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Moscow University Physics Bulletin

Heron-Inspired Method to Solve a Relativistic Wave Equation for a Spin-$$\boldsymbol{1/2}$$ Massive Particle

A. M. Kalitenko, P. I. Pronin

Published Feb 1, 2026• 0 citations
Publisher

Abstract

This article presents a novel method for solving a relativistic wave equation for a spin- $$1/2$$ massive particle. Relativistic wavefunctions are obtained using neural networks, and a novel approach to solving the Klein–Gordon–Fock equation has been developed, inspired by the ancient method of Heron. Unlike the four-component Dirac bispinor, this approach uses two-component wavefunctions. The method has been validated through application to two well-known physical systems: a hydrogen-like atom and a charged particle in a uniform magnetic field. The obtained energy spectra are in complete agreement with the analytical solutions derived from the Dirac equation. The method can be applied to more complex problems in quantum physics and chemistry.

DOI
10.3103/s0027134926700074

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This article presents a novel method for solving a relativistic wave equation for a spin- 1/21/21/2 massive particle. Relativistic wavefunctions are obtained using neural networks, and a novel approach to solving the Klein–Gordon–Fock equation has been developed, inspired by the ancient method of Heron. Unlike the four-component Dirac bispinor, this approach uses two-component wavefunctions. The method has been validated through application to two well-known physical systems: a hydrogen-like atom and a charged particle in a uniform magnetic field. The obtained energy spectra are in complete agreement with the analytical solutions derived from the Dirac equation. The method can be applied to more complex problems in quantum physics and chemistry.
This thesis is concerned with the development of a new spin-1/2 wave equation in relativistic quantum mechanics. This equation is the spin-1/2 analogue of the spin-0 Feshbach-Villars equation. The thesis begins in chapter 1 with a review of the subject of relativistic wave equations and its place in the development of modern physics. Chapter 2 sees the derivation of the new equation (hereafter referred to as the FV1/2 equation). This derivation is presented at three levels of understanding, and is combined with an analysis of the equation using elements of the underlying mathematics of the theories of special relativity and quantum mechanics. The analysis justifies the methods of derivation of the equation, and its suitability as a wave equation in relativistic quantum mechanics. It is shown th a t the FV1/2 equation is different to the Dirac equation and other equations in the literature. One of the striking features of the FV1/2 equation is th a t it contains a less restrictive dynamics than the conventional Dirac equation. Chapter 3 examines the solution space and proves some results using the discrete symmetries C, P and T, which enable a subset of the solution space to be used for physical applications. It is possible to use a subset of solutions which contain the usual Dirac negative energy states, or it is possible to use a subset which contains only positive energy states, but includes states of opposite signs of charge. The second subset corresponds to the physical properties of particles and antiparticles and suggests th a t the FV1/2 equation could be useful in atomic physics. Chapter 4 represents a start on the development of the FV1/2 equation for use in atomic physics. It contains a study of hydrogenic atoms. The energy spectra and wavefunctions for hydrogenic atoms are derived using the FV1/2 equation. The method of solution is of comparable difficulty to th a t using the Dirac equation. The spectra are found to be identical to th a t of the Dirac equation, but the wavefunctions differ. A calculation of the expectation value of the Coulomb energy is performed and the results obtained differ from the Dirac equation results by 6 x 10-3% for Z — 1, and 40% for Z = 70. This suggests th a t physically measureable quantities should be calculated and compared using the FV1/2 and Dirac equations. A calculation is presented to show the energy spectra and wavefunctions obtained are consistent with the literature. The results obtained in this chapter (again) suggest th a t the FV1/2 equation could be useful for calculations in atomic physics. Chapter 5 discusses important further work which should be undertaken if the FV1/2 equation is to become useful to physics. The development in the previous chapters concentrated on the FV1/2 equation in relativistic quantum mechanics. It should be also considered as a quantised field equation. The two electron problem and the calculation of physical quantities are crucial tests of the FV1/2 equation’s applicability in atomic physics. The mathematical analysis in chapter 2 can be extended. Suggestions for gaining a deeper mathematical understanding of the less restrictive dynamics of the FV1/2 equation are given. Finally some conclusions on what has been achieved in the thesis are presented in chapter 6.
An intriguing quasi-relativistic wave equation, which is useful between the range of applications of the Schrödinger and the Klein-Gordon equations, is discussed. This equation allows for a quantum description of a constant number of spin-0 particles moving at quasi-relativistic energies. It is shown how to obtain a Pauli-like version of this equation from the Dirac equation. This Pauli-like quasi-relativistic wave equation allows for a quantum description of a constant number of spin-1/2 particles moving at quasi-relativistic energies and interacting with an external electromagnetic field. In addition, it was found an excellent agreement between the energies of the electron in heavy Hydrogen-like atoms obtained using the Dirac equation, and the energies calculated using a perturbation approach based on the quasi-relativistic wave equation. Finally, it is argued that the notable quasi-relativistic wave equation discussed in this work provides interesting pedagogical opportunities for a fresh approach to the introduction to relativistic effects in introductory quantum mechanics courses.
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.
The eight-component relativistic wave equation for spin- particles derived in the preceding paper is studied from a mathematical viewpoint. It is shown that it is relativistically covariant, albeit not manifestly so. It has an enlarged solution space when compared to the Dirac equation. A derivation of the equation in an arbitrary gamma matrix representation is presented. Kronecker products are used to display the analogy between the eight-component equation and the two-component spin-0 equation. The linearization procedure used to obtain the eight-component equation from the second-order spin- equation is found to have a natural mathematical validation in the study of indefinite inner product spaces. The (relativistic) quantum mechanical formalism for the eight-component equation is constructed. This features the use of an indefinite inner product and some results are presented to show how the usual quantum mechanical formalism is generalized to account for this. The reason for the decoupling of the equation into two four-component equations in the Weyl representation of the gamma matrices is given. It is shown that only one of the decoupled parts has to be solved for any problem of a single particle coupled to an external electromagnetic field. Comments are made on the solution of the hydrogen atom presented in the preceding paper.
In this article, we present a method for finding quantum stationary states of the Klein–Gordon–Fock (KGF) equation using neural networks (NNs). The method has been tested on two well-known systems: a relativistic spinless particle in a Coulomb potential, and a one-dimensional relativistic harmonic oscillator. The results of training the neural network for these two systems are presented, as well as the analysis of the training process. The neural network method shows a good agreement with analytical calculations (if they can be found explicitly), providing a promising approach for solving more complex problems in quantum physics and quantum chemistry. В этой статье мы представляем метод вычисления стационарных состояний уравнения Клейна-Гордона–Фока с помощью нейронных сетей. Метод был апробирован на двух хорошо известных системах: релятивистской бесспиновой частице в кулоновском потенциале и одномерном релятивистском гармоническом осцилляторе. Представлены результаты обучения нейронной сети для этих двух систем, а также анализ процесса обучения. Метод нейронных сетей показывает хорошее соответствие с результатами аналитических вычислений (если они могут быть найдены в явном виде), что открывает перспективы для решения более сложных задач в области квантовой физики и квантовой химии.