arXiv subjects

Timothy H. Boyer

Publications and source records attributed to Timothy H. Boyer.

At least 37 records · Page 2Linked to original sources

arXiv2021

Relativity and Radiation Balance for the Classical Hydrogen Atom in Classical Electromagnetic Zero-Point Radiation

Here we review the understanding of the classical hydrogen atom in classical electromagnetic zero-point radiation, and emphasize the importance of special relativity. The crucial missing ingredient in earlier calculational attempts (both numerical and analytic) is the use of valid approximations to the full relativistic analysis. It is pointed out that the nonrelativistic time Fourier expansion coefficients given by Landau and Lifshitz are in error as the electromagnetic description of a charged particle in a Coulomb potential, and, because of this error, Marshall and Claverie's conclusion regarding the failure of radiation balance is invalid. Rather, using Marshall and Claverie's calculations, but restricted to lowest nonvanishing order in the orbital eccentricity (where the nonrelativistic orbit is a valid approximation to the fully relativistic electromagnetic orbit) radiation balance for classical electromagnetic zero-point radiation is shown to hold at the fundamental frequencies and associated first overtones.

physics.class-ph↗

arXiv2020

Conflict Between Classical Mechanics and Electromagnetism: The Harmonic Oscillator in Equilibrium with a Bath

It is pointed out that an electric charge oscillating in a one-dimensional purely-harmonic potential is in detailed balance at its harmonics with a radiation bath whose energy $U_{rad}$ per normal mode is linear in frequency $ω$, $U_{rad}=const\timesω,$ and hence is Lorentz invariant, as seems appropriate for relativistic electromagnetism. The oscillating charge is NOT in equilibrium with the Rayleigh-Jeans spectrum which arises from energy-sharing equipartition ideas which are valid only in nonrelativistic mechanics. Here we explore the contrasting behavior of harmonic oscillators connected to baths in classical mechanics and electromagnetism. It is emphasized that modern physics text are in error in suggesting that the Rayleigh-Jeans spectrum corresponds to the equilibrium spectrum of random classical radiation, and in ignoring Lorentz-invariant classical zero-point radiation which is indeed a classical equilibrium spectrum.

physics.class-ph↗

arXiv2020

Dirac's Classical-Quantum Analogy for the Harmonic Oscillator: Classical Aspects in Thermal Radiation Including Zero-Point Radiation

Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant $\hbar$) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution which arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant $\hbar$. All mechanical systems in thermal radiation will inherit the constant $\hbar$ from thermal radiation. Here we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions.

quant-ph↗

arXiv2019

Stochastic Electrodynamics: The Closest Classical Approximation to Quantum Theory

Stochastic electrodynamics is the classical electrodynamic theory of interacting point charges which includes random classical radiation with a Lorentz-invariant spectrum whose scale is set by Planck's constant. Here we give a cursory overview of the basic ideas of stochastic electrodynamics, of the successes of the theory, and of its connections to quantum theory.

physics.class-ph↗

arXiv2019

Equilibrium for Classical Zero-Point Radiation: Detailed Balance Under Scattering by a Classical Charged Harmonic Oscillator

It has been shown repeatedly over a period of 50 years that the use of relativistic classical physics and the inclusion of classical electromagnetic zero-point radiation leads to the Planck blackbody spectrum for classical radiation equilibrium. However, none of this work involves scattering calculations. In contrast to this work, currently accepted physical theory connects classical physics to only the Rayleigh-Jeans spectrum. Indeed, in the past, it has been shown that a nonlinear classical oscillator (which is necessarily a nonrelativistic scattering system) achieves equilibrium only for the Rayleigh-Jeans spectrum where the random radiation present at the frequency of the second harmonic of the oscillator motion has the same energy per normal mode as the radiation present at the fundamental frequency. Here we continue work emphasizing the importance of relativistic versus nonrelativistic analysis. We consider the scattering of random classical radiation by a charged harmonic oscillator of small but non-zero oscillatory amplitude (which can be considered as a relativistic scattering system) and show that detailed radiation balance holds not only at the fundamental frequency of the oscillator but through the first harmonic corresponding to quadrupole scattering, provided that the radiation energy per normal mode at the first harmonic is double the radiation energy per normal mode at the fundamental frequency. This condition corresponds exactly to the zero-point radiation spectrum which is linear in frequency. It is suggested that for this relativistic scattering system, the detailed balance for zero-point radiation holds not only for the fundamental and first harmonic but extends to all harmonics. Here we have the first example of the detailed balance of zero-point radiation under relativistic scattering.

physics.class-ph↗

arXiv2018

Thermodynamics of the Harmonic Oscillator: Derivation of the Planck Blackbody Spectrum from Pure Thermodynamics

In 1893, Wien applied the first two laws of thermodynamics to blackbody radiation and derived his displacement theorem. Believing that the information from thermodynamics had been exhausted, Planck turned to statistical ideas in 1900 in order to provide a physical understanding for his experimental--data-based interpolation giving the Planck spectrum without zero-point radiation. Here we point out that the third law of thermodynamics (which was developed in the early years of the 20th century) introduces additional thermodynamic information regarding thermal radiation. The Planck spectrum for thermal radiation can be derived from purely thermodynamic ideas applied to the classical simple harmonic oscillator, since every radiation mode takes a simple oscillator form. Thermodynamics alone implies the Planck spectrum including zero-point energy without any need for quantum theory or statistical ideas. It is pointed out that the inclusion of zero-point energy involves more natural thermodynamics than its exclusion.

cond-mat.stat-mech↗

arXiv2018

Contrasting Interactions Between Dipole Oscillators in Classical and Quantum Theories: Illustrations of Unretarded van der Waals Forces

Students encounter harmonic-oscillator models in many aspects of basic physics, within widely-varying theoretical contexts. Here we highlight the interconnections and varying points of view. We start with the classical mechanics of masses coupled by springs and trace how the same essential systems are reanalyzed in the unretarded van der Waals interactions between dipole oscillators within classical and quantum theories. We note how classical mechanical ideas from kinetic theory lead to energy equipartition which determines the high-temperature van der Waals forces of atoms and molecules modeled as dipole oscillators. In this case, colliding heat-bath particles can be regarded as providing local hidden variables for the statistical mechanical behavior of the oscillators. Next we note how relativistic classical electrodynamical ideas conflict with the assumptions of nonrelativistic classical statistical mechanics. Classical electrodynamics which includes classical zero-point radiation leads to van der Waals forces between dipole oscillators, and these classical forces agree at all temperatures with the forces derived from quantum theory. However, the classical theory providing this agreement is not a local theory, but rather a non-local hidden-variables theory. The classical theory can be regarded as involving hidden variables in the random phases of the plane waves spreading throughout space which provide the source-free random radiation.

physics.gen-ph↗

arXiv2018

Derivations of the Planck Blackbody Spectrum from Thermodynamic Ideas in Classical Physics with Classical Zero-Point Radiation

Based upon thermodynamic ideas, two new derivations of the Planck blackbody spectrum are given within classical physics which includes classical zero-point radiation. The first and second laws of thermodynamics, applied to a harmonic oscillator or a radiation normal mode, require that the canonical potential $ϕ(ω/T)$ is a function of a single variable corresponding to the ratio of the oscillation frequency to the temperature. The second law of thermodynamics involves extremum ideas which may be applied to thermal radiation. Our first derivation of the Planck spectrum is based upon the idea that the canonical potential $ϕ(ω/T)$ is a monotonic function and all its derivatives are monotonic when interpolating between zero-point energy at low temperature and energy equipartition at high temperature; the monotonic behavior precludes the canonical potential from giving a preferred value for the ratio $ω/T.$ Our second derivation of the Planck spectrum is based upon the requirement that the change in the Helmholtz free energy of the radiation in a partitioned box held at constant temperature should be a minimum at thermal equilibrium. Finally, the change in Casimir energy with change in partition position for the radiation in a partitioned box is shown to correspond at high temperature to the absence of zero-point energy when the spectral energy per normal mode is chosen as the traditional Planck spectrum which omits zero-point energy at low temperature; thus the idea of zero-point energy is embedded in the traditional Planck spectrum. It is emphasized that thermal radiation is intimately connected with zero-point radiation and the structure of spacetime in classical physics.

physics.gen-ph↗

arXiv2017

Blackbody Radiation in Classical Physics: A Historical Perspective

We point out that current textbooks of modern physics are a century out-of-date in their treatment of blackbody radiation within classical physics. Relativistic classical electrodynamics including classical electromagnetic zero-point radiation gives the Planck spectrum with zero-point radiation as the blackbody radiation spectrum. In contrast, nonrelativistic mechanics cannot support the idea of zero-point energy; therefore if nonrelativistic classical statistical mechanics or nonrelativistic mechanical scatterers are invoked for radiation equilibrium, one arrives at only the low-frequency Rayleigh-Jeans part of the spectrum which involves no zero-point energy, and does not include the high-frequency part of the spectrum involving relativistically-invariant classical zero-point radiation. Here we first discuss the correct understanding of blackbody radiation within relativistic classical physics, and then we review the historical treatment. Finally, we point out how the presence of Lorentz-invariant classical zero-point radiation and the use of relativistic particle interactions transform the previous historical arguments so as now to give the Planck spectrum including classical zero-point radiation. Within relativistic classical electromagnetic theory, Planck's constant h appears as the scale of source-free zero-point radiation.

physics.class-ph↗

arXiv2017

The Contrasting Roles of Planck's Constant in Classical and Quantum Theories

We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum theories. In both theories, Planck's constant sets the scale of atomic phenomena. However, the roles played in the foundations of the theories are sharply different. In quantum theory, Planck's constant is crucial to the structure of the theory. On the other hand, in classical electrodynamics, Planck's constant is optional, since it appears only as the scale factor for the (homogeneous) source-free contribution to the general solution of Maxwell's equations. Since classical electrodynamics can be solved while taking the homogenous source-free contribution in the solution as zero or non-zero, there are naturally two different theories of classical electrodynamics, one in which Planck's constant is taken as zero and one where it is taken as non-zero. The textbooks of classical electromagnetism present only the version in which Planck's constant is taken to vanish.

physics.gen-ph↗

arXiv2017

Scaling, Scattering, and Blackbody Radiation in Classical Physics

Here we discuss blackbody radiation within the context of classical theory. We note that nonrelativistic classical mechanics and relativistic classical electrodynamics have contrasting scaling symmetries which influence the scattering of radiation. Also, nonrelativistic mechanical systems can be accurately combined with relativistic electromagnetic radiation only provided the nonrelativistic mechanical systems are the low-velocity limits of fully relativistic systems. Application of the no-interaction theorem for relativistic systems limits the scattering mechanical systems for thermal radiation to relativistic classical electrodynamic systems, which involve the Coulomb potential. Whereas the naive use of nonrelativistic scatterers or nonrelativistic classical statistical mechanics leads to the Rayleigh-Jeans spectrum, the use of fully relativistic scatterers leads to the Planck spectrum for blackbody radiation within classical physics.

physics.class-ph↗

arXiv2017

Interference Between Source-Free Radiation and Radiation from Sources: Particle-Like Behavior for Classical Radiation

A simple junior-level electrodynamics problem is used to illustrate the interference between a source-free standing plane wave and a wave generated by a pulse in a current sheet. Depending upon the relative phases between the standing wave and the current pulse and also upon the relative magnitudes, we can find quite different patterns of emitted energy and momentum. If the source gives a large radiation pulse so that the source-free plane wave can be neglected, then the radiation spreads out symmetrically on either side of the current sheet. However, if the radiation sheet gives a pulse with fields comparable to those of the standing wave, then we can find a single radiation pulse moving to the right while the current sheet recoils to the left, or the situation with the directions reversed. The example is a crude illustration of particle-like behavior arising from conventional classical electromagnetic behavior in the presence of source-free radiation. The discussion makes contact with the ideas of photons in modern physics.

physics.class-ph↗

arXiv2016

The Planck Blackbody Spectrum Follows from the Structure of Relativistic Spacetime

Here we show that within classical physics, the Planck blackbody spectrum can be derived directly from the structure of relativistic spacetime. In noninertial frames, thermal radiation at positive temperature is connected directly to zero-point radiation whose spectrum follows from the geodesic structure of the spacetime. The connection between zero-point radiation and thermal radiation at postive temperature is through a time-dilating conformal transformation in the noninertial frame. Transferring the spectrum back to Minskowski spacetime, the Planck spectrum is obtained.

physics.class-ph↗

arXiv2016

Understanding the Planck Blackbody Spectrum and Landau Diamagnetism within Classical Electromagnetism

Electromagnetism is a \textit{relativistic} theory and one must exercise care in coupling this theory with \textit{nonrelativistic} classical mechanics and with \textit{nonrelativistic} classical statistical mechanics. Indeed historically, both the blackbody radiation spectrum and diamagnetism within classical theory have been misunderstood because of two crucial failures: 1)the neglect of classical electromagnetic zero-point radiation, and 2) the use of erroneous combinations of nonrelativistic mechanics with relativistic electrodynamics. Here we review the treatment of classical blackbody radiation, and show that use of Lorentz-invariant classical electromagnetic zero-point radiation can be used to explain both the Planck blackbody spectrum and Landau diamagnetism at thermal equilibrium within classical electromagnetic theory. The analysis requires that relativistic electromagnetism is joined appropriately with simple nonrelativistic mechanical systems which can be regarded as the zero-velocity limits of relativistic systems, and that nonrelativistic classical statistical mechanics is applied only in the low-frequency limit when zero-point energy makes no contribution.

physics.class-ph↗

arXiv2015

Understanding Zero-Point Energy in the Context of Classical Electromagnetism

Today's textbooks of electromagnetism give the particular solution to Maxwell's equations involving the integral over the charge and current sources at retarded times. However, the texts fail to emphasize the role played by the choice of the boundary conditions corresponding to solutions of the homogeneous Maxwell equations. Here we discuss the role of these boundary conditions for an experimenter with a hypothetical charged harmonic oscillator as his equipment. We describe the observations of the experimenter when located near a radio station or immersed in thermal radiation at temperature T. The classical physicists at the end of the 19th century chose the homogeneous boundary conditions for Maxwell's equation based upon the experimental observations of Lummer and Pringsheim which measured only the thermal radiation which exceeded the random radiation surrounding their measuring equipment. Today at the beginning of the 21st century, classical physicists must choose the homogeneous boundary conditions for Maxell's equations to correspond to the full radiation spectrum revealed by the recent Casimir force measurements which detect all the radiation surrounding conducting parallel plates, including the radiation absorbed and emitted by the plates themselves. The random classical radiation spectrum revealed by the Casimir force measurements includes electromagnetic zero-point radiation, which is missing from the spectrum measured by Lummer and Pringsheim, and which cannot be eliminated by going to zero temperature. This zero-point radiation will lead to zero-point energy for all systems which have electromagnetic interactions. Thus the choice of the boundary conditions on the homogeneous Maxwell equations is intimately related to the ideas of zero-point energy and non-radiating ground states which are introduced in classes of modern physics.

physics.class-ph↗

arXiv2015

Classical Zero-Point Radiation and Relativity: The Problem of Blackbody Radiation Revisited

The physicists of the early 20th century were unaware of two ideas which are vital to understanding some aspects of modern physics within classical theory. The two ideas are: 1) the presence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity. In classes of modern physics today, the problem of blackbody radiation within classical physics is still described in the historical context of the early 20th century. However, the inclusion of classical zero-point radiation and of relativity now allows a completely satisfactory classical understanding of blackbody radiation with the Planck spectrum, as well as of some other aspects of modern physics. Here we sketch the current classical understanding of blackbody radiation, pointing out that thermodynamics allows the presence of classical zero-point radiation, and that use of nonrelativistic physics leads to the Rayleigh-Jeans spectrum while relativistic physics gives the Planck spectrum. The current textbooks of modern physics are a century out of date in presenting the connections between classical and quantum physics.

physics.class-ph↗

arXiv2015

Classical Zero-Point Radiation and Relativity: The Problem of Atomic Collapse Revisited

The physicists of the early 20th century were unaware of two aspects which are vital to understanding some aspects of modern physics within classical theory. The two aspects are: 1) the presence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity. In classes in modern physics today, the problem of atomic collapse is still mentioned in the historical context of the early 20th century. However, the classical problem of atomic collapse is currently being treated in the presence of classical zero-point radiation where the problem has been transformed. The presence of classical zero-point radiation indeed keeps the electron from falling into the Coulomb potential center. However, the old collapse problem has been replaced by a new problem where the zero-point radiation may give too much energy to the electron so as to cause self-ionization. Special relativity may play a role in understanding this modern variation on the atomic collapse problem, just as relativity has proved crucial for a classical understanding of blackbody radiation.

physics.class-ph↗

arXiv2014

Interaction of a Magnet and a Point Charge: Unrecognized Internal Electromagnetic Momentum

Whereas nonrelativistic mechanics always connects the total momentum of a system to the motion of the center of mass, relativistic systems, such as interacting electromagnetic charges, can have internal linear momentum in the absence of motion of the center of energy of the system. This internal linear momentum of the system is related to the controversial concept of "hidden momentum." We suggest that the term "hidden momentum" be abandoned. Here we use the relativistic conservation law for the center of energy to give an unambiguous definition of the "internal momentum of a system," and then we exhibit this internal momentum for the system of a magnet (modeled as a circular ring of moving charges) and a distant static point charge. The calculations provide clear illustrations of this system for three cases: a) the moving charges of the magnet are assumed to continue in their unperturbed motion, b) the moving charges of the magnet are free to accelerate but have no mutual interactions, and c) the moving charges of the magnet are free to accelerate and also interact with each other. It is noted that when the current-carrying charges of the magnet are allowed to interact, the magnet itself will contain internal electromagnetic linear momentum, something which has not been presented clearly in the research and teaching literature.

physics.class-ph↗