Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Applied Mathematics and Physics

A Simple Remark Leading to a Basic Precision Estimate for Non-Relativistic (NR) Real Values of Quantum Mechanics Operators

Amaury de Kertanguy

Published Jan 1, 2018• 1 citation
Publisher

Abstract

Starting with a very basic statement that any physical constants cannot be written with an infinite precision, it is shown how to introduce this uncertainty into the Hamiltonian of non-relativistic atomic (NR) physics and how to estimate errors on quantum operators (energy, frequency, momenta) when an uncertainty is assigned to . The Schrödinger equation is written and the kinetic energy term is transformed into a Laplacian: . This transformation leads (as known since 1926) to the wave equation, whose solutions are wave functions. The relativity correction to the kinetic energy term is introduced and its effect is discussed. (h constant has an uncertainty value taken from CODATA.)

DOI
10.4236/jamp.2018.64071

Related papers

Basic quantum Hamiltonian's relativistic corrections

Gintautas Kamuntavičius

After analyzing Dirac's equation, one can suggest that a well-known quantum-mechanical momentum operator is associated with relativistic momentum, rather than with non-relativistic one. Consideration of relativistic energy and momentum expressions allows us to define the non-relativistic, relativistic and pseudo-relativistic (present in Schrödinger equation) kinetic energy operators. Consequences of kinetic energy operator's correction for spectra of basic quantum Hamiltonians are investigated. In some cases this correction can produce remarkable spectra modifications.

arXiv:1302.0491•Feb 3, 2013•0 citations

On Quantum Physics without Schrödinger Equation

Xiao En Wang

In the early 20th century, Planck, Einstein, Bohr, Sommerfeld, and de Broglie became the pioneers of quantum physics. Because the Schrödinger equation used the wrong energy relationship, it misled people’s understanding of quantum theory. Newtonian mechanics and Leibniz dynamics are correct. But Newton’s first law was tampered with by the British mathematician Mott during his translation, which excluded uniform circular motion from inertia; Leibniz’s kinetic energy mυ2 was modified by French engineer Coriolis to an engineering approximation of mυ2/2. Hamilton mistakenly applied the approximate kinetic energy formula of Coriolis in engineering mechanics to the dynamic analysis in theoretical physics, and established an incorrect Hamiltonian energy equation. Schrödinger ignored Sommerfeld’s research on the constant velocity of electrons in atoms, υ = αc, and thus the constant momentum and constant kinetic energy of electrons. He expressed the constant kinetic energy term of electrons as an operator that hides the velocity variable, and established the wave equation based on the incorrect Hamiltonian energy relationship. The Schrödinger equation deals with the ground state of hydrogen atoms, where electrons die when they reach twice the Bohr radius, which contradicts the stable existence of atoms. The Schrödinger ‎equation has been around for almost a century, and the quantum physics of the new millennium ‎should be a new quantum physics without the Schrödinger equation.

Journal of Physics & Optics Sciences•Apr 25, 2025•0 citations

Kinematic relativity of quantum Hamiltonian

Gintautas Kamuntavičius

It is shown that the well-known relativistic correction of quantum Hamiltonian that is present in textbooks appears after quantization of oversimplified relativistic kinetic energy decomposition. Using the proper expression one obtains the kinetic energy operator which relation with the ususal one is consistent with corresponding relation of kinetic energies in special relativity theory. The Schrödinger equation with this operator gives new interesting possibilities for quantum phenomena description.

arXiv:1401.1098•Jan 6, 2014•1 citation

Perceptions of the Schrodinger Equation

Spyros Efthimiades

The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.

arXiv:1307.6108•Bulletin of the American Physical Society•Jul 23, 2013•0 citations

Relativistic Approximations for Quantization and Harmony in the Schrödinger Equation, and Why Mechanics Is Quantized

Antony J. Bourdillon

The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.

Journal of Modern Physics•Jan 1, 2020•4 citations

Quantizing the Quantum Uncertainty

Etera R. Livine

The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the $\textrm{sl}_{2}$ Lie algebra formed by the operators $\hat{x}^{2}$, $\hat{p}^{2}$ and $\widehat{xp}$. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?

arXiv:2307.01061•Jul 3, 2023•0 citations
Starting with a very basic statement that any physical constants cannot be written with an infinite precision, it is shown how to introduce this uncertainty into the Hamiltonian of non-relativistic atomic (NR) physics and how to estimate errors on quantum operators (energy, frequency, momenta) when an uncertainty is assigned to . The Schrödinger equation is written and the kinetic energy term is transformed into a Laplacian: . This transformation leads (as known since 1926) to the wave equation, whose solutions are wave functions. The relativity correction to the kinetic energy term is introduced and its effect is discussed. (h constant has an uncertainty value taken from CODATA.)
After analyzing Dirac's equation, one can suggest that a well-known quantum-mechanical momentum operator is associated with relativistic momentum, rather than with non-relativistic one. Consideration of relativistic energy and momentum expressions allows us to define the non-relativistic, relativistic and pseudo-relativistic (present in Schrödinger equation) kinetic energy operators. Consequences of kinetic energy operator's correction for spectra of basic quantum Hamiltonians are investigated. In some cases this correction can produce remarkable spectra modifications.
In the early 20th century, Planck, Einstein, Bohr, Sommerfeld, and de Broglie became the pioneers of quantum physics. Because the Schrödinger equation used the wrong energy relationship, it misled people’s understanding of quantum theory. Newtonian mechanics and Leibniz dynamics are correct. But Newton’s first law was tampered with by the British mathematician Mott during his translation, which excluded uniform circular motion from inertia; Leibniz’s kinetic energy mυ2 was modified by French engineer Coriolis to an engineering approximation of mυ2/2. Hamilton mistakenly applied the approximate kinetic energy formula of Coriolis in engineering mechanics to the dynamic analysis in theoretical physics, and established an incorrect Hamiltonian energy equation. Schrödinger ignored Sommerfeld’s research on the constant velocity of electrons in atoms, υ = αc, and thus the constant momentum and constant kinetic energy of electrons. He expressed the constant kinetic energy term of electrons as an operator that hides the velocity variable, and established the wave equation based on the incorrect Hamiltonian energy relationship. The Schrödinger equation deals with the ground state of hydrogen atoms, where electrons die when they reach twice the Bohr radius, which contradicts the stable existence of atoms. The Schrödinger ‎equation has been around for almost a century, and the quantum physics of the new millennium ‎should be a new quantum physics without the Schrödinger equation.
It is shown that the well-known relativistic correction of quantum Hamiltonian that is present in textbooks appears after quantization of oversimplified relativistic kinetic energy decomposition. Using the proper expression one obtains the kinetic energy operator which relation with the ususal one is consistent with corresponding relation of kinetic energies in special relativity theory. The Schrödinger equation with this operator gives new interesting possibilities for quantum phenomena description.
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2\textrm{sl}_{2}sl2​ Lie algebra formed by the operators x^2\hat{x}^{2}x^2, p^2\hat{p}^{2}p^​2 and xp^\widehat{xp}xp​. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?