: boundary condi-tion of (4). For this purpose one has to take into, account, that uis the phase-speed of a material particle having the rest-mass r0 ,moving with the velocity v. Supposing v -- c, we get from(2), m o -^ 0, while the wave length is determined as the

limit of the expression A =11(1

—vul c2/ 1/2 ' when simultaneously'nov

Z --> c and m,, --- 0.Regarding this assertion, we should like to emphasize that (1)

and (2) had been deduced assuming v (c and on o > 0, whereas itis undoubtedly true that to every such value of v resp. m0 belongsa well determined Wave length. Applying the principle of conti

-nuity we have therefore ' the right to suppose the existence ora well determined wave. length as limit of (1) when simultaneouslyv --^ c, m --> 0, all the more because the existence of electro-magnetic waves iss undeniable fact.

Considering that the particle for which rno = 0 and v = c isthe photon and the De Broglie wave belonging to it is therefore theusual electromagnetic wave whose phase,speed is c, we get for (4rthe following boundary condition

u (v) --^ c, when v---c.Integrating (4) we obtain the following formula:

v

(5) A (v.) u (v) = u (vo) + A (vo) f (1 (v 2/c )3/2

in which A (v) — — v(l —v$/c9 1 1

/z '°e v and vo varying indepen-

dently are submitted to the condition. Q < v G c.Taking into account the relation a,n4ong phase-speed, wave

length and frequency u — A v we can with the , help of (4) and (1)deduce a new and interesting equation. Differentiating the aboverelation with respect to v. and substituting the value of u resp. u'in (4), we get namely

v(' u* (v) d v_

(6

) h (v — vo) = moJ (i v2/c2)3/2vo l

Let us now suppose the validity of the relation

m0 C2

E_(1 —v21c2)'12 h.In this case we get with the help of (1) and considering the

relation u = A v3

34 A. KOMJA'IHY:

C2u= —

Vfrom 'which it follows with regard to (4) u* = v.

On the other hand, putting in (6) u* = v, : we obtain_ mo c2

(7) E (i — v2/C2)lJ2 ` h v

2

i. e. either of the relations h v = m0 C resp. u* =-v is a con_(1— v2/c2) 1'

sequence of the other.Let us now put in (6) u* = k v, k $1 being an arbitrary para-

meter. We getm c2 =^

hy_kE=k^1 —v2/C2)'I$

It is natural to raise the question why. do we regard equation(7) as physically correct and 'not equation (8) containing the para

-meter k ?Putting in (4) u* = k v, its solution will be

(9) u (v) _ k c2v

According, however, to our boundary condition u (v) --* c,when v -- c, which involves on the ground of (9) k =1, i. e. equa-tion (7) is to be regarded as physically correct, because the velo-city of light is c.

Let us now return to ,our equation (5) in which we put u* = k, kbeing a constant.

We obtain the following relation

(10)u (v) — k u(v°)—k

A(v0) A (v)

Our aim is the determination of k.

First. let us take into account that ,v) is bounded andl $ 0a

when v„ is submitted to the condition c — e S r e, G c— 8, e and Sbeing fix satisfying the relation c> E > 8 > 0. Moreover as a eou-sequence of our boundary condition u (vo) — k, where u (vo) is toregarded as continous function of v 0, is bounded for the same valuesof vo supposing E and 8 to be sufficiently small.

if under such conditions v -- c, then at the same time 1A (v)--^0, involving with regard) to (10), that

u (v) — k —> 0, when v --- c.

Aiccording, however, to our boundary condition u (v) --^ C,when v -- c, from which it follows u* = k = c, i. e. if the group-speed of the De Broglie wave belonging to a particle is constant.the value of this constant is necessarily c.

ELECTROMAGNETIC- AND DE BROGLIE WAVES 35

Putting in (6) u* = c we obtain the following relation:

hi' maC li J Cv (1-?/C2)'1.

We assert this formula having physical meaning under condi-tions examined by us only, in the case, when v —^- c, which invol-ves with regard to (2), m a -- 0.

The supposition u* = c means namely that the wave belongingto the particle moving with uniform velocity v is transportingenergy, whose speed of propagation is c, i. e. it. is of electrdmagnetic-origin.

According, however, to classical electrodynamics does a par-ticle moving with velocity v <C radiate electromagnetic energyonly in the case, when its movement is accelerated.

There is only one well known case in physics when a particle-though moving with uniform velocity the wave belonging to , ittransports notwithstanding electromagnetic energy, that of the

:photon.In this case we have to put •v = c, which involves according

-to (2) mo = 0.Writing in (11) v = c, we , get the following relation:

by2=mC

.m being the mass of the photon defined by

mom =tim ^Z

(1 — v2/c2) 1

-when simultaneously v =--- c, m,, -- 0.Putting in (4) v = c, we obtain with regard to the relation

u*—c, also u=c.Let us now take u = k, k being a constant. We get from (4)

u* — jc, involving, as we have seen u* = c, and all the consequences-proved above.

On the other hand if we put u = u*, there are ;according to (4)two possibilities. Either we have v = c, involving n, = 0, this beingthe case of the photon, or we must suppose u' = 0 from which itfollows u = u* = k, involving, as it has been shown, the relationßc1=u*—c.