MECHANICAL AND ELECTROMAGNETIC PROPERTIES OF LIGHT ATOMS (QUANTA).

Ya. I. Frenkel, Leningrad.

§ 1. Mass and Weight of Light Atoms.

In 1905, after nearly a century and a half of oblivion, Newton’s hypothesis of the “emission of light” reappeared on the physical stage. It is very remarkable that Einstein was Newton’s successor here as well, just as in mechanics (including the theory of gravitation). True, the new corpuscular theory of light revived Newton’s theory in a somewhat transformed form. Namely, whereas in Newton’s theory the light atoms were treated as particles of a certain matter and, accordingly, possessed the basic principle of materiality—indestructibility, in Einstein’s theory they acquired a new, quite peculiar meaning—atoms, or “quanta,” of energy. Einstein’s light quanta, in contrast to material particles, can be created and disappear—at the expense of the mechanical energy of the atoms emitting them or, so to speak, “in favor of” the mechanical energy of the atoms absorbing them. Here the energy of a light quantum \((\varepsilon)\) is a measure of that quantity \((\nu)\) which in the language of the wave theory is called the frequency of light, i.e., the number of oscillations per second. The relation between these two quantities is expressed by the well-known formula

\[ \varepsilon = h\nu, \tag{1} \]

where \(h\) is Planck’s constant.

Newton’s theory was concerned chiefly with the question of the propagation of light and, in particular, with the reflection and refraction of light rays at the boundary of two media, which in this treatment were regarded as continuous bodies devoid of internal structure. This ma-

the macroscopic point of view gave way in Einstein’s theory to the microscopic one; in connection with this, and also in connection with the immateriality of light atoms, the new optics concerned itself chiefly with the question of the emission (creation) and absorption (destruction) of these atoms. As for the question of their propagation, it was resolved quite simply: in the absence of gravitational forces, light quanta move rectilinearly and uniformly with the limiting velocity

\[ c = 3 \cdot 10^{10}\ \text{cm/sec}. \]

(It goes without saying that what is meant here is motion in a vacuum, from one atom—the emitting one—to another—the absorbing one.) In a gravitational field the motion of a light quantum coincides with the motion of a material particle having the same limiting velocity. The latter principle follows from the fact that the motion of any particle in a gravitational field does not depend on the mass of this particle. Therefore it seems quite natural to assume that the laws established for the motion of material particles in a gravitational field remain valid also for immaterial particles, since the latter possess mass. According to the general principle established by Einstein on the basis of the theory of relativity, mass and energy are equivalent concepts, so that a light quantum possessing energy \(\varepsilon\) possesses, at the same time, the mass

\[ m = \frac{\varepsilon}{c^{2}}, \tag{2} \]

In Einstein’s theory, light atoms, as in Newton’s, are just as ponderable as material atoms. This ponderability of light finds its most direct and graphic manifestation in the deflection experienced by rays emitted by stars when passing near the solar disk.

Fig. 1.

Without dwelling on a rigorous derivation of this deflection, based on Einsteinian mechanics and the theory of gravitation, we shall give a calculation based on Newton’s own theory.

Let us imagine that, in the absence of gravitational forces, a light quantum would move uniformly along the straight line \(MN\) with velocity \(c\). Under the influence of the gravitational field emanating from the center \(S\) (the sun), the path of the quantum is curved, taking the form \(MN'\). Owing to the extreme smallness of this deflection (angle \(\alpha\)), in calculating it one may operate with the force acting on the quantum during the unperturbed motion

along the straight line \(MN\). At the point \(Q\), a force \(\dfrac{k}{r^2}\) acts per unit mass,

where \(k\) is a constant, and \(r=SQ\); the transverse component of this force (perpendicular to \(MN\)) is equal to \(\dfrac{k}{r^2}\sin\varphi\) \((\varphi=\angle SQN)\). Denoting the corresponding (transverse) component of the quantum’s velocity by \(v\), we have the equality:

\[ \frac{dv}{dt}=\frac{k}{r^2}\sin\varphi . \]

If \(r_0=SP\) is the shortest distance of the light particle from the attracting center, and \(-x=QP\) is its distance from the point \(P\), then from the triangle \(SQP\) we obtain:

\[ r=\frac{r_0}{\sin\varphi}, \qquad -x=r_0\cot\varphi \]

and further

\[ dx=c\,dt=\frac{r_0}{\sin^2\varphi}\,d\varphi . \]

The preceding equation is therefore transformed into

\[ \frac{c\sin^2\varphi}{r_0}\cdot\frac{dv}{d\varphi} = \frac{k}{r_0^2}\sin^3\varphi \]

i.e. into

\[ \frac{dv}{d\varphi}=\frac{k}{cr_0}\sin\varphi . \]

The total magnitude of the transverse velocity acquired by the light particle along its entire path \(MN'\) (or \(MN\)) is obtained by integrating the right-hand side of this equation from \(\varphi=0\) to \(\varphi=\pi\). Thus, finally, we obtain \(v=\dfrac{2k}{cr_0}\), and for the angular deflection

\[ \alpha=\frac{v}{c}=\frac{2k}{c^2r_0}. \]

In fact, the deflection of light rays turns out, both in theory and in experiment, to be twice as large. The preceding formula gives for it, in any case, the correct order of magnitude and the proper dependence on the distance \(r_0\).

§ 2. Energy and momentum of light quanta and waves.

In Newtonian mechanics, mass was regarded as the fundamental attribute of matter, an invariable property of every material particle. The resul-

the result of this representation was the very widespread—even to this day—identification of the concepts of “matter” and mass. In Einstein’s mechanics these concepts, so to speak, broke away from one another, and the concept of mass proved to be directly connected not with the concept of matter, but with the concept of energy, according to formula (2). In the case of elementary particles of matter—electrons—the role of mass, as the principal “invariant” attribute of matter, passed to electric charge. However, alongside the latter there arose the so-called rest mass \(m_0\), i.e. the mass of a particle in the state of rest. The mass of a particle moving with velocity \(v\) \((< c)\) is expressed through this “rest” mass by the formula

\[ m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \tag{3} \]

Correspondingly, for the energy \(\varepsilon = mc^2\) one obtains an expression which, when \(v\) is small in comparison with \(c\), takes the form

\[ c^2 m = c^2 m_0 + \frac{1}{2} m_0 v^2, \]

where \(\frac{1}{2}m_0v^2\) represents the ordinary kinetic energy of the particle (in Newtonian mechanics), while \(c^2m_0\) is its “internal energy,” the origin and essence of which still remain enigmatic.

It is necessary to note that the mass defined by formula (3) is treated in the theory of relativity not as the ratio between the force \((F)\) and the acceleration \(\left(\frac{\overrightarrow{dv}}{dt}\right)\), but as the coefficient of the velocity in the expression for the particle’s quantity of motion

\[ \vec{g}=m\vec{v}; \tag{3a} \]

so that the equation of motion has the form \(d(\overrightarrow{mv})=\vec{F}\), and not \(m\frac{\overrightarrow{dv}}{dt}=\vec{F}\), as in Newtonian mechanics. True, Newton’s second law in the Principia is textually expressed precisely in the first form, but in fact this considerably broader formulation was never applied by him.

If in formula (3) one puts \(v=c\) and \(m_0=0\), then it turns into an indeterminacy of the form \(m=\frac{0}{0}\). This circumstance shows that, from the point of view of the theory of relativity, the existence of particles moving with the velocity of light and having

nevertheless finite mass and, consequently, finite energy \(\varepsilon\). Such particles are the light quanta. In contrast to ordinary material particles, they do not possess a mass at rest (and also have no charge). They exist only insofar as they move with velocity \(c\). Stopping means disappearance for them. Not possessing the indestructibility characteristic of particles of matter, they must therefore be treated as immaterial particles—as atoms of energy.

Applying this idea to light quanta, we must evidently endow them, along with energy (1) and mass (2), also with momentum

\[ \vec{g}=m\vec{c}=\frac{\varepsilon}{c}=\frac{\vec{h}}{\lambda}, \tag{4} \]

where \(\lambda=\dfrac{c}{\nu}\) is the quantity which, in the language of wave optics, denotes the wavelength\(^1\).

Relation (4) is in complete agreement not only with Einstein’s mechanics, but also with electrodynamics, more precisely, with the electromagnetic theory of light. In classical electrodynamics energy is treated as a certain continuous substance distributed in space, with volume density

\[ \eta=\frac{1}{8\pi}\left(E^2+H^2\right), \]

where \(E\) is the electric and \(H\) the magnetic field strength. Corresponding to this conception of energy is the conception of momentum, whose volume density is determined by the vector product of \(E\) and \(H\) according to the formula

\[ \vec{g}=\frac{1}{4\pi c}\,\vec{E}\times\vec{H}. \]

In this connection it proves necessary to supplement the concept of energy with the concept of an energy flux, defined by the Poynting vector

\[ \vec{S}=\frac{c}{4\pi}\,\vec{E}\times\vec{H}. \]

Comparison of the latter with the vector \(\vec{g}\) leads to the relation

\[ \vec{g}=\frac{\vec{S}}{c^2}, \]

which in a certain sense corresponds to relation (2) between energy and mass. This correspondence becomes complete in the case when the energy flux can be represented in the form \(\vec{S}=\eta\vec{c}\), where \(\vec{c}\) is a vector numerically equal to the velocity of light. In that case, for the density of electromagnetic momentum one obtains the formula

\[ \vec{g}=\frac{\eta}{c^2}\vec{c}, \]

from which it follows that the quantity \(\mu=\dfrac{\eta}{c^2}\) should be treated as an ordinary

\(^1\) Arrows over letters denote the corresponding vector quantities.

density, i.e. the density of mass.—In fact the relation

\[ \vec S=\eta \vec c \]

is exactly fulfilled only for plane electromagnetic waves (or, approximately, at large distances from the sources of the field). Namely, in this case, as is known, the vectors \(\vec E\) and \(\vec H\) are numerically equal and perpendicular both to each other and to the direction of propagation of the waves. Characterizing this direction (corresponding to the vector product \(\vec E \times \vec H\)) by the unit vector \(\vec n\), we have:

\[ \vec S=\frac{c}{4\pi}\vec E\times \vec H =\frac{c}{4\pi}E^2\vec n =\frac{c}{8\pi}(E^2+H^2)\vec n =\frac{E^2+H^2}{8\pi}\vec c, \]

where \(\vec c=c\vec n\), i.e.

\[ \vec S=\eta \vec c \]

Thus, insofar as the propagation of light (in a vacuum) is concerned, the difference between the electromagnetic (wave) theory and Einstein’s quantum theory reduces to the fact that in the former light energy is represented as distributed continuously in space, whereas in the latter it is concentrated practically at separate points (or very small elements of volume).

This difference has a twofold significance. First, it means that atoms can lose energy (in the emission of light) or acquire it (in absorption) only in definite finite portions, and not continuously, as classical theory supposed. Secondly, it follows from it that in the emission of light, as well as in its absorption, we are dealing with a directed act (Einstein’s “Nadelstrahlung”). Applying to this act the law of conservation of momentum, we arrive at the conclusion that, upon the emission of light of frequency \(\nu\) in the form of a quantum with energy \(h\nu\) and momentum

\[ \frac{h\nu}{c} \]

the atom must experience a recoil in the opposite direction, receiving thereby the momentum

\[ -\frac{\overrightarrow{h\nu}}{c} \]

(“light recoil”). Conversely, upon absorption of the indicated quantum, the momentum of the latter, along with its energy, is imparted to the corresponding atom.

These conceptions must be supplemented by the following, based on the analogy between the quantum and wave theories. According to the latter, the absorption of light takes place in the following manner. The electric force of the incident waves causes an oscillatory motion of the electrons in the atoms. This oscillatory motion in turn causes

secondary electromagnetic waves. If the energy of the atom increases, i.e. the amplitude of the forced electronic oscillations increases, then the secondary electric forces prove to be opposite to the primary ones and weaken them. In this case the atom experiences a positive light pressure, directed in the direction of propagation of the waves. If, on the contrary, the energy of the atom decreases, i.e. the amplitude of the electronic oscillations decreases, then the secondary electric forces coincide in direction with the primary ones and increase them; in this case the atom experiences a negative light pressure, opposite to the direction of propagation of the waves. This relation follows directly from the formula \(\vec g = \dfrac{\vec S}{c^2}\), which connects the flux of energy with the density of electromagnetic momentum. It was retained by Einstein also in his quantum theory of light. Namely, alongside the ordinary absorption of a quantum, he introduced into consideration the so-called negative absorption, or forced emission, in which the quantum of light is not absorbed by the atom, but emerges from it together with another quantum of the same magnitude and direction, emitted by the atom itself. In this case the latter, just as in ordinary “spontaneous” radiation, experiences a push in the direction opposite to the emission of the quantum, receiving a momentum \(-\dfrac{h\nu}{c}\).

§ 3. Some applications of the quantum theory of light.

It is not difficult to show that these ideas, in connection with relation (1), lead to the same formulae for the Doppler effect as does the wave theory of light.

Let us imagine an atom of mass \(M\), moving with respect to the coordinate system under consideration with velocity \(\vec v\) (the latter we shall assume small in comparison with the velocity of light). Suppose that the atom possesses some internal energy \(\Sigma_0\), which it, remaining at rest, would have had to emit in the form of a quantum of frequency \(\nu_0 = \dfrac{\varepsilon_0}{h}\). Alongside this internal energy it possesses kinetic energy \(\dfrac{1}{2}Mv^2\). The emission of light is connected not only with a loss of internal energy, but also with a change of kinetic energy. Thus the frequency \(\nu\) of the emitted quantum must, generally speaking, differ from the quantity \(\nu_0\). Denoting the velocity of the atom after emission by \(v'\), we have, on the basis of the law of conservation of energy,

\[ \frac{1}{2} Mv^2 + \varepsilon_0 = \frac{1}{2} Mv'^2 + h\nu . \]

On the other hand, from the law of conservation of momentum there follows the equality:

\[ M\vec v=M\vec v'+\frac{h\vec\nu}{c}. \]

Combining it with the preceding one and noting that

\[ v^2-v'^2=(\vec v+\vec v')(\vec v-\vec v'), \]

we obtain:

\[ \frac12(\vec v+\vec v')\frac{h\vec\nu}{c}=h\nu-h\nu_0 \]

or, since

\[ \frac12(\vec v+\vec v')=\vec v-\frac12(\vec v-\vec v')=\vec v-\frac{1}{2M}\frac{h\vec\nu}{c}, \]

\[ \frac{\vec v\cdot h\vec\nu}{c}-\frac{1}{2M}\left(\frac{h\nu}{c}\right)^2=h\nu-h\nu_0 \]

Denoting the angle between the direction of the initial velocity and the direction of the emitted quantum by \(\vartheta\), we finally obtain, for determining the dependence of \(\nu\) on \(\vartheta\), the following equation:

\[ \nu_0=\nu\left(1-\frac{v}{c}\cos\vartheta+\frac{h\nu}{2Mc^2}\right), \tag{5} \]

which, to a first approximation (if one neglects the term \(\frac{h\nu}{Mc^2}\), which is very small under ordinary conditions), becomes the well-known Doppler equation for the frequency of light emitted by a moving source.—In a completely analogous manner one may compute the Doppler effect for the frequency of light absorbed by a moving atom. For this it is only necessary, in the preceding calculation, to regard \(\nu\) as the frequency of the incident light, \(\vec v'\) as the initial velocity, \(\vec v\) as the final velocity, and, finally, to replace the angle \(\vartheta\) between \(v\) and \(\frac{h\vec\nu}{c}\) by the angle \(\vartheta'\) between \(\vec v'\) and \(\frac{h\vec\nu}{c}\). Assuming in this case

\[ \frac12(\vec v+\vec v')=\vec v'+\frac12(\vec v-\vec v')=\vec v'+\frac{1}{2M}\frac{h\vec\nu}{c}, \]

we obtain

\[ \vec v'\cdot\frac{h\vec\nu}{c}+\frac{1}{2M}\left(\frac{h\nu}{c}\right)^2=h\nu-h\nu_0, \]

i.e., instead of (5)

\[ \nu_0=\nu\left(1-\frac{v'}{c}\cos\psi'-\frac{h\nu}{2Mc^2}\right). \tag{5a} \]

The absorption of light may be connected with the splitting of the atom into two independently moving parts: a positive ion and a negative electron (the photoelectric effect). Considering the atom immobile, i.e. neglecting the Doppler phenomenon, in this case we obtain Einstein’s well-known equation for the photoelectric effect:

\[ h\nu=\frac{1}{2}mv^2+\varepsilon, \]

where \(m\) is the mass, \(v\) the velocity of the photoelectron, and \(\varepsilon\) the work necessary to tear it out of the atom. Here \(v\) is assumed small in comparison with \(c\); the exact expression for the kinetic energy has the form:

\[ c^2(m-m_0)=c^2m_0\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right). \]

In the case of the action of light on a free electron, i.e. on a system whose internal energy \((m_0c^2)\) remains unchanged, the complete disappearance of the quantum, i.e. the transformation of its energy and momentum entirely into the energy and momentum of the electron, proves impossible. Thus, for example, if the electron before collision with the quantum is considered at rest, then after the collision it would have to acquire the kinetic energy \(c^2(m-m_0)=h\nu\) and the momentum

\[ mv=\frac{h\nu}{c}. \]

From this there would follow the relation

\[ \frac{v}{c}=\frac{m-m_0}{m}=1-\sqrt{1-\frac{v^2}{c^2}}, \]

that is,

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

This equation has two solutions, \(v=c\) and \(v=0\); both of them are obviously devoid of physical meaning.

In the case under consideration, as Compton and Debye have shown, scattering of light occurs; having given part of its energy and its momentum to the electron, the quantum as it were rebounds from it, while having a somewhat lower frequency \((\nu')\) and a new direction of motion. This phenomenon, connected with the scattering of light by free electrons,

a decrease in the frequency of the oscillations and constitutes the essence of the Compton effect. Let us note that such a lowering of the frequency occurs only in the case when the electron is initially at rest; in the general case it is complicated by the Doppler phenomenon and may even, under certain conditions, be replaced by an increase in the frequency of the scattered quantum.

Let us denote the velocity of the electron before the “collision” with the quantum by \(v_1=c\beta_1\), and after the collision by \(v_2=c\beta_2\); let us denote the frequencies of the incident and scattered quantum by \(\nu_1\) and \(\nu_2\).

Let us further suppose that the initial direction of the quantum coincides with the axis \(X\); the cosines of the angles formed by the scattered quantum with the rectangular axes \(X,Y,Z\) we shall denote by \(p,q,r\), and the corresponding cosines for the vectors \(\vec v_1\) and \(\vec v_2\) by \(a_1,b_1,c_1\) and \(a_2,b_2,c_2\).

In this case the law of conservation of energy is expressed by the equation

\[ h\nu_1+\frac{m_0c^2}{\sqrt{1-\beta_1^2}} = h\nu_2+\frac{m_0c^2}{\sqrt{1-\beta_2^2}}, \]

and the law of conservation of momentum by the equations

\[ \frac{h\nu_1}{c} + \frac{m_0\beta_1c}{\sqrt{1-\beta_1^2}}\,a_1 = \frac{h\nu_2}{c}\,p + \frac{m_0\beta_2c}{\sqrt{1-\beta_2^2}}\,a_2 \]

\[ \frac{m_0\beta_1c}{\sqrt{1-\beta_1^2}}\,b_1 = \frac{h\nu_2}{c}\,q + \frac{m_0\beta_2c}{\sqrt{1-\beta_2^2}}\,b_2 \]

\[ \frac{m_0\beta_1c}{\sqrt{1-\beta_1^2}}\,c_1 = \frac{h\nu_2}{c}\,r + \frac{m_0\beta_2c}{\sqrt{1-\beta_2^2}}\,c_2 . \]

If from these equations one eliminates the cosines \(a_2,b_2,c_2\) (using the relation \(a_2^2+b_2^2+c_2^2=1\)), and also the quantity \(\beta_2\), and introduces the notations \(a_1=\cos\theta_1\), \(p=\cos\theta\) and \(a_1p+b_1q+c_1r=\cos\varphi\) (\(\varphi=\angle\) between \(\frac{h\nu_2}{c}\) and \(\vec v'\)), then the following formula is obtained\(^1\):

\[ \nu_2=\nu_1 \frac{1-\beta_1\cos\theta_1} {1-\beta_1\cos\varphi+2a\sqrt{1-\beta_1^2}\sin^2\frac{\theta}{2}}, \tag{6} \]

where, for brevity, we have put

\[ a=\frac{h\nu_1}{m_0c^2}. \]

\(^1\) See L. de Broglie — Thèse; Annales de Physique, p. 101, 1925.

In the case of an electron at rest \((\beta_1 = 0)\), the preceding formula becomes the well-known Debye–Compton formula:

\[ \nu_2=\frac{\nu_1}{1+2a\sin^2\frac{\theta}{2}}. \tag{6a} \]

Neglecting the quantity \(a\), which characterizes the Compton effect, we obtain the change of frequency caused by the double Doppler effect (for the electron as if simultaneously absorbs light and emits it again):

\[ \nu_2=\nu_1\frac{1-\beta_1\cos\theta_1}{1-\beta_1\cos\varphi}. \tag{6b} \]

We must mention one further application of Einstein’s theory of radiation, namely its application to the question of the thermal equilibrium between matter and radiation. The interaction between the atoms of matter and the atoms of light must be such that the former, even in the absence of collisions among themselves, have, on the average, a kinetic energy of \(\frac{3}{2}kT\), where \(T\) is the absolute temperature.

Indeed, a calculation, which we are unable to reproduce in this article\(^2\), shows that the recoils experienced by atoms in the emission or absorption of light quanta impart to them a kinetic energy of the above-mentioned magnitude. In doing so it is necessary to start from the spectral distribution of light energy determined by the well-known Planck formula.

§ 4. Polarization of Light and the Electromagnetic Moment of Light Quanta.

We saw above (§ 2) that the mechanical properties of light quanta—energy, mass, quantity of motion—are also inherent in electromagnetic waves, which in a certain sense they replace. However, these mechanical quantities are, so to speak, only secondary, “derived” properties of electromagnetic waves; the fundamental quantities by which these waves are characterized are electric and magnetic intensity. What, then, corresponds to these intensities in the case of light quanta? It is not difficult to see that this question may be paraphrased as follows: with what properties must quanta be endowed in order that they may perform the function of polarized light rays?

\(^2\) See the remarkable paper by Einstein—Phys. ZS 1917, p. 127.

Let us first imagine that we are dealing with linearly polarized light. In this case, according to wave theory, the electric and magnetic vectors oscillate in two fixed directions, perpendicular both to each other and to the rays themselves. These directions can be detected experimentally, for example, by the direction in which photoelectrons are ejected; as is known, the maximum number of the latter falls in the direction corresponding to the electric vector. If we wish to interpret the photoelectric effect as the result of the absorption of light quanta, we must endow the latter with some vector property corresponding to the electric intensity. The simplest property of this kind is the electric moment. It is therefore natural to suppose that a light quantum behaves as an elementary dipole, i.e. as a combination of two electric charges of opposite sign1. The electric moment of the quantum, characterizing linearly polarized light, we shall treat as a constant vector, and not as an oscillating quantity, like the electric intensity, which—or rather whose amplitude, so to speak—is “represented” by it. For the time being we shall leave the numerical value of this vector \((p)\) undetermined.

In exactly the same way, the magnetic intensity of a light wave may be “represented” in the quantum theory of light by the magnetic moment of the quantum \((\mu)\), perpendicular to the electric one and numerically equal to it. Both vectors \(\vec p\) and \(\vec\mu\) must be perpendicular to the direction of motion of the quantum (this motion taking place in the direction in which a right-hand screw, rotated from \(\vec p\) to \(\vec\mu\), moves).

Leaving aside the question of how the presence of an electric and magnetic moment gives the light quantum the ability to act directionally (with respect to one of them) on material particles, we shall try briefly to clarify another—purely formal—question: the compatibility of the preceding representation with the requirements of the theory of relativity.

In the theory of relativity the electric and magnetic intensities are combined into a single four-dimensional quantity with 6 different components (“Sechservektor”) \(F_{\alpha\beta}=-F_{\beta\alpha}\) \((\alpha,\beta=1,2,3,4)\) according to the following scheme:

\[ \begin{array}{c|c|c|c|c|c} F_{23} & F_{31} & F_{12} & F_{14} & F_{24} & F_{34} \\ H_1 & H_2 & H_3 & -iE_1 & -iE_2 & -iE_3, \end{array} \]

where the indices \(1,2,3\) refer to the three mutually perpendicular spatial axes, and \(4\) to the time axis (multiplied by \(C\sqrt{-1}\)). In this case the electric and magnetic intensities are not themselves

…by themselves invariant quantities; they depend on the choice of the coordinate system in which they are defined. Suppose, for example, that in some “resting” coordinate system \(K^0\) the magnetic intensity is equal to zero, and the electric one is \(\vec E^{\,0}\). Considering the same field from the point of view of another system \(K\), with respect to which \(K^0\) moves with the constant velocity \(\vec v\), we obtain, along with a certain electric field \(\vec E\), slightly different from \(E^0\), a magnetic field determined by the formula

\[ \vec H=\frac{\vec v}{c}\times \vec E. \]

In the first approximation, i.e. to accuracy up to quantities of order \(\dfrac{v}{c}\), \(E\) thereby coincides with \(E^0\).

If, conversely, in the system \(K^0\) the magnetic intensity \(H^0\) is nonzero, while the electric one is equal to zero, then in the moving system \(K\) we obtain

\[ \vec E=-\frac{\vec v}{c}\times \vec H \]

with \(\vec H\), coinciding in the first approximation with \(H^0\).

We see, therefore, that the electric and magnetic intensities are not, taken separately, a certain reality in themselves, but only when taken together: one of them cannot exist without the other.

The same applies to those quantities—the electric and magnetic moments—which, in the atomic theory of light, “represent” the electric and magnetic intensities. The electric and magnetic moments do not form independent realities, but have a definite invariant meaning only when taken together. If in the resting system \(K^0\) the particle under consideration is an electric dipole with moment \(\vec p^{\,0}\), then from the point of view of the moving system it behaves as a combination of an electric dipole with (approximately) the same moment \(\vec p\) and a magnetic dipole with moment

\[ \vec\mu=-\frac{\vec v}{c}\times\vec p. \tag{7} \]

Conversely, if in the system \(K^0\) the particle behaves as a magnetic dipole with moment \(\vec\mu^{\,0}\), then in the system \(K\) it acquires, in addition, the properties of an electric dipole with moment

\[ \vec p=+\frac{\vec v}{c}\times\vec\mu, \tag{7a} \]

where \(\vec\mu \simeq \vec\mu^{\,0}\).

Let us now imagine that the particle under consideration moves with velocity \(c\). In this case relations (7) and (7a)—generally speaking, i.e. for \(v<c\), incompatible with one another—can be satisfied simultaneously. For this, as is not difficult to verify, it is necessary and sufficient that the vectors \(\vec p\) and \(\vec\mu\) be numerically equal to one another, mutually perpendicular, and perpendicular to the direction of motion. We thus obtain, on the basis of the theory of relativity, precisely that relation between the electric and magnetic moments of light quanta which is necessary in order to ensure correspondence between them and the waves of the electromagnetic theory of light.

We shall not go more deeply here into the investigation of this question. Let us note only that electrically polarized light can be represented by quanta with complex values of the vectors \(\vec p\) and \(\vec\mu\). In any case, one may think that all properties of light not connected with its oscillatory character can be interpreted in terms of the theory of light quanta. As for the oscillatory properties, i.e. the periodicity in time and space that characterizes light, here quantum theory appears, evidently, to be powerless and must be supplemented by wave conceptions, as is done in the theory of L. de Broglie, or even, perhaps, be completely replaced by them, in accordance with Schrödinger’s wave mechanics.

1. This problem was formulated by Prof. P. Ehrenfest. ↩