Volume 102A, number 8 PHYSICS LETTERS 4 June 1984

DE BROGLIE WAVE AND COMFFON WAVE

S.N. DAS Theoretical Physics Centre, Department of Physics, Midnapore College, Midnapore- 721101, India

Received 10 June 1983 Revised manuscript received 7 March 1984

In contrast to the three-wave hypothesis (TWH) presented earlier [ l ], it is argued in this letter that a massive particle in motion in a Lorentz frame will actually be associated with only two types of waves: (i) a transformed Compton wave and (ii) a superluminal de Broglie wave (B-wave). The subluminal wave (D-wave or D'-wave [2]) cannot be simultaneously cor- related with the particle under consideration.

In the three-wave hypothesis (TWH) presented ear- lier [ 1 ], it has been assumed that in a Lorentz frame where the particle is at rest, it is associated with an in- trinsic nondispersive Compton wave (C-wave). However, in a Lorentz frame where the particle moves with ve- locity o , i t is assumed to be associated with [ apart from the existing superluminal de Broglie wave (B- wave)] two more waves, viz. (i) a transformed C-wave and (ii) a subluminal wave whose phase velocity is the particle velocity o (D-wave).

It may be remarked that both in refs. [1] and [2], the main emphasis was on the introduction o f the concept of a dual wave (D-wave or D'-wave [2] but no theoretical grounds except the statement have been given to introduce the idea of an intrinsic Compton wave (C-wave) and the transformed C-wave that may be associated with the particle respectively at rest and in motion in a Lorentz frame. In view of this fact, this letter presents a simple derivation which leads to the result that the above-mentioned wave properties may consistently be associated with the particle. Further it will be argued that in a Lorentz frame, where the par- ticle is moving, it may be associated with only two waves: (i) a transformed Compton wave (C-wave) and (ii) a superluminal de Broglie wave (B-wave). The sub- luminal D-wave or D'-wave [2] cannot be simulta- neously associated with the particle under consider- ation.

It is well know [3] that the fundamental expres-

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sions for the wave-par t ic le dualism can be written in the form

E = h v , IPBI =h/X B . (1,2)

Here E is the energy of the particle and v is its de Broglie frequency, PB is the momentum of the particle and X B is the wavelength of the de Broglie wave (B- wave). Although the above expressions are true both for matter and radiation, an interesting observation may be noted that E cc v, IPBI ~ I/~.B and E oc 1/XB for photons, but E cc v, IPBI cx 1/XB and E q: 1/X B for massive particles.

So in order to restore the symmetry between the relations valid for photons as well as massive particles, we now propose that there might exist a new kind of wave of wavelength X k which is supposed to be pro- portional to the energy of the particle and that X k = ~'B for photons but X k :~ X B for the massive particles. Then one can have E cc v, IPI31 ~ l/X13 and E ~x 1/X k for photons (with X k = XB); E cx v, LPBI ~ 1/'hi3 and E o: 1/hk for massive particles (with X k ~TB)-

Actually we define here that apart from the relation between the energy and frequency of radiation or massive particles, a new wavelength h k can also be re- lated with the energy such that

E = O/X k , (3)

where the constant of proport ionali ty D is determined from the consideration that ~'k = ~-B for photons but

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Volume 102A, number 8 PHYSICS LETTERS 4 June 1984

X k ~ X B for massive particles. It is significantly noted that the ene rgy -momen tum relation E 2 = p2c2

+ m 2 c 4 provides the proper hints. From this relation one can have

E2/h2c 2 = p2B/h2 + E2 /h2c 2 , (4)

where E 0 = mOc2 is the energy of the particle. Now putting eq. (3) in eq. (4), one gets

x2 /X 2 = 1/~,2B + x2/( ' , /k) 2 ,

where x = D/he and in particular, i f x = 1 i.e. D = he,

the above relation turns out to be

1/X~ = 1/XB 2 + 1/('fk) 2 . (5)

Eq. (5) fulfils the condit ion that X k 4: X B for mas- sive particles and X k = X B for photons (m 0 = 0) and thus it seems that the value of D should equal hc. Therefore, one can have from eq. (3)

E = hc/X k .

That is,

X k = hc /E = h/ (p 2 + m2c2) I /2 (6a)

and

(Xk ) 0 = h / m o c . (6b)

Needless to say, X k and (Xk) are connected in the form

X k = (Xk) 0 (1 - t32) 1/2 , (7)

where/3 = o/c. This new wave aspect of matter differs from the de

Broglie wave in the sense that the de Broglie wave (B- wave) becomes mathematically undefined for a par- ticle at rest whereas the latter one suggests a finite value even for the rest particle and given (Xk) 0 = h/moc. It is interesting to note that (Xk) 0 is identified with the Compton wavelength ~C = h / m a c of the particle arid so X k = (Xk) 0 (1 - /32)1/2 = XC(1 _/32)1/2 for the moving particle may be termed as the transformed Compton wave.

We have thus presented a simple derivation that has consistently supported the statement [1 ] that a particle at rest and in motion in a Lorentz frame may respec- tively be associated with an intrinsic non-dispersive Compton wave [X C = (Xk) 0 = h / m o c ] and the trans- formed Compton wave [X k = XC(I - /32)1 /2] . From the above discussion, we then find that any massive particle in motion may actually be associated with

two types of matter waves viz. (i) the de Broglie wave (B-wave) that corresponds to the three-momentum PB of the particle [X B = IPB1-1 ] and (ii) the trans- formed Compton wave that corresponds to the energy E of the particle [X k = E - 1 ] , the proport ionali ty constants being different and are respectively h and hc.

Suppose we now define a space (S') dual to real space (S) such that in the dual space, the energy and momentum of the particle becomes E' = pB c and IP'I = E/c. It is then easy to see that the phase velocity of of the matter-waves associated with the particle in S'- space is v and consequently the group velocity O'g = c2/v. Needless to say, the four-momentum of the

• ! t S - -

particle P u = ( E / c, p ) = (PB, E /c) is tl,en a s p a c e - time vector.

The wavelength of the de Broglie wave (X~) and the transformed Compton wave (X~) of the particle moving in dual space (S') are now given as

x' B = h~ ~'1 = he~L" = Xk, (8a)

and

X' k = hc /E ' = h/[PBI = XB, (Sb)

implying that the de Broglie wavelength and the trans- formed Compton wavelength of matter waves (with of = v, Og = c2/v) in the dual space are respectively the same as the transformed Compton wavelength of matter waves (with of = c2/v, Vg = o)in the real space ,x. This fact thus leads to the conclusion that the concept of the three-wave hypothesis (TWH) [1] seems irrelevant. In fact, a massive particle in motion will be associated in the absolute sense with only two types of waves (XB, Xk) whether the phase velocity of the matter waves is of = c2/v or of = o.

The author is grateful to Professor S.R. Maiti of Midnapore College for helpful discussions.

,1 It is gratifying to note that in ref. [2] the wavelength of the D'-wave (with of = o) is also nothing but the trans- formed Compton wavelength h k of matter waves with of = c2/v.

[1] R. Horodecki, Phys. Lett. 87A (1981) 95. [21 R. Horodecki, Phys• Lett. 91A (1982) 269. [3] L. de Broglie, C.R. Acad Sci. 180 (1925) 498. [4] J. Aharoni, The special theory of relativity (Clarendon,

Oxford).

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