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History: Facts and the Theory Behind Them

Everything started with de Broglie’s little note to Comptes Rendus from 1923 “Ondes et Quanta” [4]. What really was the problem triggering that Note? De Broglie did not mention it explicitly – he starts directly with the technical considerations that became historical ever since – but he makes a short reference right from the title, in a footnote: “Au sujet de la présente Note, voir M. Brillouin, Comptes Rendus, t. 168, 1919, p. 1318″. In other words, if one wants to see the reason of the present commitment, go to that work [5]! And indeed, if we go there, the reason is plain in front of us, giving full meaning to the words of M. Lochak when referring to de Broglie: “Il pensait q’on pourait rendre compte de la quantification de l’atome en attachant une fréquence interne à l’électron, qui serrait en resonance sur sa trajectoire.” Indeed, Marcel Brillouin was always preoccupied to ascertaining physical reasons to the different classical models of atom. In the work referred to by de Broglie he was concerned with Rutherford atom, trying to give physical basis to, by then still fresh, Bohr model. In short, Brillouin found that the finite size of the atom might be able to explain the quantization, by a natural phenomenon: the electron can find itself periodically under the influence of its own waves somehow emitted at prior times along the trajectory. Reading this Note, one can understand the hardship in accomplishing the task along the classical ideas, in that it entices a host of uncontrollable statements regarding the nature of the medium within which the electron moves, the waves in this medium, the relation between these waves and electron, etc. It is perhaps here the point where de Broglie noticed that the concept of energy is sufficiently mature to offer a solution to the problem only in terms of controllable quantities strictly related to electron, without taking into consideration any relationship with the medium within which it moves, or the waves it creates, etc. This is the point where de Broglie’s work started. At the risk of repeating well-known facts, let us go into theoretical details, in order to see where they lead.

De Broglie noticed that if, from energetic point of view, the Theory of Quanta is to be compatible with Mechanics, the fact cannot be stated but via Special Relativity. Specifically, one can tentatively associate a periodic phenomenon (not necessarily a wave at this time!) to a particle at rest by the formula

(1)

where h denotes the Planck’s constant and ν₀ is the frequency of the periodic phenomenon to be associated to the particle whose rest mass is m₀. Now, the trouble starts, and thus the Wave Mechanics came into light. For if one takes the equation (1) as a definition of the frequency of the periodic phenomenon associated to particle, then this frequency transforms relativistically like mass itself, i.e. according to formula

(2)

On the other hand, if one takes the classical definition of the frequency as the reciprocal of a time period according to its definition, then in view of the relativistic time transformation we must have

(3)

which is, evidently, different from (2). As M. Lochak puts it [3], the frequency associated to a particle is at once covariant and contravariant. The contradiction can be solved, as de Broglie noticed, only if we are more specific about the periodic process associated to particle: this process has to be a wave in phase with the motion of particle. Indeed, let us describe the periodic process by the function

(4)

in a reference frame where the particle is at rest. In a reference frame where the particle moves with its velocity, the phase being invariant, the periodic process is described by the function

(5)

Thus, for a moving observer, the periodic phenomenon appears as a wave having the frequency given by (1) and the speed given by v_ph. It is then easy to see that the phase from (5) can be written in the form ν₁t, where ν₁ is given by (2) and t = x/v. In other words, in (5) we have to do with a wave in phase with the motion of the particle but having a frequency as given by (2). Roughly, de Broglie’s conclusion is: the periodic phenomenon has frequency ν₁ and is seen by a stationary observer as constantly in phase with a wave of frequency ν moving with a speed v_ph.

The trouble is not altogether eliminated by this ‘phase harmony’ as de Broglie calls it, but only moved to another level. This becomes obvious as we ask about the nature of the wave associated to particle this way. For this wave cannot be physical inasmuch as v_ph is not physical, being greater than the speed of light. Here de Broglie removes the contradiction by referring to the experience regarding periodic phenomena. Specifically, the phase formalism requires a one-to-one correspondence between waves and frequencies in the same reference frame in order to be able to identify the waves, and this fact cannot be experimentally realized. Rather, experimentally a frequency is the central value of a group of very close frequencies. Whence the idea that a particle, as experimentally observable, has to have associated not a single wave, but a group of waves. And if these waves are all in phase with the motion of particle, as the previous theory shows, then they have to be somehow in phase with each other by simple transitivity. Thus, the important thing is to find a space point where the phases of a group of waves coincide. Partially, this can be done as follows [6]: let us represent the phase of a generic wave from the group we are seeking for by the equation

(6)

Here the dependence of phase velocity on frequency is transferred into a dependence of relative velocity on frequency. Assume that ν is the central frequency of the group of waves, and that each wave of the group is characterized by a frequency (ν + ε) where ε is a small quantity by comparison with the frequency width of the group. Then two different waves of this group have the phases

(7)

Now, if in the time interval dt the particle moves with dx, the difference of variations of the two phases is, neglecting terms in ε² and higher,

(8)

Here the prime denotes derivative with respect to frequency. As we assumed the waves to be different, this phase difference is zero if, and only if, we have

(9)

In other words, as the particle moves, the associated waves (the group of waves) vary in the sense that their phases change. However, for a point whose velocity is defined by equation (9), the phases of the waves in our group are the same. Thus the point moving with the velocity

(10)

is the point where all the waves of the group are in phase. Now, if the function v(ν) is given by equation (2), we have

(11)

Thus the function (νv(ν))’ is the phase velocity of the group of waves associated to a particle, for which the formula (2) provides the so-called dispersion law.

Notice, for the sake of completeness, that if we consider equation (11) as an ordinary differential equation for the group velocity, then the relativistic formula (2) of transformation for frequency is just a particular solution. One can obtain this solution assuming that the group velocity cannot be greater than the speed of light in vacuum. Then the initial frequency is that associated to the particle at rest.

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