On electromagnetic- and De Broglie waves

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  • ON ELECTROMAGNETIC- AND DE BROGLIE WAVESBY A. KOMJTHY, 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.

    t 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 _ (m 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 theparticle to be equal to, the groupsped 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 WAVES33

    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 thelimit 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 )3/2in 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, wavelength 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 lLet 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= V

    from '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

    2i. 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 get

    m 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)kA(v0) A (v)Our aim is the determination of k.First. let us take into account that ,v) is bounded andl $ 0

    awhen 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 WAVES35

    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

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