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

Australian Journal of Physics

Quantum-mechanical Wave Equation for Two Particles of Spin 0

Kwong-Chuen Tam

Published Oct 1, 1975• 1 citation
Publisher

Abstract

A quantum-mechanical wave equation for two particles of spin 0 is presented in Hamiltonian formalism and is then simplified and discussed. Solutions are found for square-well and Coulomb interactions, and energy levels are determined. It is shown that, for the Coulomb interaction, the energy levels to the lowest order agree with those given by the hydrogen atom formula.

DOI
10.1071/ph750495

Related papers

A model with two quantum particles similar to the hydrogen atom

Ruslan Abdulovich Sharipov

The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.

arXiv:1308.0221•Aug 1, 2013•0 citations

Spin - 1/2 wave equations in relativistic quantum mechanics

D S Staudte

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.

•Jan 1, 1993•0 citations

Wavefunction and energy level formula for two charged particles with magnetic interaction

Hong-Yi Fan, Liang Fu

We derive the wavefunction and energy level formula for two charged particles with magnetic interaction, i.e., the Hamiltonian includes both two-body Coulomb interaction and a kinetic coupling. It is by virtue of the EPR entangled state representation we can conveniently derive the exact result.

Journal of Physics A Mathematical and General•Jul 9, 2003•2 citations

An eight-component relativistic wave equation for spin- particles II

D S Staudte

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.

Journal of Physics A Mathematical and General•Jan 7, 1996•23 citations

A Notable Quasi-Relativistic Wave Equation and Its Relation to the Schrödinger, Klein-Gordon, and Dirac Equations

Luis Grave de Peralta, Hira Farooq

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.

Journal of Modern Physics•Jan 1, 2021•7 citations

Relativistic Schrödinger Wave Equation for Hydrogen Atom Using Factorization Method

Mohammad Reza Pahlavani, Hossein Rahbar, Mohsen Ghezelbash

In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

Open Journal of Microphysics•Jan 1, 2013•4 citations
A quantum-mechanical wave equation for two particles of spin 0 is presented in Hamiltonian formalism and is then simplified and discussed. Solutions are found for square-well and Coulomb interactions, and energy levels are determined. It is shown that, for the Coulomb interaction, the energy levels to the lowest order agree with those given by the hydrogen atom formula.
The hydrogen atom with the Coulomb interaction is one of the exactly solvable non-relativistic quantum models. Unlike many other exactly solvable models it describes a real physical object providing the formulas for energy levels and stationary state wave functions of a real hydrogen atom. In this paper we modify the model replacing the Coulomb interaction by the interaction of the proton and the electron with the classical electromagnetic field serving as an intermediary transmitting the electromagnetic interaction of these two charged quantum particles.
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.
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.
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.
In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.