FROM CURRENT LITERATURE

AN ATTEMPT AT A THEORY OF LIGHT QUANTA.

L. de Broglie. A Tentative Theory of Light Quanta. — Phil. Mag. 47, 446, 1924.

The paper under review is a very ingenious and interesting attempt to connect together the wave and corpuscular theories of light, whose extreme representatives at the present time are Einstein, on the one hand, and Bohr, on the other. The synthetic theory proposed by the author is based on the following principles.

1. The opposition of particles to waves is incorrect: every particle is associated with a “phase” or “guiding” wave and, conversely, every electromagnetic wave is associated with some particle moving in the direction of one of its “rays.” At the same time the waves themselves have no energy whatsoever1; their action reduces to causing, in previously “excited” atoms, those regressive transitions associated with a decrease of mechanical energy and with radiation which, in Bohr’s theory, are treated as “spontaneous,” i.e. self-occurring. Transitions of the opposite character (“progressive”), associated with an increase in the mechanical energy of atoms, i.e. with the photoelectric effect, ionization, etc., are caused not by waves, but by the corresponding particles, which may be either “light quanta” or electrons, and moreover at the expense of the kinetic energy of these particles. Thus the sole carriers of energy are material charges—electrons and “quanta”—whereas the waves associated with them play the role of a trigger mechanism with respect to mechanical systems “charged” with energy.

2. The frequency \((\nu)\) of the oscillatory process associated with each material particle is proportional to its energy \((\varepsilon)\), or—which is the same—to its mass \((m)\), the ratio \(\frac{\varepsilon}{\nu}\) being equal to Planck’s constant \(h\).

Combining these two principles with the special theory of relativity as a guiding method, L. de Broglie constructs his theory, which for the time being is not so much a new theory of light phenomena as a new corpuscular-wave dynamics of matter.

In contrast to Einstein, who regards light quanta as atoms of energy not connected with any material substratum, de Broglie treats them as ordinary material particles, deprived, unlike electrons and protons, of electric charge, but possessing, on a par with them, a quite definite “rest mass.” This mass is assumed to be so small \((< 10^{-50}\ \mathrm{g})\) that negligible forces prove sufficient to impart to it a velocity \(v\) very close to the “critical” velocity \(c = 3 \cdot 10^{10}\ \mathrm{cm/sec}\), which

is regarded as “the velocity of light in a vacuum.” In fact, therefore, the velocity of “light quanta” is a variable quantity, approaching, but never attaining, \(c\).

A material particle (proton, electron, or “neutron,” i.e., quantum), possessing a rest mass \(m_0\), is connected with an oscillatory process whose frequency \(\nu_0\) is determined by the formula

\[ m_0 c^2=\varepsilon_0=h\nu_0 \tag{1} \]

and whose phase, as a function of time \(t_0\), may be represented by an expression of the form

\[ \varphi=\sin 2\pi\nu_0 t_0 . \]

When the particle itself moves (with respect to the observer) with velocity \(v\), then, according to the theory of relativity, its mass and energy increase in the ratio

\[ 1:\sqrt{1-\frac{v^2}{c^2}} . \]

It is not hard to see that the frequency of the corresponding oscillatory process increases in the same ratio. Indeed, denoting by \(x_0,t_0\) the space-time coordinate system connected with the particle under consideration, and by \(x,t\) that connected with the observer, we write, according to one of Lorentz’s transformation formulas,

\[ t_0=\frac{t-\dfrac{vx}{c^2}}{\sqrt{1-\dfrac{v^2}{c^2}}}. \]

Substituting this expression into the preceding formula for \(\varphi\), we obtain

\[ \varphi=\sin 2\pi\,\frac{\nu_0}{\sqrt{1-\dfrac{v^2}{c^2}}} \left(t-\frac{x}{c^2/v}\right) \tag{2} \]

i.e., the equation of an oscillatory process with frequency

\[ \nu=\frac{\nu_0}{\sqrt{1-\dfrac{v^2}{c^2}}}. \]

Thus the equality (1) is transformed into

\[ mc^2=\varepsilon=h\nu=\frac{h\nu_0}{\sqrt{1-\dfrac{v^2}{c^2}}}. \tag{2a} \]

Formula (2) has the form of the equation of a wave propagating in the direction of motion of the particle (i.e., along the \(x\)-axis) with velocity

\[ v'=\frac{c^2}{v}, \tag{2b} \]

exceeding \(c\) by as many times as \(c\) exceeds \(v\). This wave is what de Broglie regards as the “guiding” or “phase” wave, while the question of its physical structure and properties is left by him, for the time being, entirely open (see below). If we imagine a group of similar waves corresponding to

to a series of infinitely close values \(v\), then the resulting velocity of this group is expressed, as is known, by the formula

\[ u=\frac{dv}{d\left(\frac{v}{v'}\right)} . \]

Substituting here the preceding values of \(v\) and \(v'\), we find

\[ u=\frac{d\left(1-\frac{v^2}{c^2}\right)^{-\frac12}} {d\left[\frac{v}{c^2}\left(1-\frac{v^2}{c^2}\right)^{-\frac12}\right]} =v . \]

Thus the velocity of the particle is equal to the group velocity of a set of waves with frequencies

\[ \nu=\frac{1}{h}\,\frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}} \]

and phase velocities \(v'=\frac{c^2}{v}\), for slightly different values of \(v\).

The corpuscular-wave dynamics developed by de Broglie is based, as has already been mentioned above, on the identity of all possible trajectories (orbits) of the particle under consideration in the given force field with the orthogonal trajectories (rays) of the corresponding phase waves. In this connection, with respect to the phase waves, the force field (i.e. the space in which electromagnetic forces act) plays the role of the dispersive medium of ordinary wave optics. Thus the wavelength \(\lambda\) at that point of the force field through which the corresponding particle passes, having (or which would have) velocity \(v\), is equal to

\[ \lambda=\frac{v'}{\nu}=\frac{c^2}{\nu v} =\frac{c^2\sqrt{1-\frac{v^2}{c^2}}}{\nu v_0}, \quad \text{i.e.} \quad \lambda=h\,\frac{\sqrt{1-\frac{v^2}{c^2}}}{m_0v}. \tag{3} \]

Denoting by \(ds\) the element of the arc of a ray passing through two given points (1, 2), we can determine the form of this ray with the aid of Fermat’s principle (of quickest passage)

\[ \delta\int_1^2 \frac{ds}{\lambda}=0 \tag{3a} \]

which, as is not difficult to verify, coincides in the case under consideration with Maupertuis’ principle (of least action)

\[ \int_1^2 m_0c^2\left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} -\sqrt{1-\frac{v^2}{c^2}} \right)dt = \delta\int \frac{m_0v^2}{v\sqrt{1-\frac{v^2}{c^2}}}\,dt = \delta\int \frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}\,ds =0, \]

which determines the trajectory of the corresponding material particle.

Let us imagine an electron rotating in a circle of radius \(r\) around a positive nucleus. According to Bohr’s theory, stable orbits of this kind are determined by the condition: \(mvr=n\cdot \frac{h}{2\pi}\), where \(n\) is an integer, i.e. according to (3)

\[ 2\pi r\cdot \frac{m_0v}{h\sqrt{1-\frac{v^2}{c^2}}} = \frac{2\pi r}{\lambda} =n . \]

Thus, from the point of view of de Broglie’s theory, the meaning of this condition reduces to the requirement that the “phase wave” accompanying the electron “be tuned in resonance with one of the overtones of the length of the orbit” \((2\pi r)\), i.e., in other words, that the latter contain an integral number of waves. This result is easily generalized to the case of closed orbits of arbitrary type. Namely, the condition of “resonance,” i.e. of stability of the motion, is then expressed, according to de Broglie, by the equality

\[ \int \frac{ds}{\lambda}=n \qquad (n\text{—an integer}), \tag{3b} \]

where the integral is taken over a closed curve. Substituting here the value of \(\lambda\) from (3), we obtain

\[ \int \frac{m_0 v^2}{h\sqrt{1-\frac{v^2}{c^2}}}\,dt = \int \frac{m_0}{h\sqrt{1-\frac{v^2}{c^2}}} \left(v_x^2+v_y^2+v_z^2\right)\,dt = \frac{1}{h}\int\left(mv_x\,dx+mv_y\,dy+mv_z\,dz\right)=n \]

which coincides with Bohr’s stability condition in the general form given to it by Einstein1. Thus this condition for the first time acquires a definite physical meaning. The latter will become clearer to us if we take into account that, when condition (3b) is present, the phase wave overtaking the electron in fact merges with the one which at that very moment “for the first time” issues from it.

Without dwelling on a detailed exposition of the optical side of de Broglie’s theory, which is only outlined by him, we shall confine ourselves to a few general remarks.

First of all, the combination of light quanta with phase waves makes it possible to explain interference phenomena, which in Einstein’s conception remained entirely incomprehensible. The essence of the explanation proposed by de Broglie reduces to the following. When a phase wave passes through an excited atom, the latter has a certain probability, proportional to the intensity of the wave, of emitting an “atom of light” of the corresponding frequency, of course only together with a spherical phase wave, which in turn can cause radiation in neighboring atoms. On the other hand, the ability of several light quanta to produce a photoelectric effect, or in general progressive transitions, is likewise determined by the resultant intensity at the corresponding points associated with these quanta of phase waves. As for this intensity, it must stand in a definite correspondence with the quantity which in Maxwell’s theory is regarded as the energy of the electromagnetic field.

Such energy in fact does not exist. We must return (as Bohr’s theory already partly does) to the former mechanical conception of energy, as the sum of kinetic and potential energy, i.e. of two parts depending on the motion and interaction of the material particles under consideration. Until now only protons (nuclei) and electrons have been counted as such. De Broglie adds to them “light quanta,” whose kinetic energy must replace the obscure “radiant energy” that still continues to figure in Bohr’s theory; the radiation field is transformed into an energyless field of phase waves determining the probability of one or another (regressive or progressive) material process. Let us note that an analogous interpretation of the electromagnetic field was recently proposed by Bohr himself (Phil. Mag., May 1924), who, however, does not introduce the concept of light quanta as material particles associated with waves, and

therefore completely renounces the principle of conservation of energy. In the de Broglie theory this principle remains in force. However, the nature of light quanta, the laws of their interaction with one another, and also with protons and electrons, and finally the mechanism of their emission and absorption, remain completely unexplained1. Despite all its incompleteness and indefiniteness—which may in time be eliminated—this theory appears to us a very successful attempt to connect both sides of light phenomena—the interference side and the photoelectric side—and, moreover, to do so organically, and not formally, as in Bohr’s theory, to merge optics with dynamics into a single continuous whole.

Ya. Frenkel.