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On the Foundations of Quantum Mechanics

Richard Joseph Oldani

Published Dec 2, 2025• 0 citations
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Abstract

Matrix mechanics describes the spectral emissions of hydrogen gas, but quantum mechanics concerns the emission and absorption of radiation by a single atom. To obtain the spectral emissions of a single hydrogen atom we introduce thermal energy in infinitesimal increments registering the spectral lines one by one until a complete spectrum is obtained. We describe the gradual increase of the internal quantum mechanical energy by integrating between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the conditions necessary for Hamilton’s principle. Continuous time is thereby introduced to quantum mechanics during energy absorption by means of a Lagrangian formulation using energy-time conjugate variables, independently of Hamiltonian formulations that use position-momentum conjugate variables.

DOI
10.20944/preprints202510.0504.v2

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Matrix mechanics describes the spectral emissions of hydrogen gas, but quantum mechanics concerns the emission and absorption of radiation by a single atom. To obtain the spectral emissions of a single hydrogen atom we introduce thermal energy in infinitesimal increments registering the spectral lines one by one until a complete spectrum is obtained. We describe the gradual increase of the internal quantum mechanical energy by integrating between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the conditions necessary for Hamilton’s principle. Continuous time is thereby introduced to quantum mechanics during energy absorption by means of a Lagrangian formulation using energy-time conjugate variables, independently of Hamiltonian formulations that use position-momentum conjugate variables.
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Matrix mechanics describes the spectral emissions of hydrogen gas, but quantum mechanics concerns the emission and absorption of radiation by a single atom. To obtain the spectral emissions of a single hydrogen atom we reformulate matrix mechanics by introducing thermal energy in infinitesimal increments and registering the spectral lines one by one until a complete spectrum is obtained. Wave mechanics is reformulated from a conservative to a non-conservative system by linking the output of matrix mechanics to the input of wave mechanics. To describe the gradual increase and decrease of energy by a quantum system we integrate between two specific endpoints, the atomic orbitals, and two specific times, the period τ, which are the conditions necessary for Hamilton’s principle. The Lagrangian method is shown to be superior for analyzing quantum theory because it describes quantum systems relativistically as evolutions in time.
We consider three long-standing conceptual difficulties of quantum mechanics; the time variable, the quantum-classical divide, and gravitational fields. We investigate possible solutions by extending matrix mechanics from single points in time to continuous time in order to describe the absorption energy of a quantum system. Due to the conservation of energy absorption is equal to emission, as described by wave functions; which is an illustration of the classical-quantum divide. Hamilton’s principle of least action proves to be a natural fit for describing the resulting two-part quantum system and it specifies the true paths to be the ones with action equal to ћ. Energy absorption followed by discrete, four-dimensional changes in structure and energy emission is shown to be a pattern that is repeated countless times throughout nature by phase transitions and other phenomena. Experimental evidence is cited to show that the resultant mathematical model varies consistently with changes in gravitational potential. A universal physical interpretation is proposed.
We reexamine the original derivations of quantum mechanics, published 100 years ago, to test the accuracy of the formulations and the logic underlying them, and we find that attempts were not made to describe atomic processes continuously in time. To correct this omission we show that matrix mechanics clearly describes the continuous absorption of thermal energy that causes the hydrogen line spectra, where energy and cause follow the same path. When the same methods are used to inspect the emission process as determined by the wave function we find that energy and cause do not follow the same path. We conclude that failure to follow strict causality is the reason that the wave function is neither physical nor real. Energy-cause superposition is hypothesized to be a universal principle of nature, true not only for atomic systems but for material systems throughout the cosmos and for life as well. It is also equivalent to the action principle, ∫Ldt, which is fundamental to all of physics.
Although quantum mechanics correctly predicts what it is possible to observe (the emissions) it ignores the other half of natural phenomena, what cannot be observed (the absorptions), thereby violating the conservation laws. By describing only one-half of quantum mechanics conceptual difficulties such as wave function collapse, infinite paths, non-locality, and abstract mathematical models appear out of nowhere. The deficiencies are corrected in non-relativistic theory by showing that the unobserved absorption energies of off-diagonal elements in matrix mechanics are essential and that matrix and wave mechanics are equally important for formulating a complete quantum mechanics. Due to the conservation of energy absorption must be carried out to completion before energy emission can begin. When relativistically correct equations of motion are derived they show that the wave function is incomplete because it describes the excitation and decay of an electron with twice the allowable action minimum. Einstein’s unfinished quantum theory forms the mathematical foundation leading up to and underlying the proposed mathematical model. Keywords Conservation laws; Quantum mechanics; Hamilton’s principle; Energy; Momentum; Path integral formulation; Matrix mechanics
We hypothesize that quantum mechanics is incomplete because it does not take into consideration the absorption of energy by quantum systems as described by matrix mechanics. To be complete quantum mechanics must include both matrix and wave mechanics, and due to the conservation of energy the first (absorption) must be carried out to completion before the second one (emission) can begin. The combined model of matrix and wave mechanics integrates naturally with Einstein’s use of independent coordinate systems K and K’ to describe the absorption and emission of radiation. All subsequent mathematical models are versions of this physical model.