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Electromagnetic Radiation Under Phase Symmetry Breaking in Quantum Systems

Dhiräj Sinha

Published Feb 10, 2021• 0 citations
Publisher

Abstract

Abstract According to classical mechanics, electron acceleration results in electromagnetic radiation while in quantum mechanics radiation is considered to be arising out of a transition of the charged particle from a higher to a lower energy state. A different narrative is presented in quantum field theory, which considers radiation as an outcome of the perturbation of zero point energy of quantum harmonic oscillator which results in a change in density of electrons in a given state. The theoretical disconnect in the phenomenological aspect of radiation in classical and quantum mechanics remains an unresolved theoretical challenge. As a charged particle changes its energy state, its wavefunction undergoes a spatial phase change, hence, we argue that the spatial phase symmetry breaking of the wavefunction is a critical aspect of radiation in quantum mechanics. This is also observed in Josephson junction, where a static voltage induces spatial phase symmetry breaking of current resulting in emission of electromagnetic waves. As temporal symmetry breaking of the magnetic vector potential generates classical radiation and a wave-function of a charged particle can always be associated with a specific magnetic vector potential; the concept of radiation under spatial phase symmetry breaking offers a novel perspective towards unifying the phenomenon of radiation in quantum mechanics and classical electromagnetism.

DOI
10.21203/rs.3.rs-175077/v1

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Abstract According to classical mechanics, electron acceleration results in electromagnetic radiation while in quantum mechanics radiation is considered to be arising out of a transition of the charged particle from a higher to a lower energy state. A different narrative is presented in quantum field theory, which considers radiation as an outcome of the perturbation of zero point energy of quantum harmonic oscillator which results in a change in density of electrons in a given state. The theoretical disconnect in the phenomenological aspect of radiation in classical and quantum mechanics remains an unresolved theoretical challenge. As a charged particle changes its energy state, its wavefunction undergoes a spatial phase change, hence, we argue that the spatial phase symmetry breaking of the wavefunction is a critical aspect of radiation in quantum mechanics. This is also observed in Josephson junction, where a static voltage induces spatial phase symmetry breaking of current resulting in emission of electromagnetic waves. As temporal symmetry breaking of the magnetic vector potential generates classical radiation and a wave-function of a charged particle can always be associated with a specific magnetic vector potential; the concept of radiation under spatial phase symmetry breaking offers a novel perspective towards unifying the phenomenon of radiation in quantum mechanics and classical electromagnetism.
This chapter reviews the classical description of radiation. It describes how classical electromagnetic radiation interacts with a quantum mechanical particle. The general rule is that, even though the wave functions and the electromagnetic potentials change with the gauge, all physically observable quantities are independent of the gauge. The chapter also describes the process of the absorption of light by a system in terms of the effect of an external classical electromagnetic field on the system. The total emission rate is the sum of the induced and the spontaneous rates. Spontaneous emission is just the quantum mechanical version of the classical phenomenon of radiation from an accelerating charge. A very important consequence of the fact that the electromagnetic field is quantized is that there are vacuum fluctuations of the field, analogous to the zero point motion of a harmonic oscillator. Einstein gave a simple statistical argument to determine the rate at which spontaneous emission takes place.
It is in accordance with such experiments as single photon self-interference that a photon, conveying one radiation energy quantum " h × frequency", is spatially extensive and stretches an electromagnetic wave train. A wave train, hence an energy quantum, can only be emitted (or absorbed) by its source (or absorber) gradually. In both two processes the wave and "particle" attributes of the radiation field are simultaneously prominent, where an overall satisfactory theory has been lacking; for the latter process no known theoretical description currently exists. This paper presents a first principles treatment, in a unified framework of the classical and quantum mechanics, of the latter process, the emission (similarly absorption) of a single radiation quantum based on the dynamics of the radiation-emitting source, a charged oscillator, which is itself extensive across the potential well in which it oscillates. During the emission of one single radiation quantum, the extensive charged oscillator undergoes a continuous radiation damping and is non-stationary. This process is in this work treated using a quasi stationary approach, whereby the classical equation of motion, which directly facilitates the correspondence principle for a particle oscillator, and the quantum wave equation are established for each sufficiently brief time interval. As an inevitable consequence of the division of the total time for emitting one single quantum, a fractional Planck constant h is introduced. The solutions to the two simultaneous equations yield for the charged oscillator a continuously exponentially decaying Hamiltonian that is at the same time quantised with respect to the fractional- h at any instant of time; and the radiation wave field emitted over time stretches a wave train of finite length. The total system of the source and radiation field maintains at any time (integer n times) one whole energy quantum, ( n × ) h × frequency, in complete accordance with the standard notion of quantum mechanics and experiment.
The magnetic Aharonov-Bohm effect shows that charged particles may be affected by the vector potential in regions without any electric or magnetic fields [1]. The Aharonov-Bohm effect was experimentally confirmed [2-3] and has been found in many situations [4-6]. A common explanation is based on quantum mechanics, which states that the wavefunctions associated with the charges will accumulate a phase shift due to the vector potential. However, consensus about its nature and interpretation has not been achieved [7-14]. We here propose a simple but reasonable interpretation based on the theory for electromagnetic radiation and couplings [15]. The energy associated with a pulse radiator is divided into a Coulomb-velocity energy and a radiative energy, together with a macroscopic Schott energy accounting for the energy exchange between them. All these energies are expressed with terms including the potentials, so are the mutual coupling energies. There exists a force acting on the moving charges even though the fields completely vanish. This force makes the charges pass through the magnetic solenoid in different velocity with different path length, causing a phase shift the same as that obtained with quantum mechanics. The theory is originally aimed for providing an interpretation for electromagnetic radiation and mutual coupling. It is derived directly from the Maxwell theory with no modification but only substitution and reorganization.
Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the zzz axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.
The quantum theory of radiation, formulated in wave-mechanical language but in terms of running waves, is shown to lead under suitable conditions to the classical theory of emission by a particle as a first approximation. Non-relativistic mechanics is used, and the limitations of the method are brought out in terms of classical theory. The characteristic quantum phenomenon of the Compton effect requires that the Compton recoil shall exceed the spread in momentum of the particle, and so that the wave-packet shall greatly exceed in diameter the wave-length of the incident radiation; this condition is realizable on any scale of magnitude. For quasi-classical motion just the reverse is required; furthermore, if the motion is to last for a number of waveperiods, the wave-length must greatly exceed hmc\frac{h}{\mathrm{mc}}mch​ (m=massoftheparticlem=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{particle}m=massoftheparticle). The classical packet-motion arises from combinations between quantum-states of the particle-field system; this fact suggests that the photon is not a fundamental constituent of the field but only one possible aspect of its action. The infinite "zero-energy" associated apparently even with the lowest energy-state of the field implies no electromagnetic field intensities at any point. Whereas for problems on energy or momentum it is customary to represent an incident wave-train by a single typical quantum state, to obtain motional phenomena of classical type a more complete expression such as that given in the paper must be used. The physically empty field likewise corresponds to a mixture of quantum states but without correlation among their amplitudes.
Classical electromagnetic radiation from quantum currents and densities are calculated. For the free Schrodinger equation with no external force it's found that the classical radiation is zero to all orders of the multipole expansion. This is true of mixed or pure states for the charged particle. It is a non-trivial and surprising result. A similar result is found for the Klein-Gordon currents when the wave function consists of only positive energy solutions. For the Dirac equation it is found that radiation is suppressed at lower frequencies but is not zero at all frequencies. Implications of these results for the interpretation of quantum mechanics are discussed.