Electromagnetic radiation and resonance phenomena in quantum mechanics V. A. Golovko Moscow State Evening Metallurgical Institute, Lefortovsky Val 26, Moscow 111250, Russia. Abstract It is demonstrated that, if one remains in the framework of quantum mechanics taken alone, stationary states (energy eigenstates) are in no way singled out with respect to nonstationary ones, and moreover the stationary states would be difficult if possible to realize in practice. Owing to the nonstationary states any quantum system can absorb or emit energy in arbitrary continuous amounts. The peculiarity of the stationary states appears only if electromagnetic radiation that must always accompany nonstationary processes in real systems is taken into account. On the other hand, when the quantum system absorbs or emits energy in the form of a wave the determining role is played by resonance interaction of the system with the wave. Here again the stationary states manifest themselves. These facts and influence of the resonator upon the incident wave enable one to explain all effects ascribed to manifestation of the corpuscular properties of light (the photoelectric effect, the Compton effect etc.) solely on a base of the wave concept of light. 2 1. Introduction Amongst elementary particles, the most unusual properties pertain apparently to the particle under number one, namely, the photon. As to other elementary particles, there exist some assumptions concerning their structure or at least one could estimate their size. In the matter of the photon, not only do any conjectures as to its structure lack but even there are no reasonable guesses about its dimensions. In this connection it should be remarked that the dual (particle-like and wave-like) properties of an object require two parameters with a dimension of length. For such particles as an electron or a proton, the size and de Broglie wavelength are quite different things, the latter being dependent in addition on the velocity even in the nonrelativistic case. If the object is characterized by only one parameter of length dimension, the object is either a particle or wave. Particularly striking are the properties of the photon once this notion is applied to radio waves that represent a special case of electromagnetic radiation. What could be said, for example, about the size of the photon (regarded as an elementary particle) in the case where the wavelength of a radio wave amounts to several kilometres? Once the photons of an electromagnetic wave are separate particles with different masses, if the wave is not monochromatic, a question may also be raised as to what forces keep the photons from diverging when the wave travels a huge distance in space without distorting its shape and without losing the information encoded in it. Contradictions in the notion of the photon and the fact that this notion is understood differently in different situations were observed time and again in the literature [1-3]. It is generally believed that convincing corroboration to the fact that light is made up of particles, i.e. the photons, is furnished by the photoelectric and Compton effects. At the same time, attempts were made to explain these phenomena on the base of undulatory concepts (see, e.g., [4-6]). Such attempts, however, suffer from an essential shortcoming: in order to explain these (and other) effects one is obliged to invent an individual mechanism for each effect, the mechanism being rather involved sometimes, whereas the explanation of all these effects with use made of the photonic concept alone indicates that a single mechanism lies at the root of the effects. Besides, the undulatory explanations mentioned sidestep often the fact that the photoelectric and Compton effects are observed practically without time delay [2,5]. The photon is one of the basic notions in quantum electrodynamics. In quantum electrodynamics, however, the photon is by no means a tiny particle. The photon is an elementary excitation of the electromagnetic field, the excitation that concerns the whole of the volume occupied by the system in question. Berestetskii et al. [7] underline that the wave function of a photon does not altogether signify the same thing as does the wave function of an ordinary particle in quantum mechanics, and the notion of coordinates of the photon has no physical meaning at all (see also [8], Sec. 5c). According to Bogoliubov and Shirkov [9], Sec. 6.2, by a particle in quantum field theory is meant an excitation of an oscillator, so that a nth excited state of the oscillator is referred to by convention as a state with n particles, the photons in our case. From [9] it follows as well that, in a coordinate space, to the photon corresponds a plane wave exp[i(ωt−kx)] filling the overall volume of a big cube V. Nowhere in 3 quantum electrodynamics does one resort to the concept of a point-like photon, and the photon in quantum electrodynamics is a wave pure and simple. In this connection Lamb [3] proposes to give up the use of the word ‘photon’ that often leads to confusion. An adequate terminology is always useful, of course, while a more important point emerges from the foregoing. Strange though it might appear at first thought, it is quantum electrodynamics that points to the fact that there must exist a wave explanation of the processes relevant to the photoelectric and Compton effects (quantum electrodynamics itself does not consider time development of the processes inasmuch as its methods are intended to calculate only the result of the processes). The question of another sort arises in connection with the concept of the photon. It would seem rather strange that two very different physical realms such as an electromagnetic field and a material quantum system (say, an atom) are characterized by one and the same constant, namely, the Planck constant =. Let us write down Bohr’s relation relevant to emission or absorption of electromagnetic radiation by the quantum system: =ω = En − Em. (1.1) On the left, one has the energy of a photon with angular frequency ω. The energy levels En and Em of the system on the right depend also upon the same constant =. Besides, the surprising thing is that different quantum systems create and emit identical photons whatever the nature and structure of a quantum system may be. It is worth noting that the interaction between the electromagnetic field and the quantum system is characterized by another constant, the charge, that pertains to the quantum system alone. Moreover, spectral lines have a natural breadth various for different lines, so that the radiation contains a continuous set of frequencies in the vicinity of the frequency given by (1.1) rather than one frequency ω. The same quantum system can be in a nonstationary state, in which case its energy does not equal any En. No relation is known which could replace Bohr’s rule of (1.1) when the system passes between nonstationary states or from a nonstationary state into a stationary one. At the same time, some radiation must exist in these situations too. The preceding paragraph suggests that, before proceeding to the problem of the photon, it is necessary to revisit the nonstationary states in quantum mechanics insofar as the stationary states alone are usually discussed in detail. In this paper it will be shown that the resulting logical chain that invokes electromagnetic radiation and resonance phenomena leads to the conclusions that all effects ascribed to manifestation of the corpuscular properties of light can be completely explained on a base of the wave concept of light alone and therefore there are no photons in nature. The question as to why quantum electrodynamics employs the notion of a photon (and the notion is helpful there) will also be discussed in the concluding section. 4 2. Nonstationary states Let us present first a brief summary of well-known facts that will be needed for what follows. A special role in quantum mechanics is ascribed to stationary states whose energy will be denoted as En and the corresponding eigenfunctions as ψn(q) where q is a set of coordinates. We suppose that the system in question is isolated and the Hamiltonian does not depend on the time. For the sake of simplicity we assume En to be discrete. If one knows ψn(q), one can also write down the general solution of the Schrödinger equation for an arbitrary nonstationary state of the system (see, e.g., [10,11]): i − Et = n Ψ(q,t) = ∑anneψ ()q, (2.1) n the coefficients an satisfying the normalization condition 2 ∑ an =1. (2.2) n The coefficients can be calculated if the initial wave function Ψ(q,0) is known with use made of the fact that the functions ψn(q) are orthonormal. The mean energy E of the system in the nonstationary state described by the wave function Ψ(q,t) is computed by one of the formulae ∗∗⌠ ∂Ψ E H dqi=⎮ d q, (2.3) =Ψ∫ Ψ =⎮ Ψ ⌡ ∂t where H(p,q) is the Hamiltonian of the system and the Schrödinger equation has been used. Substituting (2.1) into the last expression of (2.3) yields 2 E = ∑ aEnn. (2.4) n We can also calculate the mean square deviation from the mean energy: ⌠ 2 2222∗∗⎮ ∂Ψ 2 Δ≡−E EE =Ψ−H E Ψ=−Ψd q = ⎮ dqE − . (2.5) ()∫ () ⎮ 2 ⌡ ∂t In the case of (2.1) one has 222 2 Δ=E ∑ aEnn − E. (2.6) n If the system is in a stationary state where all an = 0 except for a coefficient am = 1, then (2.4) and 2 (2.6) give EE=Δ=m , E 0.