A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a hydrogen atom together with the derived relativistic Hamiltonian to quantify the impact of finite proton mass to the fine structure, correction to the fine structure is obtained as a result of the entanglement of the relative motion and the center-of-mass motion, which can be used to verify the correctness of the proposed eigenvalue equation. Applying the equation to the measurement of electron double-slit interference, it is analyzed that the photon packets with Lorentzian spectral lineshape, the domain and state density of the sub-system, and the energy of the incident electron together determines the spontaneous emission rate of the incident electron. Photons generated in this process excite electrons from the valence band to the conduction band of the detector. Corresponding to any emitted or absorbed photon, the sub-system is found to be uniquely determined by maximizing the transition rate. This new principle is valid for atoms too. Closed-form expressions are obtained for the transition rates and example numerical results show good correlation between calculation and the position measurement experiment. The discovered common mechanisms that determine the sub-systems, the preferred bases, and transition rates form the foundation of a new, general, and consistent theory of quantum measurement.
A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a hydrogen atom together with the derived relativistic Hamiltonian to quantify the impact of finite proton mass to the fine structure, correction to the fine structure is obtained as a result of the entanglement of the relative motion and the center-of-mass motion, which can be used to verify the correctness of the proposed eigenvalue equation. Applying the equation to the measurement of electron double-slit interference, it is analyzed that the photon packets with Lorentzian spectral lineshape, the domain and state density of the sub-system, and the energy of the incident electron together determines the spontaneous emission rate of the incident electron. Photons generated in this process excite electrons from the valence band to the conduction band of the detector. Corresponding to any emitted or absorbed photon, the sub-system is found to be uniquely determined by maximizing the transition rate. This new principle is valid for atoms too. Closed-form expressions are obtained for the transition rates and example numerical results show good correlation between calculation and the position measurement experiment. The discovered common mechanisms that determine the sub-systems, the preferred bases, and transition rates form the foundation of a new, general, and consistent theory of quantum measurement.
This paper presents a unified pedagogical derivation of the hydrogen atom spectrum and radiative transition structure by organizing three historically separate frameworks into a single mathematical chain: Schrödinger’s non-relativistic wave mechanics, Sommerfeld’s relativistic fine-structure theory, and Maxwell’s classical electrodynamics. The analysis shows how Bohr energy levels arise as the non-relativistic bound-state limit, how Sommerfeld-type fine structure appears when relativistic corrections are included, and how Maxwell radiation theory connects naturally to photon emission rates through transition dipole moments and spontaneous decay constants. The completed framework is advanced by adding the necessary modern ingredients of spin, Lorentz covariance, minimal electromagnetic coupling, and field quantization. With these additions, the chain reaches the Dirac equation coupled to the quantized electromagnetic field, the standard foundation of quantum electrodynamics. The resulting Scretching Master Equation is presented as a structured bridge from introductory quantum mechanics to QED, linking energy eigenvalues, angular-frequency scaling, transition dipole strength μ², spontaneous emission Γ = A₂₁, lifetime τ, and photon radiation into one coherent derivational system.
This report investigates the flawed foundations of standard quantum theory based on a misunderstanding of the role of measurement. The report is major part of a research programme (UET) based on a new theory of the electromagnetism (EM), centered exclusively on the interaction between electrons. All the previous papers to date in this series have presented a realistic view of the dynamics of two or more electrons as they interact only between themselves. This paper now posits a theory of how this microscopic activity is perceived by human beings in attempting to extract information about atomic systems. The standard theory of quantum mechanics (QM) is constructed on only how the micro-world appears to macro measurements - as such, it cannot offer any view of how the foundations of the world are acting when humans are not observing it (the vast majority of the time) - this has generated almost 100 years of confusion and contradiction at the very heart of physics. All human beings (and all our instruments) are vast collections of electrons, our information about atomic-scale can only be obtained destructively and statistically. This theory now extends the realistic model of digital electrons by adding an explicit measurement model of how our macro instruments interfere with nature's micro-systems when such attempts result in human-scale information. The focus here is on the connection between the micro-world (when left to itself) and our mental models of this sphere of material reality, via the mechanism of atomic measurements. The mathematics of quantum mechanics reflects the eigenvalues of the combined target system plus equipment used for measurement together. Therefore, QM has constructed a theory that inseparably conflates the ontological and epistemological views of nature. This standard approach fails to examine isolated target systems alone. It is metaphysically deficient. This investigation concludes that the Quantum State function (Ψ) is not a representation of physical reality, within a single atom, but a generator function for producing the average statistical results on many atoms of this type. In contrast, the present theory builds on the physical reality of micro-states of single atoms, where (in the case of hydrogen), a single electron executes a series of fixed segments (corresponding to the micro-states) across the atom between a finite number of discrete interactions between the electron and one of the positrons in the nucleus. The set of serial segments form closed trajectories with real temporal periods, contra to Heisenberg’s ‘papal’ decree banning such reality because of his need to measure position and momentum at all times. This is the first paper in the multi-paper series (UET7n), which is dedicated to analyzing, criticizing and replacing existing theories of quantum mechanics.
We describe a quantum mechanical measurement as a variational principle including interaction between the system under measurement and the measurement apparatus. Augmenting the action with a nonlocal term (a double integration over the duration of the interaction) results in a theory capable of describing both the measurement process (agreement between system state and pointer state) and the collapse of both systems into a single eigenstate (or superposition of degenerate eigenstates) of the relevant operator. In the absence of the interaction, a superposition of states is stable, and the theory agrees with the predictions of standard quantum theory. Because the theory is nonlocal, the resulting wave equation is an integrodifferential equation (IDE). We demonstrate these ideas using a simple Lagrangian for both systems, as proof of principle. The variational principle is time-symmetric and retrocausal, so the solution for the measurement process is determined by boundary conditions at both initial and final times; the initial condition is determined by the experimental preparation and the final condition is the natural boundary condition of variational calculus. We hypothesize that one or more hidden variables (not ruled out by Bell's Theorem, due both to the retrocausality and the nonlocality of the theory) influence the outcome of the measurement, and that distributions of the hidden variables that arise plausibly in a typical ensemble of experimental realizations give rise to outcome frequencies consistent with Born's rule. We outline steps in a theoretical validation of the hypothesis. We discuss the role of both initial and final conditions to determine a solution at intermediate times, the mechanism by which a system responds to measurement, time symmetry of the new theory, causality concerns, and issues surrounding solution of the IDE.
Purpose of the study: The goal of this work is to present a comparative study between the semi-relativistic theory of Sakho and the relativistic theory of Dirac applied to calculations of the eigenvalue of the Hamiltonian fine structure of the hydrogen like systems. The three small perturbative corrections (relativistic mass correction, spin-orbit coupling, and the Darwin term) of the Hamiltonian are considered. Methodology: This research is based on correction of the semi-classical Bohr’s theory on the hydrogen like systems considering the relativistic mass correction by ignoring the electron’s spin. This approach referred to as semi-relativistic theory allows one to express the eigenvalues of the Hamiltonian fine structure to be compared to that from the relativistic wave function of Dirac Main Findings: The research gives the first expression of the quantized energy of hydrogen like systems taking only into account, the relativistic mass correction. This result provides exactly the eigenvalues of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states n2Lj (j = l ± s) of the hydrogen like systems (i.e. 1 2s1/2, 2 2p3/2, 3 2d5/2, 42f7/2, and so on). Novelty/Originality of this study: Theoretical determination of the eigenvalue of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states is obtained with a soft semi-relativistic theory without including in the calculations the spin-orbit coupling and the Darwin term corrections.
Abstract In Non-Relativistic Quantum Mechanics, the Schr¨odinger equation describes the evolution of a quantum system as a superposition of different states, but cannot explain why the system is only ever measured to be in one of these states. Here we show a new dynamical wave equation for the evolution of the wave function, and then apply this equation to a variety of different physical cases, showing how the wave function changes when we move from one potential to another. We then show how a measurement in quantum mechanics is an example of a potential changing the possible eigenvalues associated with a quantum system. The dynamical process of a measurement that evolves the wave function not only returns a single eigenvalue for the basis it is measuring, but also returns a new Hamiltonian with corresponding eigenvalues. This shows, contrary to some interpretations, that the wave function is continuous after measurement.
This paper presents a unified pedagogical derivation of the hydrogen atom spectrum and radiative transition structure by organizing three historically separate frameworks into a single mathematical chain: Schrödinger’s non-relativistic wave mechanics, Sommerfeld’s relativistic fine-structure theory, and Maxwell’s classical electrodynamics. The analysis shows how Bohr energy levels arise as the non-relativistic bound-state limit, how Sommerfeld-type fine structure appears when relativistic corrections are included, and how Maxwell radiation theory connects naturally to photon emission through transition dipole moments and spontaneous decay constants. The framework is then advanced by adding the necessary modern ingredients: spin, Lorentz covariance, minimal electromagnetic coupling, gauge invariance, and electromagnetic field quantization. With these additions, the derivational chain reaches the Dirac equation coupled to the quantized electromagnetic field, which forms the standard foundation of quantum electrodynamics. The resulting Scretching Master Equation is presented as a structured bridge from introductory quantum mechanics to QED. It links energy eigenvalues, angular-frequency scaling, transition dipole strength μ², spontaneous emission Γ = A₂₁, lifetime τ, and photon radiation into one coherent derivational system. No new physical law is claimed. The contribution is a corrected synthesis, notation system, and teaching route through established physics.