In a recent paper [J. Math. Phys. 35, 3333 (1994)] Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyze the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the ‘‘mass spectrum’’ for the Klein–Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative ‘‘masses.’’ Finally, two major reasons are presented against the viability of this alternative proposal: (a) It does not lead to the correct energy spectrum for the hydrogen atom. (b) For field models, the standard quantum theory cannot be recovered from this alternative description.
In a recent paper [J. Math. Phys. 35, 3333 (1994)] Good postulated new rules of quantization, one of the major features of which is that the quantum evolution of the wave function is always given by ordinary differential equations. In this paper we analyze the proposal in some detail and discuss its viability and its relationship with the standard quantum theory. As a byproduct, a simple derivation of the ‘‘mass spectrum’’ for the Klein–Gordon field is presented, but it is also shown that there is a complete additional spectrum of negative ‘‘masses.’’ Finally, two major reasons are presented against the viability of this alternative proposal: (a) It does not lead to the correct energy spectrum for the hydrogen atom. (b) For field models, the standard quantum theory cannot be recovered from this alternative description.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
In this paper, we discuss in brief the most common wave equations in quantum mechanics and some recent development in wave mechanics. We also present two new quantum wave mechanics equations based on the Compton momentum. We also question the idea that energy and mass are scalars, and we claim they are vectors instead. We have good reasons to think that the standard momentum is a mathematical derivative of the more fundamental Compton momentum. This will hopefully simplify interpretations of quantum mechanics significantly; our new relativistic wave equations look promising, but need further investigation into what they predict. This way of looking at quantum mechanics in new light is not in conflict with existing equations, but they are supplemental to the collection of existing wave equations. We prove mathematically that if one satisfies our new relativistic energy Compton momentum relation, one also satisfies the standard relativistic energy momentum relation automatically. They are two sides of the same coin, where the relations to the Compton wavelength likely represent the deeper reality, so we have reasons to think our new wave mechanics addresses a deeper level of understanding than the existing conception. We also look at our new wave equation in relation to hydrogen-like atoms; we follow the “standard approach” used for the Schro dinger equation of putting it in polar coordinate form and, by change of variables, finding three ODEs and their solutions. We give a table summary of our new ODEs and their solutions compared to the well-known solutions of the Schro dinger equation.
Abstract Wave mechanics triumphed when Schrödinger published his now famous equation and showed how to describe hydrogen-like atoms using it. However, while looking for the right equation, Schrödinger first explored, but did not publish, the equation that we today call the Klein–Gordon equation. An alternative possible choice is explored in this work. It is shown a quasi-relativistic wave equation which solutions match the Schrödinger’s results at electron energies much smaller than the energy associated to the electron’s mass, but include, at higher energies, the relativity energy’s correction calculated in traditional first order perturbation theory. A discussion is presented about several consequences that would follow from using this quasi-relativistic wave equation as a quantum mechanics foundational equation. It is also suggested the academic use of this equation for introducing the students to the implications of the special theory of relativity in introductory quantum mechanics courses.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
Bohr’s model of the hydrogen atom was successful in that it gave us a radically new way to look at atoms. However, it has serious shortcomings. It could not be used to explain the spectra of He or any multielectron atom. It could not predict the intensities of the H absorption and emission lines. With de Broglie’s hypothesis that matter was also wavelike,1 there arose a question at the 1925 Solvey conference: What is the wave equation? De Broglie could not answer this; however, over the next year Erwin Schro¨dinger, working in Vienna, published a series of papers in which he deduced the general form of the equation that bears his name and applied it successfully to the hydrogen atom.2,3 What emerged was a new set of postulates, much like Newton’s, that laid the foundations of quantum theory.