Classical-field description of the quantum effects in the light-atom interaction

In this paper I show that light-atom interaction can be described using purely classical field theory without any quantization. In particular, atom excitation by light that accounts for damping due to spontaneous emission is fully described in the framework of classical field theory. I show that three well-known laws of the photoelectric effect can also be derived and that all of its basic properties can be described within classical field theory.

Using this viewpoint, it was shown in [11] that all of the basic optical properties of the hydrogen atom have a simple and clear explanation in the framework of classical electrodynamics without any quantization. In particular, it was shown that the atom can be in a pure state indefinitely.
This arrangement means that the atom has a discrete set of stationary states, which correspond to all possible pure states, but only the pure state that corresponds to the lowest eigenfrequency is stable. Precisely this state is the ground state of the atom. The remaining pure states are unstable, although they are the stationary states. Any mixed state of an atom in which several eigenmodes are excited simultaneously is non-stationary, and according to classical electrodynamics, the atom that is in that state continuously emits electromagnetic waves of the discrete spectrum, which is interpreted as a spontaneous emission.
In reference [11], a fully classical description of spontaneous emission was given, and all of its basic properties that are traditionally described within the framework of quantum electrodynamics were obtained. It is shown that the "jump-like quantum transitions between the discrete energy levels of the atom" do not exist, and the spontaneous emission of an atom occurs not in the form of discrete quanta but continuously.
As is well known, the linear wave equation, e.g., the Schrödinger equation, cannot explain the spontaneous emission and the changes that occur in the atom in the process of spontaneous emission (so-called "quantum transitions"). To explain spontaneous transitions, quantum mechanics must be extended to quantum electrodynamics, which introduces such an object as a QED vacuum, the fluctuations of which are considered to be the cause of the "quantum transitions".
In reference [11], it was shown that the Schrödinger equation, which describes the electron wave as a classical field, is sufficient for a description of the spontaneous emission of a hydrogen atom. However, it should be complemented by a term that accounts for the inverse action of selfelectromagnetic radiation on the electron wave. In the framework of classical electrodynamics, it was shown that the electron wave as a classical field is described in the hydrogen atom by a nonlinear equation [11] where the last term on the right-hand side describes the inverse action of the self-electromagnetic radiation on the electron wave and is responsible for the degeneration of any mixed state of the hydrogen atom. Precisely, this term "provides" a degeneration of the mixed state of the hydrogen atom to a pure state, which corresponds to the lower excited eigenmodes of an atom. As shown in [11], this term has a fully classical meaning and fits into the concept developed in [7][8][9][10][11] in that the photons and electrons as particles do not exist, and there are only electromagnetic and electron waves, which are classical (continuous) fields.
In this paper, using equation (1), the light-atom interactions are considered from the standpoint of classical field theory. In particular, two effects will be considered: (i) an excitation of the mixed state of the atom by an electromagnetic wave without ionization of the atom; and (ii) the photoelectric effect, in which the action of the electromagnetic wave on the atom causes its ionization.

Optical equations with damping due to spontaneous emission
Let us consider a hydrogen atom that is in a light field, while accounting for its spontaneous emission.
Under the action of the light field, the excitation of some eigenmodes of the electron wave in the hydrogen atom at the expense of others occurs. This phenomenon will be manifested in a stimulated redistribution of the electric charge of the electron wave in the atom between its eigenmodes that occur under the action of the nonstationary electromagnetic field [11]. As a result, the atom, as an open volume resonator, will go into a mixed state in which, in full compliance with classical electrodynamics, electric dipole radiation will occur [11], which is customarily called a spontaneous emission. As shown in [11], the emission will be accompanied by a spontaneous "crossflow" of the electric charge of the electron wave from an eigenmode that has a greater frequency to an eigenmode that has a lower frequency. Under certain conditions, between the spontaneous and stimulated redistribution of the electric charge of an electron wave within the atom, a detailed equilibrium can be established, in which the amount of electric charge that crossflows from an eigenmode into an eigenmode per unit time will be equal to the amount of electric charge that crossflows from eigenmode into eigenmode per unit time.
Clearly, such an equilibrium will be forced and will depend on the intensity of the light field. This is easily seen when the light field is instantaneously removed. As shown in [11], the atom, which was previously in a mixed state, will spontaneously emit electromagnetic waves, and this spontaneous emission will be accompanied by a spontaneous redistribution of the electric charge of the electron wave between the excited eigenmodes of the atom. As a result, over time, the whole electric charge of the electron wave "crossflows" into a lower eigenmode (i.e., with a lower frequency) of the initially excited eigenmodes and the atom enters into a pure state which can be retained indefinitely because spontaneous emission is absent in this state.
Let us consider this process in more detail by the example of the hydrogen atom.
If an atom is in an external electromagnetic field, then equation (1) should be supplemented by the terms which take into account the external action. As a result, one obtains the equation where ( , ) and ( , ) are the scalar and vector potentials of an external electromagnetic field.
We consider a linearly polarized electromagnetic wave with the wavelength , which is much more than the characteristic size of the hydrogen atom: In this case, one can consider the electric field of the electromagnetic wave in the vicinity of the hydrogen atom to be homogeneous but nonstationary: ( , ) = 0 cos , where 0 and are constants. For such a field, one can select the gauge [12], at which In this case, equation (2) takes the form As usual, the solution of equation (5) can be found in the form where the constants and the functions ( ) are the eigenvalues and eigenfunctions of the linear Schrödinger equation The functions ( ) form the orthonormal system: Substituting (6) into (5) while accounting for (7) and (8), we obtain is the electric dipole moment of the electron wave in the hydrogen atom: Considering expressions (6), (11) and (12), we can write Let us consider the so-called "two-level atom", i.e., the case in which only two eigenmodes and of an atom are excited simultaneously.
For definiteness, one assumes that > , and correspondingly, > 0 (14) In this case, equations (9) and (11) take the form In general, differentiating vector with respect to time, we must account for the fact that the parameters are functions of time. As shown below, the parameters are changed with time much more slowly than the oscillating factor exp( ). This arrangement means that there is the condition Considering (17) and (18), we obtain approximately Substituting expression (19) into equations (15) and (16), we obtain where | | 2 = ( * ), ( Equations (20) and (21) contain rapidly oscillating terms with frequencies of , and 2 , which in view of (18) can be removed by averaging over the fast oscillations. As a result, we obtain the equations Equations (22) and (23) describe the Rabi oscillations with damping due to spontaneous emission. The first term on the right-hand side of equation (22) describes the excitation of mode due to the impact of the incident electromagnetic wave on mode . In quantum mechanics, it is traditionally interpreted to be an induced transition from a lower energy level to a higher energy level due to "absorption of the photon". The first term on the right-hand side of equation (23) describes the excitation of mode due to the impact of the incident electromagnetic wave on mode . In quantum mechanics, it is traditionally interpreted as an induced transition from a higher energy level to a lower energy level due to the "emission of a photon". The last term on the right-hand side of equations (22) and (23) is traditionally interpreted as a spontaneous transition from a higher level to a lower level , which is accompanied by an " emission of a photon" with a frequency of . Obviously, Condition (25) according to [11] expresses the law of conservation of electric charge, which means that in the process under consideration, the electric charge of the electron wave is simply redistributed between modes and . Note that in this section, we do not consider the photoelectric effect, i.e., an emission of an electron wave by the atom under the influence of an incident electromagnetic wave; the photoelectric effect will be discussed below.
Using equations (22) and (23) for the parameters in (24), we obtain the equation Equations (27) and (28) describe the interaction of electromagnetic waves with a hydrogen atom while accounting for a spontaneous emission. They, in fact, are the optical equations with damping due to spontaneous emission [13]. However, there are fundamental differences in equations (27) and (28) from the conventional (linear) optical equations [13]. Thus, in quantum optics, the optical equations that account for damping due to spontaneous emission are not turn into the conventional (linear) optical equations [13]. Moreover, the damping rate cannot be obtained within the framework of the linear Schrödinger equation and is introduced into the linear optical equations phenomenologically [13]; its value (29) is derived only in the framework of quantum electrodynamics. In the theory under consideration, the damping rate (29) is a direct and natural consequence of the nonlinear Schrödinger equation (5).
Let us consider a conventional approach for such systems in the case in which | − | ≪ [13]. Substituting cos = where is a constant.
Substituting expression (36) into equations (34) and (35), we obtain the equations Hence, using (25), we obtain In particular, Equation (40) can be rewritten in the form The solution of equation (42) exists only if its right-hand side is positive. This arrangement is One can rewrite this condition in the form which is satisfied for any if or if The solutions of equation (42) for different values of the parameter | |⁄ are shown in Fig.   1. Equation (42) has two solutions for the same Ω (see Fig. 1). Their sum is equal to one, which means that one root can be considered to be , while the other is considered to be .
Theoretically, each of these roots can correspond to . Recall that condition (14) was accepted.
Therefore, due to the spontaneous emission, mode always loses an electric charge, while mode receives it. This arrangement means that only the smaller root of equation (42) should correspond to mode , while the larger root corresponds to mode . Such a solution will obviously be always stable: there are no small perturbations that could violate this condition because the system will always return to it. In contrast, the second solution, in which corresponds to a larger root of equation (42), will be unstable, and any small perturbations lead to the system spontaneously returning to the first stable state due to crossflow of the electric charge of the electron wave from the upper excited mode to the lower mode. Thus, in the stable state, the smaller of the two roots of equation (42) always corresponds to , while the larger root corresponds to .
Then, for the case in (46), the maximum value of , which can be achieved at resonance The corresponding minimum value of will be ( ) =

Light scattering by an atom
Using the results of the previous section, it is easy to calculate the secondary radiation (induced and spontaneous), which an atom that is in the field of a classical electromagnetic wave creates.
This radiation will be perceived as the scattering of the incident electromagnetic wave.
The intensity of the electric dipole radiation according to classical electrodynamics is defined by the expression [12] where is the electric dipole moment of the electron wave in a hydrogen atom; the bar denotes averaging over time.
For a two-level atom, as discussed in the previous section, accounting for expressions (17), (18) and (24) implies that If the atom is in a stationary forced excited state, then the parameter is determined by the expressions (36) and (39), and it oscillates at the frequency given in (33).
In this case, and for the intensity of the scattered radiation, we obtain or when accounting for expressions (29) and (41), we obtain According to the expression in (52), in the approximation under consideration, we are contending with Rayleigh scattering. However, if we account for the rapidly oscillating terms in equations (27) and (28), the non-Rayleigh components in the scattering spectrum will be detected, but their intensity will be negligible.
Let us calculate the scattering cross-section where [12] =̈2 4 3 sin 2 (56) is the amount of energy that is emitted (scattered) by the atom per unit time per unit solid angle ; is the angle between the vector ̈ and the direction of the scattering; and is the energy flux density of the incident electromagnetic wave.
In our case, Then, accounting for expressions (29) and (41), we obtain Using expressions (29) and (40), we can also write = 9ℏ 2 2 2 The scattering pattern will be determined by the parameter .
In that case, when the vector is real-valued, based on definition (30), we can write where is the angle between the vectors and 0 .
Then, we obtain In other cases (Ω ≠ 0), in the calculation of the right-hand side of expression (62), it is necessary to account for the fact that parameter , which is a solution of equation (42), will depend on the nondimensional parameter | | 2 2 ⁄ (see Fig. 1). Accounting for expressions (61) and (29) At a low intensity of the incident electromagnetic wave, when we obtain ≪ 1, and then, Then, using expression (60), we obtain In particular, at the resonance frequency (Ω = 0), = 9 2 4 2 sin 2 (cos 2 cos 2 + sin 2 cos 2 ) In other cases (Ω ≠ 0), in the calculation of the right-hand side of expression (68), it is necessary to account for the fact that the parameter , which is a solution of equation (42), will depend on the nondimensional parameter | | 2 2 ⁄ (see Fig. 1). Using the expressions (67) and (29) Thus, the Rayleigh scattering of an electromagnetic wave by a hydrogen atom is fully described within the framework of classical field theory without any quantization.
Until now, we have assumed that under the influence of an incident electromagnetic wave, the electron wave in an atom is only redistributed between its eigenmodes but not emitted outward by the atom. In this case, internal electric currents arise inside the atom that, however, cannot be detected by macroscopic devices. Such a situation occurs at a relatively low frequency of the incident electromagnetic wave. If this frequency is sufficiently large, then an emission of the electron wave by the atom occurs. Because the electron wave has an electric charge that is continuously distributed in space [10,11], in this case, an external electric current (photoelectric current) appears that can be detected by macroscopic devices. As a result, the photoelectric effect will be observed.
The photoelectric effect has a special place in quantum mythology because it became the first This contradiction was overcome due to the quantization of radiation, which postulates that the absorption of light occurs in the form of discrete quanta ℏ (A. Einstein, 1905). At present, in connection with this finding, it is considered to be generally accepted that the photoelectric effect provides "evidence" for the quantum nature of light.
However, in the early years of quantum mechanics, it was shown that the photoelectric effect is fully described within the framework of so-called semiclassical theory, in which light is considered to be a classical electromagnetic wave, while the atom is quantized and described by the wave equation, e.g., the Schrödinger equation or the Dirac equation [1,13,15,16]. In this case, the wave equation is solved as a typical classical field equation, whereby a continuous wave field is calculated. A "quantization" of this wave field occurs only at the stage of interpreting the solution, from which the "probability of photoelectron emission" from an atom is determined.
Because the electron in such a theory is considered to be a quantum particle and light is considered to be a classical electromagnetic field, such a theory is considered to be "semiclassical".
However, as shown in [7][8][9][10][11], there is no need to introduce the quantization of electromagnetic and electron fields because this interpretation is external to the wave equation, and it does not follow from these equations. Moreover, this approach is superfluous in explaining the many physical phenomena that before were interpreted as a result of the quantization of matter.
As will be shown below, the failures of classical electrodynamics in explaining the photoelectric effect are connected with the incorrect postulate that electrons are particles. I will show that for a consistent explanation of the photoelectric effect within the framework of classical field theory, it is sufficient to abandon this postulate and instead consider continuous classical electron waves instead of the particles-electrons [10,11]. The considered theory is fully classical because it does not contain not only the quantization of the radiation but also the quantization of the electron wave.
From the considered point of view [7][8][9][10][11], the photoelectric effect represents an emission of the continuous charged electron wave by an atom that was excited by the incident classical electromagnetic wave. Formally, the photoelectric effect is no different from the stimulated emission of electromagnetic waves by an atom [11], with the only difference being that the electron wave emitted by an atom is electrically charged while the electromagnetic wave does not carry the electric charge. Assuming that the electric charge is continuously distributed in the electron wave [10,11], one concludes that in the process of the emission of the electron wave, the atom is positively charged continuously. However, accounting for the fact that the electron wave for an as yet inexplicable reason does not "feel" its own electrostatic field [11], this process will not affect the emission of the following "portions" of the continuous electron wave because they must overcome the same electrostatic potential of the nucleus.
Let us consider the photoelectric effect for the hydrogen atom being in the classic monochromatic electromagnetic wave.
In this section, we neglect the inverse action on the electron wave of its own non-stationary electromagnetic field. For this reason, the last term in the Schrödinger equation (5), which is associated with a spontaneous emission of the electromagnetic waves, will not be considered, and we will use the conventional linear Schrödinger equation where 0 is the frequency of the incident light. We will consider here the approximation in (3), when the wavelength of the incident electromagnetic wave is substantially larger than the characteristic spatial size of the electron field in the hydrogen atom, which is of the order of the Bohr radius .
The wave function of an electron wave can be represented as in [17] = where the first sum describes that part of the electron wave that is contained in the eigenmodes of the atom (i.e., corresponding to a "finite motion" of the electron wave), and for this term, all < 0, while the integrals describe the electron waves that are emitted by an atom (i.e., which corresponds to the "infinite motion" of the electron wave), to which it is known that > 0 Because the frequencies have 0 > 0, > 0 and 1 < 0, the value 1 − − 0 is not equal to zero for any , and thus, the first term will always be limited and will describe the oscillations that are of small amplitude. At the same time, 1 − + 0 = 0 at the resonance frequency of 0 = | 1 | + , and near the resonant frequency, the second term in (86) will increase indefinitely. Therefore, the second term in (86) makes the main contribution to the effect that is under consideration. Neglecting the first term in expression (86), we obtain and integrating it over the surface of an infinite sphere whose centre is in the nucleus of the atom. However, it is more convenient to accomplish this step while using the law of conservation of charge and accounting for the fact that = − | | 2 is the electric charge that is contained in mode of the electron wave [11]. Then, ̇ is the internal electric current in the atom, by which The second and third terms on the right-hand side of expression (92) are rapidly oscillating at a frequency of 0 , and they can be discarded by averaging over the fast oscillations. Then, we Assuming that all of the orientations of the atom in space are equally probable, and therefore the vector 1 is statistically isotropic, one averages the current (93) over all possible orientations of the atom. Then, where the bar denotes averaging over all possible orientations, and the indices and are the vector indexes.
For the isotropic vector 1 , Then, Accordingly, for the mean photoelectric current (93), we obtain where the parameter does not depend on the incident light intensity | 0 | 2 , and instead, the parameter depends on the frequency 0 of the incident light.
Thus, we have obtained the first law of the photoelectric effect without using the photon hypothesis within the framework of only classical field theory while considering the electromagnetic and electron waves as classical fields.
Let us consider the dependence of the parameter on the frequency of the incident light 0 .
Let us denote Then, we obtain Here, we account for the fact that 1 < 0.
The function sin( ) has a sharp peak in the vicinity of = 0 and has a width of Δ~⁄ , and at → ∞, it behaves similar to a delta-function: ∫ sin( ) ∞ −∞ = . The function ( ) in the vicinity of = 0 is smooth and varies weakly on the interval Δ~⁄ . Therefore, with reasonable accuracy at 0 < | 1 | − 2 , we can write , it is necessary to account for the fact that a small neighbourhood of the point = 0 will make the main contribution to the integral in (101) (due to the delta-like behaviour of the integrand). As a result, for 0 − | 1 | ≫ 2 , we obtain In this case, the parameter will vary with the frequency of the incident light 0 . Fig. 3 shows, in a nondimensional form, the dependence of the parameter on the frequency difference 0 − | 1 | in the vicinity of the frequency 0 = | 1 |.
We can see that the parameter is virtually zero at 0 < | 1 | − 2 , and it almost linearly varies from zero to 3 6ℏ 2 (0) when 0 changes in the range from | 1 | − 2 to | 1 | + 2 , and it virtually equals the value in (103) at 0 > | 1 | + 2 . The width of the frequency range in which there is a noticeable change in the parameter is Δ 0~⁄ .
Assuming | 1 |~10 14 rad/s (which corresponds to visible light) for the observation time > 10 -9 s, we obtain Δ 0 <3•10 9 rad/s, which is significantly less than | 1 |: From this analysis, it follows that for the actual duration of the observation, the parameter will have almost a threshold dependence on the frequency of the incident light 0 : for 0 < | 1 |, we (w 0 -|w 1 |)t obtain ≈ 0, and the photoelectric current is almost absent, while at 0 > | 1 |, the parameter will take the value in (103), and the photoelectric current (97) will be proportional to the intensity of the incident light.
Thus, we have obtained the third law of the photoelectric effect also without using the photon hypothesis, within only the framework of classical field theory. Let When accounting for the smallness of the frequency range (106), it can be assumed that the electron wave that is emitted by an atom is almost monochromatic and has the frequency in (107), which linearly depends on the frequency of the incident light 0 and does not depend on its intensity.
Let us place on the path of the electron wave a decelerating potential. In this case, we come to the problem of propagation of the electron wave in the field of the decelerating potential, which is quite accurately described by the linear Schrödinger equation. At large distances from the atom, the electron wave can be considered to be approximately flat. To simplify the analysis, instead of the decelerating potential, having a linear dependence on the coordinates along which the electron wave propagates, let us consider the potential step (barrier) of the same "height" 0 and the same width to be the actual decelerating potential. The solution of the Schrödinger equation for the potential step is well known [17]: at ℏ > 0 , the electron wave passes through a potential step and is partially reflected from it, while when ℏ < 0 , the electron wave is mainly reflected from the potential step, although a small part goes through the potential step due to tunnelling. The transmission coefficient of the electron wave for the potential step (in our interpretation, this coefficient is the ratio of the electric current of the electron wave behind the potential step to the electric current of the electron wave arriving to the potential steps from an atom) in the limiting case ℏ = 0 is defined by the expression [17] = (1 + Here, instead of the energy of a non-relativistic quantum particle, we use a Schrödinger frequency (which is equal to the difference between the true frequency of the electron wave that is entered into the solution of the Dirac equation and its "rest frequency" = 2 ℏ ⁄ [10]).
With the increase in the width of the potential step , the transmission coefficient (108) decreases rapidly, and for an actual decelerating potential that has macroscopic sizes that substantially exceed the de Broglie wavelength = 2 √ ℏ 2 , it is almost equal to zero because, in this case, we can neglect the tunnelling.
Thus, for the macroscopic decelerating potentials that are used in the experiments, there is a threshold effect: when ℏ > 0 , the electron wave passes through the decelerating potential, while when ℏ ≤ 0 , the electron wave is fully "reflected" by the decelerating potential and the photoelectric current is not observed behind it. This arrangement means that there is a limit to the value of the decelerating potential, which is the stopping potential = ℏ above which the photoelectric current is absent.
When accounting for expression (107), we obtain This result completely coincides with the above given formulation of the second law of the photoelectric effect, and it was obtained within the framework of classical field theory without the use of such concepts as photons and electrons.
Note that expression (110) can be formally written in the form where the notations = ℏ| 1 | and = were introduced. The expression in (111) can be considered to be Einstein's equation for the photoelectric effect, and one can interpret it within the framework of the photon-electron representations in which the parameter is interpreted as the kinetic energy of the photoelectrons, while the parameter is interpreted as a work function of the atom. However, this approach is no more than an interpretation that is based on the formal similarity of the pure wave expression (110) and the mechanical law of energy conservation.
The above analysis has shown that such a corpuscular interpretation of the photoelectric effect is superfluous.
The well-known experiments by E. Meyer and W. Gerlach (1914) on the photoelectric effect on the particles of metal dust, irradiated with ultraviolet light, are considered to be one of the pieces of "irrefutable evidence" that light energy is propagated in the form of identical indivisible quanta (photons). Assuming that the electrons are particles while light is composed of continuous classical electromagnetic waves, we can calculate the time during which the metal particle will absorb a sufficient amount of energy for the ejection of an electron. In the experiments by E. Meyer and W. Gerlach, this duration was of the order of a few seconds, which means that the photoelectron cannot leave a speck of dust earlier than in a few seconds after the start of irradiation. In contrast to this conclusion, the photoelectric current in these experiments began immediately after the beginning of the irradiation. Hence, it is usually concluded that this finding is only possible if the light is a flux of photons each of which can be absorbed by the atom only entirely and, therefore, can "knock out" the electron from the atoms at the moment of its collision with the metal particle.
However, this conclusion follows only in the case in which the electrons are considered to be indivisible particles. If instead of considering the electrons to be particles we consider a continuous electron wave [10,11], then as was shown above, the photoelectric current appears almost without delay after the start of irradiation of an atom by the classical electromagnetic wave and occurs even at very low light intensities, when the light frequency exceeds the threshold frequency for the given atom. This finding is because to start the photoelectric current, the atom does not need to accumulate the energy that is equal to the ionization potential because the electron wave is emitted by the atom continuously and not in the form of discrete portions -"electrons". Note that precisely the need to explain the ejection of discrete electrons from an atom under the action of light led A. Einstein to the idea of light quanta, which when absorbed, gave to the atom sufficient energy for the liberation of a whole electron.
The above analysis shows that all three laws of the photoelectric effect only approximately reflect its actual regularities. In particular, the photoelectric current appears and disappears nonabruptly when "passing" through the threshold frequency | 1 |, and it gradually increases or decreases in the frequency range that has the width Δ 0~⁄ near the threshold frequency | 1 |.
However, this effect can be detected only for ultrashort observation times of ~10 -15 s, which is difficult to achieve in the experiments on the photoelectric effect. Moreover, consideration of the nonlinear effects in the interaction of the light wave with an atom shows [18] that the photoelectric current appears even in the case when the frequency of the incident light is significantly less than the threshold frequency | 1 |, which is predicted by the linear theory. Such effects can be observed only in a very intense laser field [19]. Strictly speaking, the theory [18], which describes the ionization of an atom in an intense laser field, is fully classical in the sense under consideration because an atom is described by the Schrödinger equation, while the light wave is considered to be a classical electromagnetic field. The true result of this finding is the photoelectric current that is created by the continuous electron wave emitted by an atom because precisely the photoelectric current is calculated in the theory [18]. However, traditionally, the results of the theory [18] are interpreted from the standpoint of photon-electron representations, which makes it necessary to interpret the main result of the theory [18] as the probability of the ionization of an atom (i.e., the probability of the liberation of an "electron" from the atom) per unit time. The representations with respect to the multiphoton ionization of an atom, when the atom "absorbs simultaneously several photons", the total energy of which exceeds the ionization potential of the atom, were a consequence of such an interpretation. When there is a requirement for too many "photons" for the liberation of the "electron", talking about the simultaneous absorption of such a large number of particles becomes meaningless (because of the low probability of this process); then, the results of the theory [18] are interpreted as a tunnel ionization in which the intense laser field changes the potential field in which the "electron" is positioned, which gives it the "opportunity" to leave the atom due to tunnelling. From the point of view of the ideas that are developed in this series of papers, both "multiphoton" and "tunnel" ionization of an atom are the result of the same process -the interaction of a classical electromagnetic wave with a classical electron wave.
Finally, note that there is no difficulty in calculating the angular distribution of the photoelectric current in the framework of the theory under consideration, if we account for the fact that the continuous electric current created by the electron wave emitted by an atom under the action of light is calculated by expression (88) using the wave function in (73) and (87). Once again, note that this current is not the distribution over the directions of the particles-electrons that are emitted by an atom, but the distribution over the directions of the current of a continuous charged electron wave that is emitted by the atom. All of the known expressions that are obtained earlier for the photoelectric effect (see, e.g., [14,16]) remain valid, but they should now be interpreted from the standpoint of classical field theory.

Concluding remarks
Thus, we see that the light-atom interaction is fully described within the framework of classical In this paper, in the calculation of an atom-field interaction, we did not account for a property of an electron wave, such as the spin, i.e., the internal angular momentum and the associated internal magnetic moment of the electron wave that is continuously distributed in space [10]. To describe the atom-field interaction while accounting for the internal magnetic moment of the electron field, it is necessary to use the Dirac equation or, in the Schrödinger long-wave approximation [10], the Pauli equation, which should be supplemented by the terms that account for the inverse action of the self-radiation field on the electron wave. This issue will be considered in the subsequent papers of this series.