The Chemistry of Matter Waves

The discovery of X-ray diffraction promised to resolve the mystery of molecular structure, but a hundred years on it is fast receding into the fourth dimension. The contemporary development of quantum mechanics performed no better. It introduced, without explanation the notion of non-commuting dynamic variables, described by complex functions, failed to account for electron spin or optical activity and still appears to be at odds with special relativity. The confusion starts with Maxwell’s formulation of the electromagnetic field, interpreted differently in quantum and relativity theories, and grows with the chemical practice of reducing complex quantum functions to real classical variables. This leaves the nature of a single molecule’s structure undefined—neither classical nor non-classical.

The development of physical science over the last two millenia is traced from the summary of Lucretius, through the early Christian era, to the transformation into critical science after Copernicus. This revolution saw the birth of physics and chemistry to replace Aristotelian authority and alchemy, guided by the principles formulated by Isaac Newton and John Dalton. The new awareness blossomed into the formulation of a comprehensive theoretical mechanics and the recognition of seventy well-characterized chemical elements to replace the four elements of antiquity.

The two major achievements of 19th century science that produced the periodic table of the elements and the electromagnetic theory are reviewed. A critical analysis of Prout’s hypothesis, Newlands’ law of octaves and Nagaoka’s Saturnian model of the atom argues for a major re-assessment of the currently accepted history and interpretation of this most important chemical discovery of all time. The synthesis of concepts around chemical affinity and molecular conformation by Sommerfeld is recognized as the ultimate development of chemical theory based on Newton’s particle model.

The developments that led to the unification of magnetism, electricity and optics happened during the same period. The empirical observations that resulted in Maxwell’s synthesis are reviewed. The theory of electromagnetic radiation and the supporting theory of wave motion are critically examined.

Classical science reached maturity in the discovery of the electromagnetic field and the periodic variation of the chemical properties of atoms, for which no theoretical explanations existed. The theory of relativity and quantum theory, in the form of wave mechanics, developed in response. The details are briefly discussed and critically examined. By design, the theory of relativity provided a common basis for mechanical and electromagnetic motion, which could be refined into a model for gravitational interaction. The search for an equivalent space-time origin of the electromagnetic field resulted in the recognition of gauge fields, one of which gave birth to wave mechanics. As a theory that underpins atomic periodicity and chemistry it has only been partially successful and, reduced to a scheme of quantum chemistry, based on real linear functions, has failed completely.

Henri Poincaré, one of the pioneers of relativity theory predicted that, for the sake of simplicity, physicists would never abandon Euclidean geometry. It is argued here that chemical theory has stagnated for the same reason. It is pointed out how a fresh approach in four-dimensional non-Euclidean space-time could eliminate most of the conceptual stumbling blocks that inhibit the growth of a non-classical theory for chemistry. Immediately foreseen benefits include an understanding of four-dimensional action, recognized as the spin function, to replace the unrealistic concept of orbital angular momentum associated with standing electron waves. The controversial issues of non-local interaction and the discrepancy with relativity resolve themselves, giving new meaning to the concept of quantum potential energy. Without the debilitating assumption of point particles problematical issues such as the exclusion principle, wave-particle duality, quantum probability, the measurement problem, uncertainty principle, molecular shape and the mysterious fine-structure constant, also disappear. An alternative wave model is introduced and shown to be consistent with elemental periodicity as it occurs in projective space-time, which is briefly discussed.

The sensational aspects of quantum theory, from the wave-particle nature of electrons to Schrödinger’s cat, are the artefacts that result from describing nonlinear systems by linear differential equations. As linear waves are dispersive, a wave model of the electron is still being rejected, whereas a nonlinear wave model is shown to account for electronic behaviour in all conceivable situations. This chapter introduces the distinction between linear and nonlinear systems with examples from hydrodynamics and mechanics and applied to the wave mechanics of wave packets, solitons, electrons and lattice phonons. Special topics for discussion include the motion of free electrons, the fine-structure parameter, electron diffraction, photoelectric and Compton effects, X-ray diffraction, metallic conduction, superconductivity and elementary covalent interaction. A new innovation, introduced here, is recognition of the quantum potential as a nonlinearity parameter that enables a seamless transition between classical and non-classical systems.

The concept of matter waves as a product of four-dimensionally curved space-time is examined. A vital step in the analysis is taking cognisance of the controversial concept of an all-pervading aether. The discrepancy between relativity and quantum theory is traced to the three-dimensional linear equations of wave mechanics, in contrast to Minkowski space-time. The notion of space-like interaction is re-examined and shown to arise from a superficial interpretation of space-time curvature. The more appropriate projective topology is shown to be suitable, in principle, to define four-dimensional matter waves. The transformation from the more general underlying space-time to the familiar three-dimensional affine space is shown to be mediated by the golden ratio, which is further characterized in terms of Fibonacci numbers, Farey sequences and other concepts of number theory. It is demonstrated conclusively that the observed periodic table of the elements and the wave-mechanical approximation are correctly simulated by number theory, with a clear distinction of the respective four- and three-dimensional bases of the two models.

The wave structure of the electron lends itself to the formulation of chemical phenomena in terms of number theory. Without a particle concept the behaviour of elementary units of matter, in the form of solitons, is described directly in the wave formalism originally proposed by Schrödinger and Madelung in hydrodynamic analogy. The quantum condition appears naturally as a minimum action principle. All atoms are alike with nuclei bathed in a uniform electronic fluid, the spherical wave structure of which is revealed by optimization on a logarithmic spiral. The density distribution pattern has much in common with the Bohr–de Broglie model of atomic structure and predicts a number of important atomic properties, including atomic size, ionization radius, electronegativity and atomic polarizability. The intimate connection between atomic properties and space-time curvature is convincingly demonstrated by derivation of atomic radii as a periodic function optimized on Fibonacci spirals. Details of covalent interaction are elucidated by the manipulation of ionization radii and the golden ratio as parameters to predict interatomic distance, bond order, dissociation energy, stretching force constant and dipole moments. Extended to molecules the matter-wave approach demonstrates that the concepts of structure and shape of a free molecule are strictly four-dimensional. Molecular structure observed and modelled in three dimensions only applies to condensed phases. Molecules involved in chemical change are essentially in the free state and their mode of interaction is not always obvious as a function of assumed three-dimensional structure. Proposed mechanisms for synthetic processes serve to rationalize the apparent discrepancies.

The quantum theory as formulated almost a hundred years ago appears outdated in view of new developments. The firm belief in quantum magic persists and chemical practice appears irrevocably committed to it, despite many failures. By way of re-assessment the assumptions behind the Copenhagen interpretation of the theory are shown to be indefensible on closer scrutiny. It offers no reasonable atomic model nor an explanation of stationary states. The belief in a quantum theory of chemistry appears baseless, and a more useful theory is needed. The extension of chemical modelling by number theory into a general physically meaningful theory is explored through the simulation of the unexplained phenomena of high-temperature superconductivity and low-temperature nuclear activity. The prospect of number-theory analysis in nanoscience is explored.