Nonrelativistic Quantum Mechanics Research Articles

Exact solutions of Dirac equation for hydrogen atom using the linear combination of orthogonal Laguerre basis functions

The associated Laguerre polynomials are very popular in the Schrödinger equation of nonrelativistic quantum mechanics. However, the situation is quite different in the case of the Dirac equation of relativistic quantum mechanics. The exact solutions of the Dirac equation for three-dimensional (3D) hydrogen atom have been obtained using the wavefunction expansion method. The exact solutions of the radial wavefunctions are found in the form of a linear combination of the orthogonal Laguerre basis function spinors. According to the equality of coefficients in Laguerre series on both sides of the equations, the exact eigenenergies and exact expressions of the corresponding eigenstates of the Dirac equation for 3D hydrogen atom are obtained, which are the same as the power series solutions. Additionally, a comparison with the results from Dirac and Schrödinger equations for 3D hydrogen atom is discussed. Due to the relativistic effects, the Dirac bispinor occurs, and there are the tiny differences of the results between Dirac and Schrödinger equations. For the Dirac states with nr = 0, there is one set of the Dirac bispinor. However for the Dirac states with nr ≠ 0, there is a combination of two sets of the Dirac bispinor. For degenerate Dirac states, their exact expansion coefficients are commutative symmetric. These results show that the linear combination of the orthogonal Laguerre basis functions is the exact solutions of the Dirac equation for 3D hydrogen atom.

The Quantum Theory of the Electromagnetic Field in the Weyl–Wigner Representation as a Local Realistic Model

I revisit the Wigner (or Weyl–Wigner, WW) representation of the quantum electromagnetic field. I show that, assuming that Fock states are just mathematical concepts devoid of physical reality, WW suggests a realistic interpretation which turns out to be (classical) Maxwell theory with the assumption that there is a random radiation filling space, the vacuum field. I elucidate why, in sharp contrast, non-relativistic quantum mechanics of particles does not admit a realistic interpretation via WW. I interpret experiments involving entangled light beams within WW, in particular optical tests of Bell inequalities. I show that WW provides clues in order to construct local models for those experiments. I give arguments why Bell definition of local realism is not general enough.

Complete solutions of the Schrödinger equation with improved exponential Kratzer–Feus potential

This study rigorously investigates the improved exponential Kratzer–Feus potential (IEKFP) model combined with exponential Kratzer–Feus potential (EKFP) and other terms produced by the effect of phase–space deformation [Formula: see text]. The new energy equation in three-dimensional non-relativistic non-commutative phase–space (3D(NR-NCPS)) symmetries is obtained using the parametric generalized Bopp’s shifts method and standard independent time perturbation theory of hydrogen-related molecules (CH, H2, NO, HCL, and LiH) diatomic molecules. This is achieved by applying a Green–Aldrich approximation scheme to the centrifugal terms. The new non-relativistic energy equation for the IEKFP in the presence of deformation phase–space is dependent on the discrete atomic quantum numbers ([Formula: see text] and [Formula: see text], the dissociation energy [Formula: see text], the equilibrium bond length [Formula: see text], the screening parameter [Formula: see text], and the deformation phase–space parameters ([Formula: see text]. The new resulting energy equation is utilized to calculate the partition function, from which thermodynamic properties such as mean energy, specific heat capacity, entropy, and free energy are derived in both three-dimensional non-relativistic quantum mechanics 3D(NR-QM) and (3D(NR-NCPS)) symmetries. This study has multiple applications in various domains, including atomic and molecular physics. We have indicated that our work deals with the study of the possibility of the effect of various phase–space deformations on various physical values such as energy and various thermodynamic properties under the influence of exponential Kratzer–Feus potential. So we completed the study of Amadi et al.

Investigating the effects of phase-space non-commutativity coordinates on the modified Deng-Fan Yukawa potential model and thermodynamic properties in 3D(NR-NCPS) and 3D(NR-QM) symmetries

An approximate new bound state solution of the three-dimensional deformed Schrodinger equation under the deformed phase-space sym- ¨ metries for the modified Deng-Fan Yukawa potential model that is obtained from the combination of the corresponding expression in threedimensional non-relativistic quantum mechanics symmetries and some central terms [exp (−αr)/r (1 − exp (−αr)), exp (−2αr)/r (1 − exp (−αr))2 , exp (−3αr)/r (1 − exp (−αr))3 , exp (−αr)/r 2 , exp (−αr)/r 3 and 1/r 4 ] coupled with the infinitesimal non-commutativity vector Θ and the angular momentum operator L. With the help of the parametric generalized Bopp’s shifts method, the independent time perturbation theory method, and an approximation scheme, the analytical energies of the studied were obtained for both symmetries, for different quantum numbers. The new non-relativistic energy equation under the studied potential for the homogenous diatomic molecules (HODMs) (H2, I2); the heterogeneous diatomic molecules (CO, HCl, LiH); the neutral transition metal hydrides (ScH, TiH, VH, CrH); the transition-metal lithide (CuLi); the transition-metal carbides (TiC, NiC); the transition metal nitrite (ScN) and the transition metal fluoride (ScF) and in the presence of deformation phase-space are dependent on the discrete atomic quantum numbers (j, l, s and m), the dissociation energy, the equilibrium bond length, and the screening parameter (re, De, and α), the deformation phase parameters (P nc p and S nc p ). The new resulting energy equation is utilized to calculate spin-averaged mass spectra of the heavy mesons under the studied potential and Deng-Fan Yukawa potential model in three-dimensional non-relativistic quantum mechanics and 3it’s extended symmetries. Furthermore, we have calculated the partition function, from which thermodynamic properties such as mean energy, specific heat capacity, entropy, and free energy are derived in both three-dimensional non-relativistic quantum mechanics and the deformed phase-space symmetries symmetries. Notably, the two special cases, representing the modified Yukawa potential and the modified Deng-Fan potential were treated in extended phase-space symmetry for energies and thermodynamic properties. Our current study promises to apply to different areas of physics in various domains, including atomic and molecular physics.

The Unbearable Indefiniteness of Spacetime

We consider the observables describing spatiotemporal properties in the context of two of the most popular approaches to quantum gravity (QG), namely String Theory and Loop QG. In both approaches these observables are described by non-commuting operators. In analogy with recent arguments put forward in the context of non-relativistic quantum mechanics [see Calosi and Mariani (Philos. Compass 16(4):e12731, 2021) for a review], we suggest that the physical quantities corresponding to those observables may be interpreted as ontologically indeterminate—i.e., indeterminate in a way that is non-epistemic and semantic-independent. This working hypothesis has not received enough attention in the current debate on QG, and yet it may prove explanatory useful in several respects. First, it provides a clear background for understanding how some features of QG are ontologically continuous to features of quantum mechanics. Second, it sets the stage for asking new interesting questions about QG, for instance concerning the status of the so-called Eigenstate-Eigenvalue link. Third, it indirectly shows how the debate on ontological indeterminacy may extend well beyond the non-relativistic case, contrary to what seems to be assumed. Fourth, and perhaps more importantly, it provides a promising alternative to the received view on QG [Wüthrich et al. (Philosophy Beyond Spacetime: Implications from Quantum Gravity, Oxford University Press, Oxford, 2021)] according to which spacetime is not fundamental. On the view we shall suggest, spacetime may be indeterminate and yet fundamental.

The Feynman propagator within three parameters generalized Dunkl derivative: One-dimensional case

In this paper, the Feynman path integral approach within three parameters Dunkl derivative is investigated in the one-dimensional case of nonrelativistic quantum mechanics. We successfully derive the propagator in Cartesian coordinates and give its exact expression for the free particle, inverse square, harmonic oscillator and pure Coulomb potentials. The energy eigenvalues and their corresponding wave functions are determined.

Numerical Study of Thermodynamic Properties of Screened Kratzer-Eckart Potential with External Magnetic Field

Abstract Attempts to understand the behaviour of various quantum systems are still become interest for many researchers in recent years. While experimentation is too complicated to be done, analytical mathematical approach is become a reliable choice. This approach is using potential functions or models as representations of the real quantum system, such as physical force interactions within a molecule. In this study, the new combination model of screened Kratzer and screened Eckart potentials is investigated under the non-relativistic quantum mechanics framework. These potentials are well-suited for modelling atomic interactions in diatomic molecules. The primary objective is to examine how these interactions are affected by the presence of external magnetic fields. Using a specific form of the Schro dinger equation, both the energy eigenfunction and the wave function have been derived. The Nikiforov-Uvarov Functional Analysis (NUFA) method was employed for this derivation. Additionally, several thermodynamic properties were obtained from the energy equation using numerical methods. Selected diatomic molecule data were used in the computations to demonstrate the effect of the magnetic field. A detailed analysis was performed through graphical representations, revealing that an increase in the magnetic field or the associated Aharonov-Bohm (AB) flux enhances and splits the energy spectra. The effects on thermodynamic properties vary across different molecules and conditions, but overall, the magnetic field increases the thermodynamic properties of the molecules.

Numerical Solutions of Certain Time-Independent Quantum Mechanics Problems by Using the Python Environment

This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.

Exactly solvable stochastic spectator

The stochastic formalism of inflation allows us to describe the scalar-field dynamics in a non-perturbative way. The correspondence between the diffusion and Schrödinger equations makes it possible to exhaustively construct analytical solutions in stochastic inflation. Those exact statistical quantities such as distribution and correlation functions have one-to-one correspondence to the exactly solvable solutions in non-relativistic quantum mechanics in terms of classical orthogonal polynomials. A class of such solutions is presented by means of isospectral Hamiltonians with an underlying symmetry called shape invariance.

Tetraquarks at large M and large N

We study tetraquarks in large N QCD with heavy quarks, in the domain where non-relativistic quantum mechanics offers an adequate approximation. Within the regime of validity of the Born-Oppenheimer approximation, we systematically study and explicitly construct tetraquark states. At leading order in the 1/N expansion, the bound spectrum consists of free mesons, while the 1/N corrections give rise to a Born-Oppenheimer potential that can bind the mesons into tetraquarks. We find two different types of tetraquarks, each endowed with distinct color-spatial wavefunctions. These states arise in the presence of an O(N) mass hierarchy between the quarks and the antiquarks. We provide a quantitative argument indicating that only for such a hierarchy is the ground state of the system a tetraquark. We discuss what the extrapolation of our results to realistic values of the parameters may imply for the QCD tetraquark states.

A new model describing improved bound-state and statistical properties in the framework of deformed Schrödinger equation for the improved Coshine Yukawa potential in 3D-NR(NCPS) symmetries

In this paper, within the three-dimensional non-relativistic non-commutative phase–space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schrödinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) ([Formula: see text]) and other terms produced from the effect of phase–space deformation ([Formula: see text] and [Formula: see text]). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp’s shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase–space are dependent on the discrete atomic quantum numbers ([Formula: see text],m), the dissociation energy [Formula: see text], the equilibrium bond length the screening parameter [Formula: see text], the deformation phase parameters ([Formula: see text]) and the deformation space parameters ([Formula: see text]). Additionally, the thermal properties of the CYP and ICYP with 3D Schrödinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase–space. This study has multiple applications in various domains, including atomic and molecular physics.

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

Energy spectra with the Dirac equation of the q-deformed generalized Pöschl-Teller potential via the Feynman approach for .

The diatomic molecules of potassium is widely used in industrial chemicals and alternative energy. Besides that, is very useful for researching molecular interactions and energy states, especially in the context of quantum chemistry and spectroscopy. In the present work, a newly proposed diatomic potential model within relativistic and non-relativistic quantum mechanics has been considered, to obtain corresponding energy eigenvalues and related normalized eigenfunctions. The Dirac equation has been solved for an arbitrary spin-orbit quantum number using the path integral technique with the -deformed generalized Pöschl-Teller potential . By including a Pekeris-type approximation to handle the centrifugal factor, it was possible to obtain the spin and pseudospin-symmetric solution of the relativistic energy eigenvalues and wave equation. To assess the correctness of this work, Maple software was used to present some numerical findings for various values of and . With the constraint , it was shown that in the situation of pseudospin symmetry, only bound states exist with negative energy. In the non-relativistic limits, the non-relativistic ro-vibrational energy expression of the diatomic molecule is derived from the relativistic energy equation under spin symmetry. Under Varshni conditions, both vibrational and ro-vibrational energies of the molecule were computed and compared with the data. The average absolute percentage deviations from the data obtained for the potassium molecule are . This demonstrates that the model is a very consistent model to study and characterize diatomic molecules.

[formula omitted] symmetry in hydrogen-atom-like model with spin

[formula omitted] symmetry in hydrogen-atom-like model with spin

Traveling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Traveling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

A hyperbolic Kac-Moody Calogero model

A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.

Quantum ontology de-naturalized: What we can't learn from quantum mechanics

Philosophers of science commonly connect ontology and science, stating that these disciplines maintain a two-way relationship: on the one hand, we can extract ontology from scientific theories; on the other hand, ontology provides the realistic content of our scientific theories. In this article, we will critically examine the process of naturalizing ontology, i.e., confining the work of ontologists merely to the task of pointing out which entities certain theories commit themselves to. We will use non-relativistic quantum mechanics as a case study. We begin by distinguishing two roles for ontology: the first would be characterized by cataloging existing entities according to quantum mechanics; the second would be characterized by establishing more general ontological categories in which existing entities must be classified. We argue that only the first step is available for a naturalistic approach; the second step not being open for determination or anchoring in science. Finally, we also argue that metaphysics is still a step beyond ontology, not contained in either of the two tasks of ontology, being thus even farther from science.

Remarks on quantum mechanics and non-reflexive logic

In this paper we discuss and outline a version of non-relativistic quantum mechanics based on a new non-reflexive logic, where the basic entities (elementary particles) lack identity conditions. Some relationships with quantum field theories are also sketched.

From the Hamilton-Jacobi equation to the Schrödinger equation and vice versa, without additional terms and approximations

In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.

Quantum Mechanics Based on an Extended Least Action Principle and Information Metrics of Vacuum Fluctuations

We show that the formulations of non-relativistic quantum mechanics can be derived from an extended least action principle. The principle can be considered as an extension of the least action principle from classical mechanics by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a physical system needs to exhibit during its dynamics in order to be observable. Second, there is constant vacuum fluctuation along a classical trajectory. A novel method is introduced to define the information metrics to measure additional observability due to vacuum fluctuations, which is then converted to an additional action through the first assumption. Applying the variational principle to minimize the total actions allows us to recover the basic quantum formulations including the uncertainty relation and the Schrödinger equation in the position representation. In the momentum representation, the same method can be applied to obtain the Schrödinger equation for a free particle while further investigation is still needed for a particle with an external potential. Furthermore, the principle brings in new results on two fronts. At the conceptual level, we find that the information metrics for vacuum fluctuations are responsible for the origin of the Bohm quantum potential. Even though the Bohm potential for a bipartite system is inseparable, the underlying vacuum fluctuations are local. Thus, inseparability of the Bohm potential does not justify a non-local causal relation between the two subsystems. At the mathematical level, quantifying the information metrics for vacuum fluctuations using more general definitions of relative entropy results in a generalized Schrödinger equation that depends on the order of relative entropy. The extended least action principle is a new mathematical tool. It can be applied to derive other quantum formalisms such as quantum scalar field theory.