Abstract In Schrödinger's equation, the local contribution to the kinetic energy is sometimes negative, but it allows a conservative potential in contrast to an alternative, never negative, energy proposed by Ruedenberg. The total kinetic energy evaluated in certain characteristic volumes, and in all of space, is identical in the two cases. Dirac's suggestion in 1962 of a bubble‐surface representing the electron is applied in a far‐reaching, but incomplete, attempt to identify external kinetic energy with a first‐order relativistic effect and the internal energy constituting the difference between Schrödinger and Ruedenberg's proposals is related to oscillations of the surface.
Abstract In Schrödinger's equation, the local contribution to the kinetic energy is sometimes negative, but it allows a conservative potential in contrast to an alternative, never negative, energy proposed by Ruedenberg. The total kinetic energy evaluated in certain characteristic volumes, and in all of space, is identical in the two cases. Dirac's suggestion in 1962 of a bubble‐surface representing the electron is applied in a far‐reaching, but incomplete, attempt to identify external kinetic energy with a first‐order relativistic effect and the internal energy constituting the difference between Schrödinger and Ruedenberg's proposals is related to oscillations of the surface.
Abstract A fundamental belief is that the formulism of relativistic quantum mechanics equations (RQMEs) should remain in low momentum motion. However, it is found that some formulas from RQMEs were lost in Schrödinger equation. For example, a free relativistic particle has positive and negative energy branches. The former includes positive kinetic energy (PKE) and the latter negative kinetic energy (NKE). The latter should be treated on an equal footing as the former. Nevertheless, from Schrödinger equation, a free particle can have only PKE. Starting from RQMEs and taking low momentum approximation, we derive NKE Schrödinger equation which is for the cases that free particles have NKE. Thus negative energy branch of RQMEs can be retained in low momentum motion. We point out a fact that whether Schrödinger equation is applicable in a region where a particle’s energy E is less than potential V , E < V , has never been quantitatively verified. In such a region NKE Schrödinger equation should be employed. With the help of NKE Schrödinger equation, the lost formulas are recovered. The so-called difficulty of negative probability of Klein–Gordon equation for free particles is solved. A PKE (NKE) particle can have stationary motion only when it is subject to an attractive (repulsive) potential, which is determined by Virial theorem. Two NKE electrons in a potential can constitute a stable system, a new kind of possible mechanism for electron paring. The whole discussion stems from RQMEs with no any new postulation. Experiments are suggested, which may confirm that there are indeed NKE electrons.
A new relativistic quantum wave equation has been derived by applying the quantum prescription to the momentum and the kinetic energy rather than to the momentum and the total energy, since after all it is the kinetic energy that generates the momentum. The resulting equation reduces to the Schrodinger equation in the nonrelativistic limit and to the Klein-Gordon equation for “massless particles” in the relativistic limit, i.e., if the velocity of the particle approaches that of light, c. For massive particles in general, the new equation deviates from the Klein-Gordon equation. The same equation is shown to decouple according to the Dirac formalism, yielding a modified form of Dirac equation. When applied to a rest particle, the modified Dirac equation is shown to avoid a negative energy solution and instead include a constant solution. The other, the time-dependent particle solution of the modified Dirac equation, has the characteristic frequency Mc2/(ћ/2) , i.e., twice those of the Dirac solutions, Mc2/ћ.
This chapter presents the first concrete consequences of the foregoing one. The possibility of configurations, x, for which the classical expression for the potential energy, V(x), is larger than the total energy, E, implies, that this expression, V(x), does no longer represent the contribution of the configuration x to the work storage of the system. For this, a ‘limiting function’, F(x), is introduced such, that the non-classical contribution of the configuration x to the work storage, Vncl(x) = F(x)V(x), is smaller than the total energy. The same is done for the momentum configurations and the kinetic energy. Moreover, since there are no trajectories, the non-classical representation of the energies become integral expressions, in agreement with Schrödinger’s vision. Then, the general properties of the limiting functions are deduced. Limiting amplitudes (dimensionless wave functions) are introduced, in order to find an integral relationship between the motions in space and in momentum space, as envisaged by Schrödinger, again.
The Schrödinger equation, Klein‐Gordon equation (KGE), and Dirac equation are believed to be the fundamental equations of quantum mechanics. Schrödinger’s equation has a defect in that there are no negative kinetic energy (NKE) solutions. Dirac’s equation has positive kinetic energy (PKE) and NKE branches. Both branches should have low-momentum, or nonrelativistic, approximations: One is the Schrödinger equation, and the other is the NKE Schrödinger equation. The KGE has two problems: It is an equation of the second time derivative so that the calculated density is not definitely positive, and it is not a Hamiltonian form. To overcome these problems, the equation should be revised as PKE- and NKE-decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are unitary in that all contain the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as, in the author’s opinion, the matter and dark matter of our universe. The problems of one-dimensional step potentials are resolved by utilizing the modified fundamental equations for a nonrelativistic particle.
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.
\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