This paper posits the existence of, and finds a candidate for, a variable change that allows quantum mechanics to be interpreted as quantum geometry. The Bohr model of the Hydrogen atom is thought of in terms of an indeterministic electron position and a deterministic metric and the motivation for this paper is to try to change variables to have a deterministic position and momentum for the electron and nucleus but with an indeterministic (quantum) metric that reproduces the physics of the Bohr model. This mapping is achieved by allowing the metric in the Hamiltonian to be different to the metric in the space-time distance element and then representing the two metrics with vierbeins and assuming they are canonically conjugate variables. Effectively, the usual Schrödinger space-time variables have been re-interpreted as four of the potentially sixteen parameters of the metric tensor vierbein in the distance element while the metric tensor vierbein in the Hamiltonian is an operator expressible as first-order derivatives in these variables or vice versa. I then argue that this reproduces observed quantum physics at the sub-atomic level by demonstrating the energy spectrum of electron orbitals is exactly the same as the usual relativistic Bohr model for the Hydrogen atom in a certain limit. Next, by introducing a single dimensionless running coupling that shows up in the analogous place as, but in addition to, Planck's constant in the commutator definition I argue that this allows massive objects to couple to the physical space-time geometry but not massless ones - no matter coupling value. This claim is based on a fit to the Schwarzschild metric with a few simple assumptions and thus obtaining an effective theory of how the quantum geometries at nearby space-time points couple to one another. This demonstrates that this coupling constant is related to Newton's gravitational constant.
This paper posits the existence of, and finds a candidate for, a variable change that allows quantum mechanics to be interpreted as quantum geometry. The Bohr model of the Hydrogen atom is thought of in terms of an indeterministic electron position and a deterministic metric and the motivation for this paper is to try to change variables to have a deterministic position and momentum for the electron and nucleus but with an indeterministic (quantum) metric that reproduces the physics of the Bohr model. This mapping is achieved by allowing the metric in the Hamiltonian to be different to the metric in the space-time distance element and then representing the two metrics with vierbeins and assuming they are canonically conjugate variables. Effectively, the usual Schrödinger space-time variables have been re-interpreted as four of the potentially sixteen parameters of the metric tensor vierbein in the distance element while the metric tensor vierbein in the Hamiltonian is an operator expressible as first-order derivatives in these variables or vice versa. I then argue that this reproduces observed quantum physics at the sub-atomic level by demonstrating the energy spectrum of electron orbitals is exactly the same as the usual relativistic Bohr model for the Hydrogen atom in a certain limit. Next, by introducing a single dimensionless running coupling that shows up in the analogous place as, but in addition to, Planck's constant in the commutator definition I argue that this allows massive objects to couple to the physical space-time geometry but not massless ones - no matter coupling value. This claim is based on a fit to the Schwarzschild metric with a few simple assumptions and thus obtaining an effective theory of how the quantum geometries at nearby space-time points couple to one another. This demonstrates that this coupling constant is related to Newton's gravitational constant.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].
It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and L contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length λP, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator [p,x]q under the assumption of quantized space-time and contrast it with the usual commutator [p,x].
A deterministic quantum mechanics theory is presented in a series of papers starting with the present one. The proposed theory is shown to be consistent with the current mainstream statistical quantum theory as well as with classical physics. It produces solutions, which demonstrate that causality, physical reality, and determinism are restored and can explain in simple form concerns that are raised by results from the current mainstream statistical quantum theory. The meaning of particle- wave duality and complementarity, the possibility of a particle, like the electron, to cross through the nucleus as it does when the angular momentum of the electron is zero at the ground state of the hydrogen atom, the possibility of a point-size particle to have an “intrinsic spin”, the possibility of “quantum jumps” as the electron transitions instantaneously from one stable orbital to another without passing through the space in between the orbitals and does that at irregular time intervals, and the natural collapse of the wave function as part of the solution are some of the results that emerge from the proposed deterministic quantum mechanics theory. The phenomenon of entanglement is shown to be consistent with the proposed deterministic quantum mechanics theory. The Bohr-Schrödinger energy levels leading to the experimentally confirmed spectral lines, as well as the fine structure constant emerge directly from the solution of the equations governing theproposed deterministic quantum mechanics presented in Part II of these series of papers.
We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.
\begin{abstract} We propose a structural modification of the non-relativistic Schrödinger equation obtained by introducing an additional operator-valued potential term that couples nontrivially to both kinetic and potential contributions. The modified dynamical equation takes the form \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\Phi+ m\delta\Phi- \frac{\hbar^2}{2m}\,\delta\Phi\,\nabla^2\right]\psi,\end{equation} where δΦ represents an additional scalar field or structural correction that deforms the standard Hamiltonian operator. Unlike perturbative potential additions, this term modifies the kinetic structure itself, effectively generating a position-dependent renormalization of the Laplacian operator and inducing a nontrivial coupling between propagation and background geometry. We analyze the mathematical consistency of the modified Hamiltonian, deriving conditions for self-adjointness on L2(R3) and establishing criteria under which unitary time evolution and probability conservation are preserved. The deformation leads to a modified dispersion relation and alters the spectral structure of the system, suggesting an interpretation in terms of emergent or dynamically modified configuration-space geometry. Beyond its formal properties, this framework raises foundational questions regarding the ontological status of the kinetic operator in quantum mechanics. By altering the generator of time evolution itself, the proposal challenges the assumption that the Laplacian represents a fixed Euclidean background structure. Instead, quantum evolution may reflect a deeper geometric or structural layer, in which mass, potential energy, and propagation are relational rather than primitive quantities. This formulation provides a mathematically consistent extension of the Schrödinger framework and opens avenues for interpreting quantum dynamics as arising from modified geometric or higher-structural principles. \end{abstract} \noindent\textbf{Keywords:} Modified Schrödinger Equation; Operator Deformation; Quantum Geometry; Self-Adjoint Hamiltonians; Foundations of Quantum Mechanics. \section{Spacetime–Induced Modification of the Schrödinger Equation} \subsection{1. Introduction} In conventional quantum mechanics, uncertainty is treated as a fundamental property of wavefunctions. In this work we explore an alternative interpretation: that quantum uncertainty may emerge from small-scale spacetime moduli perturbations. We begin in standard 4-dimensional spacetime and then extend the formulation to a constrained multi-time geometry. \subsection{2. Quantum Dynamics in Curved Spacetime} \paragraph{Explanation of the Curved-Spacetime Klein–Gordon Equation.} The equation (□g+ℏ2m2c2)Φ=0describes the dynamics of a scalar quantum field Φ propagating in a curved spacetime background. Here, □g denotes the covariant d'Alembert operator associated with the metric tensor gμν, defined as □gΦ=−g1∂μ(−ggμν∂νΦ), where g=det(gμν) is the determinant of the spacetime metric. The factor −g ensures covariance under general coordinate transformations and accounts for the local volume element of curved spacetime. Physically, this equation generalizes the flat-spacetime Klein–Gordon equation by replacing ordinary derivatives with covariant derivatives determined by the geometry. The metric tensor gμν encodes gravitational effects, meaning that spacetime curvature directly modifies wave propagation. Thus, particle dynamics are influenced not only by mass m but also by geometric structure. When weak metric perturbations are assumed, gμν=ημν+hμν,∣hμν∣≪1,the geometry deviates slightly from Minkowski space ημν. In this regime, gravitational effects enter as small corrections to kinetic terms, enabling a perturbative analysis of how spacetime moduli fluctuations influence quantum evolution. We begin from the relativistic Klein–Gordon equation in curved spacetime: \begin{equation}\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0\end{equation} where \begin{equation}\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \partial_\nu \Phi \right)\end{equation} Assume weak metric perturbation: \begin{equation}g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1\end{equation} We separate fast rest energy oscillation: \begin{equation}\Phi(x,t) = e^{-i m c^2 t / \hbar} \psi(x,t)\end{equation} Taking the non-relativistic limit yields a modified Schrödinger equation: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m} \left(1 + \frac{2\phi}{c^2}\right)\nabla^2+ m\phi\right] \psi\end{equation} where ϕ is the weak gravitational potential. \subsection{3. Introducing Spacetime Moduli Perturbations} We decompose: \begin{equation}\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)\end{equation} where δϕ represents small-scale spacetime moduli fluctuations. The Hamiltonian becomes: \begin{equation}H = H_0 + \delta H\end{equation} with \begin{equation}H_0 = -\frac{\hbar^2}{2m}\nabla^2 + m\bar{\phi}\end{equation} and \begin{equation}\delta H \approx m\,\delta\phi(x,t)-\frac{\hbar^2}{m c^2}\delta\phi(x,t)\nabla^2\end{equation} Thus the modified Schrödinger equation is: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi\nabla^2\right]\psi\end{equation} \textbf{Interpretation:}Spacetime fluctuations enter quantum evolution as both an effective potential term and a modulation of kinetic energy. \paragraph{Physical Interpretation and Motivation.} The decomposition ϕ(x,t)=ϕˉ(x,t)+δϕ(x,t)separates spacetime structure into a smooth background component ϕˉ and a small fluctuating component δϕ. The background term represents the classical gravitational potential arising from large-scale geometry, while δϕ encodes microscopic spacetime moduli perturbations. These perturbations may originate from quantum geometric effects, discrete spacetime structure, or Planck-scale fluctuations. Physically, this decomposition allows us to isolate how small-scale spacetime irregularities influence quantum dynamics. By writing the Hamiltonian asH=H0+δH,we distinguish between ordinary quantum evolution in a smooth geometry (H0) and additional corrections induced by spacetime fluctuations (δH). The termδH≈mδϕ−mc2ℏ2δϕ∇2shows that spacetime perturbations modify both the potential energy and the kinetic structure of the particle. \paragraph{Intuitive Picture for a Non-Expert.} A helpful analogy is to imagine a wave traveling across the surface of a calm lake. If the water surface is perfectly smooth, the wave propagates cleanly. However, if the surface contains tiny ripples, the wave becomes distorted and slightly scattered. In this analogy, the smooth lake corresponds to ϕˉ, while the small ripples correspond to δϕ. Similarly, a quantum particle is described by a wavefunction. If spacetime itself contains microscopic fluctuations, then the wavefunction does not evolve in a perfectly smooth background. Instead, it experiences tiny distortions in its phase and amplitude. Over time, these small distortions accumulate and can lead to spreading of momentum and position distributions. Thus, the modified Schrödinger equation iℏ∂tψ=[−2mℏ2∇2+mϕˉ+mδϕ−mc2ℏ2δϕ∇2]ψ mathematically encodes the idea that spacetime itself actively influences quantum states. Rather than treating uncertainty as purely intrinsic, this framework allows us to explore the possibility that part of quantum indeterminacy arises from dynamical geometric perturbations at very small scales. \subsection{4. Stochastic Model of Spacetime Perturbation} \paragraph{Physical Motivation for the Stochastic Model.} In the previous section, spacetime perturbations δϕ(x,t) were introduced as small deviations from a smooth geometric background. To proceed analytically, we model these perturbations as stochastic fluctuations. This assumption does not imply that spacetime is fundamentally random; rather, it reflects our effective description after coarse-graining microscopic geometric degrees of freedom. In approaches such as Loop Quantum Gravity, spacetime geometry is fundamentally discrete at the Planck scale. When averaged over many microscopic quantum geometric states, these discrete structures manifest as effective fluctuations at larger scales. The condition⟨δϕ(x,t)⟩=0states that there is no preferred bias in the fluctuations: on average, spacetime remains centered around the smooth background ϕˉ. The two-point correlation function⟨δϕ(x,t)δϕ(x′,t′)⟩=C(x−x′)δ(t−t′)encodes how strongly spacetime fluctuations at one location are related to those at another. The spatial function C(x−x′) determines the correlation length, while the temporal delta function represents rapid (short-time) fluctuations. We introduce the interaction HamiltonianHint(t)=m∫d3xδϕ(x,t)∣x⟩⟨x∣to capture how spacetime perturbations couple locally to the mass-energy of the particle. The use of a second-order cumulant expansion is justified when fluctuations are weak but rapidly varying; it allows us to derive an effective evolution equation for the ensemble-averaged density matrix ρˉ. The resulting term∂tρˉ(x,x′,t)=−Γ(x−x′)ρˉ(x,x′,t)describes decoherence: of