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arXiv:1804.03542 · European Journal of Physics

Contrasting interactions between dipole oscillators in classical and quantum theories: illustrations of unretarded van der Waals forces

Timothy H Boyer

Published Feb 8, 2018• 2 citations
arXivHTMLPDF

Abstract

Abstract 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 reanalysed 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 modelled as dipole oscillators. In this case, colliding heat-bath particles can be regarded as providing local hidden variables for the statistical mechanical behaviour 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.

DOI
10.1088/1361-6404/aaadf5

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Abstract 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 reanalysed 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 modelled as dipole oscillators. In this case, colliding heat-bath particles can be regarded as providing local hidden variables for the statistical mechanical behaviour 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.
A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The NNN-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.
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 ℏ) 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 that arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant ℏ. All mechanical systems in thermal radiation will inherit the constant ℏ 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.
The article “Insights from the classical atom,” by Petar Grujić and Nenad Simonović (Physics Today, May 2012, page 41), was interesting but incomplete. It left out stochastic electrodynamics (SED), an approach to classical mechanics that includes the effects of a proposed zero-point radiation background.The zero-point radiation is a Lorentz-invariant random distribution of energy into different modes of electromagnetic radiation. Quantum electrodynamics (QED) uses the hypothesis that the amplitudes of the Lorentz-invariant radiation are quantized, whereas SED uses the hypothesis that they are continuous.The motions of electric charges in SED are modeled using the equations of motion for a particle, often associated with a concrete visualization. SED is considered part of classical physics because of the continuous distribution of amplitudes. However, discrete values for energy and momentum emerge from SED without the ad hoc quantization conditions of QED.Stochastic electrodynamics has been investigated as a classical physics explanation for quantum phenomena. For instance, the blackbody radiation spectrum has been derived without quantum assumptions.11. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374 Lorentz-invariant radiation provides perturbations that mimic quantum phenomena such as the Casimir force, the van der Waals force, the Lamb shift, and the radius of the Bohr atom.22. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737Physical chemistry provides many mathematical expressions that are often thought to be fundamentally quantum in nature. However, some of those expressions have been shown to emerge from both QED and SED. For example, the Schrödinger equation has been derived from SED,33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 and the wavefunction in the SED Schrödinger equation is shown to have the a priori significance of position probability amplitude. A physical model based on SED has been developed to increase our understanding of both the diffraction of particle beams and the Pauli exclusion principle.44. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707Quantum mechanics is often cited to justify chemical bonding on an elementary level. However, the visualizations used for chemical bonding are often based on mathematical expressions that can be derived from either QED or SED. Regardless of which turns out to be more accurate in physics experiments or more complete in mathematical description, the two theories overlap with regard to physical chemistry. Therefore, I propose that introductory chemistry taught in secondary schools can be rationalized as easily by references to SED as to QED.Either theory produces identical predictions for many experiments involving second-order correlations, squeezing (quadrature noise reduction), and the original Einstein-Podolsky-Rosen proposal.55. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912 However, SED is not completely equivalent to QED on a mathematical or physical level. The SED Schrödinger equation is incomplete, because there is a companion equation that has no counterpart in ordinary quantum mechanics and restricts initial conditions.33. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 G. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993 Furthermore, the predictions of QED and SED are different with regard to high-order correlations in nonlinear optics.I agree fully with Grujić and Simonović that the classical atom is alive and well. Models of the atom based on SED are sufficiently accurate for many technological and most pedagogical applications. Even if the classical atom isn’t the most fundamental description of the physical world, it is still an important one.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. T. H. Boyer, Phys. Rev. 182, 1374 (1969). https://doi.org/10.1103/PhysRev.182.1374, Google ScholarCrossref, ISI2. M. Ibison, B. Haisch, Phys. Rev. A 54, 2737 (1996). https://doi.org/10.1103/PhysRevA.54.2737, Google ScholarCrossref, ISI3. G. H. Goedecke, Found. Phys. 13, 1101 (1983); https://doi.org/10.1007/BF00728139 , Google ScholarCrossref, ISIG. H. Goedecke, Found. Phys. 13, 1195 (1983). https://doi.org/10.1007/BF00727993, , Google ScholarCrossref, ISI4. A. F. Kracklauer, Found. Phys. Lett. 12, 441 (1999). https://doi.org/10.1023/A:1021629310707, Google ScholarCrossref, ISI5. S. Chaturvedi, P. D. Drummond, Phys. Rev. A 55, 912 (1997). https://doi.org/10.1103/PhysRevA.55.912, Google ScholarCrossref, ISI© 2013 American Institute of Physics.
We study forces between two neutral atoms, modeled as three-dimensional harmonic oscillators, arising from mutual influences mediated by an electromagnetic field but not from their direct interactions. We allow as dynamical variables the center-of-mass motion of the atom, its internal degrees of freedom, and the quantum field treated relativistically. We adopt the method of nonequilibrium quantum field theory which can provide a first-principles, systematic, and unified description including the intrinsic and induced dipole fluctuations. The inclusion of self-consistent back-actions makes possible a fully dynamical description of these forces valid for general atom motion. In thermal equilibrium we recover the known forces---London, van der Waals, and Casimir-Polder---between neutral atoms in the long-time limit. We also reproduce a recently reported force between atoms when the system is out of thermal equilibrium at late times. More noteworthy is the discovery of the existence of a type of (or identification of the source of some known) interatomic force which we call the ``entanglement force,'' originating from the quantum correlations of the internal degrees of freedom of entangled atoms.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
A rigorous derivation of the Schrödinger theory from the Pauli (or Dirac) theory implies that the Schrödinger equation describes an electron in an eigenstate of spin. Furthermore, the ground-state kinetic energy is completely determined by the electron spin density. This can be explained by interpreting the spin as an orbital angular momentum, which is necessarily accompanied by a kinetic energy. Thus, the spin is a zero-point angular momentum associated with the zero-point energy of the electron. Since the dispersion in electron momentum is determined by the zero-point energy, the Heisenberg uncertainty relations for an electron can be interpreted as a property of the electronspin motion. The kinetic interpretation of spin and the statistical interpretation of quantum mechanics can be jointly sustained by regarding the electron as a point particle. It follows that stationary electron states are sources of fluctuating electric fields. There is reason to believe that these fluctuating fields are responsible for the Van der Waals force and can be identified with electromagnetic vacuum field fluctuations. The kinetic interpretation of spin then implies that the Van der Waals forces are spin dependent. These ideas are not only consistent with the conventional mathematical formalism of quantum mechanics, they provide a more complete and coherent interpretation of many details in the formalism than does the alternative Copenhagen interpretation. They present some difficulties, however, which, if the kinetic interpretation of spin is correct, probably require some modification of quantum electrodynamics to be resolved.