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On the Foundations of Quantum Mechanics

Richard Joseph Oldani

Published Feb 18, 2026• 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 in a non-conservative system 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. Wave mechanics is shown to be limited to conservative systems. A complete quantum mechanics is formulated using energy-time conjugate variables as the time integral of a Lagrangian independently of Hamiltonian formulations that use position-momentum conjugate variables.

DOI
10.20944/preprints202510.0504.v3

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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 in a non-conservative system 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. Wave mechanics is shown to be limited to conservative systems. A complete quantum mechanics is formulated using energy-time conjugate variables as the time integral of a Lagrangian independently of Hamiltonian formulations that use position-momentum conjugate variables.
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.
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 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.
We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.
We reformulate matrix mechanics by incrementally heating a single hydrogen atom in order to derive the complete, diagonalized Hamiltonian matrix. The spectral lines and transition probabilities cannot be generated by reversing time thereby demonstrating asymmetry. Experimental evidence in support of the theoretical model is obtained from experiments performed with the simplest quantum system, an electron cyclotron. Wave mechanics is also reformulated by altering the original Schrödinger equation describing a time symmetric, conservative system to a system of two independent equations describing a time asymmetric, non-conservative system.
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.