Quantum Mechanics In Phase Space Research Articles

Higher-order phase-space moments for off-diagonal rotating Morse oscillators

The present paper proposes an advancement beyond the recent mathematical formulation [Cherroud O and Yahiaoui S-A 2023 Eur. Phys. J. Plus. 138 534], by extending their approach to the evaluation of off-diagonal matrix elements 〈xˆmpˆl〉n,n′J,J′ for the rotation-vibration Morse oscillators. Calculations are conducted within the framework of phase-space quantum mechanics, considering (r → r eq). This methodology facilitates the derivation of analytical expressions for 〈xˆs〉n,n′J,J′ and 〈pˆs〉n,n′J,J′ matrix elements, where s = 1, 2, …. This approach enables the evaluation of both diagonal ( n=n′,J=J′≠0 ) and off-diagonal ( n≠n′,J=J′≠0 ) matrix elements through a more explicit and compact analytic formula, applicable for cases where J≠J′ as well as for scenarios involving non-rotating effects J=J′=0 .

The continuity equation in the phase space quantum mechanics

A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The 1-D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free motion and of scattering in a nonrelativistic and relativistic case for setups with internal degrees of freedom is discussed and illustrated. Properties of Wigner functions of unbound states are analysed.

Dispersion chain of quantum mechanics equations

Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton–Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie–Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fiber bundles

The discrete phase space and continuous time representation of relativistic quantum mechanics are further investigated here as a continuation of paper I.1 The main mathematical construct used here will be that of an area filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as [Formula: see text], a circle of radius [Formula: see text] where [Formula: see text]. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell [Formula: see text]. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a [Formula: see text]-dimensional phase space hyper-tori [Formula: see text] as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fiber bundles. We also study free scalar Bosons in the background [Formula: see text]-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein–Gordon equation. The second quantized field quanta of this system can cohabit with the tiny Planck oscillators inside the [Formula: see text] phase cells for eternity. Finally, a generalized free second quantized Klein–Gordon equation in a higher [Formula: see text]-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.

Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

Wigner functions, allowing for a reformulation of quantum mechanics in phase space, are of central importance for the study of the quantum-classical transition. A full understanding of the quantum-classical transition, however, also requires an explanation for the absence of macroscopic superpositions to solve the quantum measurement problem. Stochastic reformulations of quantum mechanics based on spontaneous collapses of the wavefunction are a popular approach to this issue. In this article, we derive the dynamic equations for the four most important spontaneous collapse models—Ghirardi–Rimini–Weber (GRW) theory, continuous spontaneous localization (CSL) model, Diósi-Penrose model, and dissipative GRW model—in the Wigner framework. The resulting master equations are approximated by Fokker–Planck equations. Moreover, we use the phase-space form of GRW theory to test, via molecular dynamics simulations, David Albert’s suggestion that the stochasticity induced by spontaneous collapses is responsible for the emergence of thermodynamic irreversibility. The simulations show that, for initial conditions leading to anti-thermodynamic behavior in the classical case, GRW-type perturbations do not lead to thermodynamic behavior. Consequently, the GRW-based equilibration mechanism proposed by Albert is not observed.

Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length [Formula: see text] is investigated. Fundamental physical constants such as [Formula: see text], [Formula: see text], and [Formula: see text] are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles [Formula: see text] of radii [Formula: see text] within the discrete [Formula: see text]-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities [Formula: see text] and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion [Formula: see text] matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein–Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group [Formula: see text].

Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study

Abstract We formulated the oscillators with position-dependent finite symmetric decreasing and increasing mass. The classical phase portraits of the systems were studied by analytical approach (He’s frequency formalism). We also study the quantum mechanical behaviour of the system and plot the quantum mechanical phase space for necessary comparison with the same obtained classically. The phase portrait in all the cases exhibited closed loop reflecting the stable system but the quantum phase portrait exhibited the inherent signature (cusp or kink) near origin associated with the mass. Although the systems possess periodic motion, the discrete eigenvalues do not possess any similarity with that of the simple harmonic oscillator having m = 1.

Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

AbstractThe phase‐space formulation of quantum mechanics has recently seen increased use in testing quantum technologies, including methods of tomography for state verification and device validation. Here, an overview of quantum mechanics in phase space is presented. The formulation to generate a generalized phase‐space function for any arbitrary quantum system is shown, such as the Wigner and Weyl functions along with the associated Q and P functions. Examples of how these different formulations are used in quantum technologies are provided, with a focus on discrete quantum systems, qubits in particular. Also provided are some results that, to the authors' knowledge, have not been published elsewhere. These results provide insight into the relation between different representations of phase space and how the phase‐space representation is a powerful tool in understanding quantum information and quantum technologies.

The geometrization of quantum mechanics, the nonlinear Klein–Gordon equation, Finsler gravity and phase spaces

The Geometrization of Quantum Mechanics proposed in this work is based on the postulate that the quantum probability density can curve the classical spacetime. It is shown that the gravitational field produced by smearing a point-mass Mo at r=0 throughout all of space (in a spherically symmetric fashion) can be interpreted as the gravitational field generated by a self-gravitating anisotropic fluid droplet of mass density 4πMor2φ∗(r)φ(r) and which is sourced by the probabilitycloud (associated with a spinless point-particle of mass Mo) permeating a 3-spatial domain region D3=∫4πr2dr at any time t. Classically one may smear the point mass in any way we wish leading to arbitrary density configurations ρ(r). However, Quantum Mechanically this is not the case because the radial mass configuration M(r) must obey a key third order nonlinear differential equation (nonlinear extension of the Klein–Gordon equation) displayed in this work and which is the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We conclude by extending our proposal to the Lagrange–Finsler and Hamilton–Cartan geometry of (co) tangent spaces and involving the relativistic version of Bohm’s Quantum Potential. By further postulating that the quasi-probability Wigner distribution W(x,p)curves phase spaces, and by encompassing the Finsler-like geometry of the cotangent-bundle with phase space quantum mechanics, one can naturally incorporate the noncommutative and non-local Moyal star product (there are also non-associative star products as well). To conclude, Phase Space is the arena where to implement the space–time–matter unification program. It is our belief this is the right platform where the quantization of spacetime and the quantization in spacetime will coalesce.

Star Product on the Euclidean Motion Group in the Plane

In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.

Phase-space quantum mechanics as a Landau-level problem

We point out the connection between the problem of formulating quantum mechanics in phase space and projecting the motion of a quantum mechanical particle onto a particular Landau level. In particular, we show that lowest Landau level wave functions, which are widely used in studies of quantum Hall effect, are actually phase space wave functions in this context. We demonstrate the usefulness of this understanding by analyzing some simple problems, and propose other utilities.

Development in Phase Space Quantum Mechancis for Constructing Different Masses’ Entangled State Representation

For developing the phase space quantum mechanics we explore how to construct different masses’ entangled state representation with the use of tthe Weyl-Wigner correspondence. We shall adopt a new approach, which uses the Radon transform of two-mode Wigner operator in normally ordering form to realize our goal. As an application of the entangled state representation the new normally ordered generalized squeezing operator is derived. The technique of integration within ordered product of operators is fully used.

Phase space Heisenberg-limited estimation of the average phase shift in a Mach–Zehnder interferometer

We address the problem of distributed quantum metrology with a single squeezed-vacuum source by using the formalism of quantum mechanics in phase space. In particular, we demonstrate Heisenberg-limited sensitivity in the measurement of the average of two arbitrary phase shifts in the arms of a Mach–Zehnder interferometer. We obtain exact results for the measurement probability at the interferometer output for any value of the phases, which give us insight into the emergence of Heisenberg-limited sensitivity for periodical values of the phases.

Ensemble in phase space: Statistical formalism of quantum mechanics

We present an alternative formalism of quantum mechanics tailored to statistical ensemble in phase space. The purpose of our work is to show that it is possible to establish an alternative autonomous formalism of quantum mechanics in phase space using statistical methodology. The adopted perspective leads to obtaining within the framework of its theory the master quantum-mechanical equation without recourse to the other formalisms of quantum mechanics, and gives the idea of operators pertaining to dynamical quantities. The derivation of this equation starts with the ensemble in phase space. We have explained with the help of this equation the structure of quantum mechanics in phase space and the approximation to the Schrodinger equation. Furthermore, we have shown that this formalism provides reasonable results of quantization by dealing with some simple cases, which confirm the validity of this formalism. In particular, we have demonstrated that this formalism can easily give the relativistic wave equation without treating the problem of linearizing the Hamiltonian operator by making the most of the point that the master equation is a first-order partial differential equation with respect to time, position and momentum variables, and makes use of the phase velocity. The ultimate outcome this formalism produces is that primary and general matters of quantum mechanics can be studied reasonably within the framework of statistical mechanics.

General approach to quantum mechanics as a statistical theory

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -- the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -- extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -- putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

Quantum Physics in Phase Space: an Analysis of Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

New Fourier transform containing a pair of complex Euler Γ-functions with a monomial: Mathematical and physical applications

One of the goals of this paper is to propose an elementary method to find a general formula for a new Fourier transform containing a pair of complex Euler Γ-functions with a monomial sm in terms of Gauss’s hypergeometric functions 2F1. Some mathematical results have been obtained and discussed using this transform. Furthermore, physical applications in the phase space quantum mechanics requiring the expectation values of position and momentum operators for a quantum system endowed with position-dependent effective mass are presented.

Derivation of one-particle semiclassical kinetic theory in the presence of non-Abelian Berry curvature

In quantum mechanics it is often required to describe in a semiclassical approximation the motion of particles moving within a given energy band. Such a representation leads to the appearance of an analogues of fictitious forces, associated with the Berry curvature, in the semiclassical equations of motion. The purpose of this paper is to derive systematically the kinetic Boltzmann equations displaying these effects in the case that the band is degenerate, and as such the Berry curvature is generically non-Abelian. We use the formalism of phase-space quantum mechanics to derive the results.

Magnetic susceptibility of Dirac fermions, Bi-Sb alloys, interacting Bloch fermions, dilute nonmagnetic alloys, and Kondo alloys

Wide ranging interest in Dirac Hamiltonian is due to the emergence of novel materials, namely, graphene, topological insulators and superconductors, the newly-discovered Weyl semimetals, and still actively-sought after Majorana fermions in real materials. We give a brief review of the relativistic Dirac quantum mechanics and its impact in the developments of modern physics. The quantum band dynamics of Dirac Hamiltonian is crucial in resolving the giant diamagnetism of bismuth and Bi-Sb alloys. Quantitative agreement of the theory with the experiments on Bi-Sb alloys has been achieved, and physically meaningful contributions to the diamagnetism has been identified. We also treat relativistic Dirac fermion as an interband dynamics in uniform magnetic fields. For the interacting Bloch electrons, the role of translation symmetry for calculating the magnetic susceptibility avoids any approximation to second order in the field. The expressions for magnetic susceptibility of dilute nonmagnetic alloys give a firm theoretical foundation of the empirical formulas used in fitting experimental results. The unified treatment of all the above calculations is based on the lattice Weyl-Wigner formulation of discrete phase-space quantum mechanics. For completeness, the magnetic susceptibility of Kondo alloys is also given since Dirac fermions in conduction band and magnetic impurities exhibit Kondo effect.