_{1}

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Special Relativity sets tight constraints on the form of the possible relations between the four-momentum of a particle and the wave four-vector. In fact, we demonstrate that there is just one way, according to Special Relativity, to relate the energy and the momentum of a corpuscle with the characteristics of a plane wave, frequency and wave vector, if the momentum has to flow in the same direction of the wave propagation: the laws must be of direct proportionality like de Broglie and Planck-Einstein equations.

In the autumn of 1924, the French physicist Louis de Broglie submitted to the judgement of Sorbonne University in Paris, one of the most famous PhD theses in the history of physics [

In his thesis de Broglie suggests that the quanta of light had to be completely comparable to other known material particles. For instance, they had to have a rest mass different from zero, although very small^{1}. Moreover, if the photons had to be put on the same conceptual framework of other particles, according to the French physicist, it was also possible to imagine that particles different from the photons could share the strange dual property of wave and corpuscle with the light. So the fundamental hypothesis of his dissertation was to consider true for all the particles, not only for the quanta of light, the Planck-Einstein law:

where

De Broglie initially imagines that the source of the frequency is related with some periodic phenomenon inside the particle [

holds, he writes that in the rest frame S_{0} of the particle there must be:

where

However, comparing equation (1) with equation (2):

and considering the condition (3), we easily obtain the relativistic formula of transformation between frequencies

which is typically linked with a wave phenomenon: _{0}.

In order to resolve this difficulty, de Broglie assumes the existence of a “fictitious” wave associated with the particle [

that is

According to the French physicist, the wave was fictitious because, being its speed greater than the speed of light, it cannot transport energy^{2}. In order to justify the assumption of equation (7), he shows that if the periodic inner phenomenon and the external wave with phase velocity in equation (7) are in phase at a given time, they will be always in phase; i.e. the particle moves within the wave maintaining its inner vibration in phase with the wave. De Broglie called that “law of the harmony of phases”.

His result, according to de Broglie, suggests that “any moving body could be accompanied by a wave, and it is impossible to disjoin the motion of the body from the wave propagation” [

Let us recall the relativistic momentum of the particle:

By comparing equation (9) with equation (5) and assuming that this holds in any frame, we can write the momentum in terms of the wave frequency:

But from equation (8) we know that

Equation (11) connects the module of the momentum p of a particle with the wavelength

De Broglie’s prediction on the wave nature of the electron, the reason for his Nobel Prize, will be confirmed a few years later in the experiments of Davisson, Germer [

Today just a few introductory textbooks to quantum theory describe the original way of de Broglie’s thinking. Some books contain simplified versions [17,18] or the successive de Broglie’s derivation with wave packets [

Perhaps, from the present perspective, many of de Broglie’s initial suppositions appear strange (but see [

In the next section we report an alternative deduction of the de Broglie relation obtained directly from the Lorentz transformations and the Planck-Einstein equation. In Section 3, following de Broglie and other authors [23-25], we show how Special Relativity puts constraints on the possible formulas that may connect energy and momentum of a particle with wavelength, frequency and wave amplitude. In particular we demonstrate that equations like de Broglie’s and Planck-Einstein’s are the only relations allowed by Special Relativity, once we assume that momentum and wave vector have the same direction.

For particles without rest mass

Putting

An elegant way to derive the de Broglie relation, for any massive or massless particle, can be achieved using directly the Lorentz transformations^{3}. We put forward the following assumptions:

1.a: Each particle is associated with a wave phenomenon.

2.a: In every inertial frame the relation

We want to show that:

Theorem a: According to Special Relativity and 1.a-2.a, between momentum and wave vector there is necessarily the relation

Let S and S’ be two inertial frames in relative motion. According to 2.a, we assume that in both frames the Planck-Einstein relation applies to any particle:

We introduce, for the particle and the wave, the fourmomentum P and the wave four-vector K, respectively:

where the wave vector k has modulus

Moreover, we suppose S’ to move with respect to S with speed

The Lorentz transformations for the four-momentum and the wave four-vector are

where

Then we have

By multiplying the second of equations (17) by the Planck constant

and by subtracting side by side the two equations (18):

(19)

from which, because of equation (13), we get

If we exclude the trivial condition in which the relative velocity V of the frames is zero (in such a case the factor

equivalent to the de Broglie equation for the x component of the momentum.

We point out as the just given demonstration, with the assumption 2.a, holds for particles of any mass, while de Broglie’s demonstration, starting from the relation (9), only holds for particles with nonzero rest mass.

Equation (21) can be easily generalized to other components of momentum and wave vector. In fact, consider a general orientation of the wave vector k with respect to the S frame_{y} and S’_{z} travelling with velocities along y and z with respect to S. Applying the correspondent relations (17) for S’_{y} and S’_{z} for the pairs of components

As already remarked by Einstein in one of his fundamental works of 1905, the energy of an electromagnetic radiation contained in a closed surface, and the frequency of the same radiation, change under the Lorentz transformations in the same way [

Therefore it would seem, from the developed reasoning in Section 2 and from the Ashby and Miller result, that, at least for the photons, both Planck-Einstein and de Broglie equations may follow from Special Relativity. Moreover it is also natural to wonder if a general constraint exists, which is valid for particles with any mass, for which the condition of relativistic invariance imposes the form of both Planck-Einstein and de Broglie relations.

As a matter of fact, here we intend to show that between the determinate four-momentum of a particle and the wave four-vector of a monochromatic wave there must be a condition of direct proportionality, if the velocity of the particle has the same direction of the associated wave propagation.

We assume from experience that every particle is in relation with a wave [

We recall that in classical mechanics a plane wave possesses infinite total momentum and infinite total energy, and then we can only define for it a flux and a density of momentum, or a flux and a density of energy [

We consider as previously, for sake of simplicity, a plane wave of angular frequency

In S,

1.b: Each particle is associated with a wave and it is impossible to disjoin the motion of the particle from its wave.

2.b: The finite energy and momentum of a free particle are associated with the characteristics of a monochromatic plane wave, amplitude, frequency and wave vector:

3.b: The momentum of the particle flows in the same direction of the wave propagation.

We intend to show:

Theorem b: The only functions (22) allowed by Special Relativity and by 1.b - 3.b, are the relations of proportionality

For the development of Theorem b we need the following lemma:

Lemma: In the frame S_{0} where the momentum of the particle is zero,

According to our assumptions, the momentum has to flow in the direction of the wave propagation, in conformity with what happens to the waves of classical mechanics. That is, for

We require that such a condition holds in every inertial system S. Let S_{0} be the frame in which the momentum of the particle is zero_{0} for the components of the momentum and the wave vector are:

where now we have represented in explicit form the dependence of β on the velocity _{0}, coincident with the rest frame of the particle (hence _{0} there is

So assuming that the momentum, i.e., the velocity of the particle, is always in the direction of the wave propagation implies the existence of a frame S_{0 }where momentum and wave vector are both zero^{4}.

Let us look for the velocity _{0 }as a function of

we get immediately:

Remembering that the phase velocity is defined as

Equation (28) is exactly equivalent to equation (8), postulated by de Broglie in order to obtain the harmony of phases between the periodic inner phenomenon and the external wave.

Coming back to Theorem b, we consider the two invariants square moduli of the four-momentum and the wave four-vector:

where _{0} and the frequency in the frames where the momentum and the wave vector are zero. We study the four different possible cases:

Case 1:

We have, from the Lorentz transformations between S and S_{0} for the four-vectors P and K:

Now, according to our Lemma, we assume that in the frame S_{0} we have

and by dividing side by side equations (31) we get:

So, being the inertial frame S arbitrary, from the first of equations (32) we deduce that the ratio

where C is for the initial hypotheses finite, positive and constant with respect to the space-time coordinates

Identifying C with the Planck constant, we can recognize in equations (34), respectively, the Planck-Einstein and the de Broglie relations.

From the definition of C in relation (33), we see that

Therefore C does not explicitly depend on the amplitude, but may depend on the inertial mass of the particle and the invariant

However, if we require, according postulate 2.b and the first of equations (32), that in the limit

Finally, if we demand that C is independent of the mass, such as the Planck constant seems to be experimentally [

Case 2:

Now, in S_{0}, we have

A way out is represented by allowing the factor _{0} and the particle travel at the speed of light. In order to avoid the infinite energy of the particle we should assume

Case 3:

In this case, since_{0} we have

Case 4:

Let us consider the Lorentz transformations between two inertial frames S and S’ for the components of the four-momentum and the wave four-vector:

If we put ^{5}:

Inserting equations (38) into equations (37):

we obtain, dividing side by side:

Being primed and non-primed arbitrary inertial frames, from the first of relations (40), we deduce that the ratio

and from equations (40) we get again the Planck-Einstein and the de Broglie relations (34).

Summing up, the physically meaningful cases are Case 1 and Case 4. Case 1 corresponds to particles with inertial mass and waves with

We have shown that, once we assume the existence of a wave phenomenon with finite energy and momentum like a classical particle, and that the momentum fluxes along the same direction of the wave propagation, from Special Relativity follows that the four-momentum P and the wave four-vector K can be related just in one way, by a rule of direct proportionality:

where C is an invariant under the Lorentz transformations.

Therefore, we have deduced de Broglie and PlanckEinstein relations for plane waves from more general assumptions than those usually considered. De Broglie uses the following hypotheses:

1) The rest energy of a particle

where h is an invariant constant.

2) The relationship between phase velocity

or 2ʹ) The frame in which the particle is at rest is the same frame in which the wave vector is zero,

Assumptions 2) and 2ʹ) are equivalent, and de Broglie himself showed it [

2‴) The momentum always flows in the same direction of the wave propagation in every inertial frame.

The fact that the particle is at rest, that is

Our result shows that the relationship between Relativity and quantum physics is closer than usually thought^{6}. It is also a meaningful example of the conditions that Special Relativity imposes to other theories, with Einstein’s words [

This work has been supported by the EU-STREP Project QIBEC within the activities of the BEC center. QSTAR is the MPQ, IIT, LENS, UniFi joint center for Quantum Science and Technology in Arcetri.