Wave Functions In Phase Space Research Articles

Phase space Schrödinger equation and quadratic friction

A general Lagrangian for a point particle that moves across a viscous field quadratically dependent on the velocity is found. Afterward, the system is quantized via the Weyl quantization rule, and an explicit expression for the wave function in phase space is found.

The Landau problem and nonclassicality

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

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.

Quantum field theory in phase space

The tilde conjugation rule in thermofield dynamics, equivalent to the modular conjugation in a [Formula: see text]-algebra, is used to develop unitary representations of the Poincaré group, where the Hilbert space has the phase space content, a symplectic Hilbert space. The state is described by a quasi-amplitude of probability, which is a sort of wave function in phase space, associated with the Wigner function. The quantum field theory in phase space is then constructed, including the quantization rules for the Klein–Gordon and the Dirac fields, the derivation of the electrodynamics in phase space and elements of a relativistic quantum kinetic theory. Towards a physical interpretation of the theory, propagators are associated with the corresponding Wigner functions. The Feynman rules follow accordingly with vertices similar to those of usual non-Abelian quantum field theories.

Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

This paper is a continuation of our previous works about coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It concerns a study on the properties of wavefunctions in the phase space representation and the momentum dispersion operator, its representations and eigenvalue equation. After the recall of some results from our previous papers, we give most of the main properties of the phase space wavefunctions and consider some examples of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any phase space wavefunction is solution of this equation.

On phase-space representations of quantum mechanics using Glauber coherent states

A phase-space formulation of quantum mechanics is proposed by constructing two representations (identified as pq and qp) in terms of the Glauber coherent states, in which phase-space wave functions (probability amplitudes) play the central role, and position q and momentum p are treated on equal footing. After finding some basic properties of the pq and qp wave functions, the quantum operators in phase-space are represented by differential operators, and the Schrodinger equation is formulated in both pictures. Afterwards, the method is generalized to work with the density operator by converting the quantum Liouville equation into pq and qp equations of motion for two-point functions in phase-space. A coordinate transformation between those points allows one to construct a cell in phase-space, whose central point can be treated as a parameter. In this way, one gets equations of motion describing the evolution of one-point functions in phase-space. Finally, it is shown that some quantities obtained in this paper are related in a natural way with cross-Wigner functions, which are constructed with either the position or the momentum wave functions.

Zwitters: Particles between quantum and classical

We describe both quantum particles and classical particles in terms of a classical statistical ensemble, with a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same quantum formalism. Quantum particles are characterized by a specific choice of observables and time evolution of the probability density. Then interference and tunneling are found within classical statistics. Zwitters are (effective) one-particle states for which the time evolution interpolates between quantum and classical particles. Experimental bounds on a small parameter can test quantum mechanics.

Phase-Space Wave Functions of Harmonic Oscillator in Nanomaterials

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator in nanomaterials within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the harmonic oscillator. We also discuss the character of wave function and the “Fourier-like” projection transformations in phase space.

Phase-Space Wave Functions of Diatomic System in One-Dimensional Nanomaterials

The exact solutions of the stationary Schrödinger equations for the diatomic system with an empirical potential function in one-dimensional nanomaterials are solved within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The wave functions in position and momentum representations can be obtained through the Fourier-like projection transformation from the phase-space wave functions.

GeneralizedQ-functions

The modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter lambda specifies orderings in a mapping from the operator psi)(sigma to the corresponding phase space wavefunction, where sigma is a given fiducial vector. The choice lambda = 0 specifies the Weyl mapping and the Q-function so obtained is the usual one when sigma is the vacuum state. More generally, any choice of of lambda in the range (-1,1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q-functions are non-negative by construction. They are shown to be proportional to expectation of the system state rho with respect to a generalized displaced squeezed state which depends on lambda and position (p,q) in phase space. Thus, when a system has been prepared in the state rho, a generalized Q-function is proportional to the probability of finding it in the generalized squeezed state. Any such Q-function can also be written as the smoothing of the Wigner function for the system state rho by convolution with the Wigner function for the generalized squeezed state.

Semi-classical wormholes are unstable

We show that Lorentzian (traversable) wormholes with semi-classical spacetimes are unstable. Semi-classicality of the energy–momentum tensor of the exotic matter used to stabilize the wormhole implies localization of its wavefunction in phase space, leading to evolution according to the classical equations of motion. Previous results related to violation of the NEC then require that the matter is unstable to small perturbations.

Wave Mechanics in Quantum Phase Space: Harmonic Oscillator

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the “Fourier-like” projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.

Role that separatrices and stochastic webs play in quantum dynamics

We numerically analyze the influence of separatrices and stochastic webs on the evolution of quantum wave functions in phase space and we compare our findings with the corresponding classical evolution. We study the dynamics in regular and nonregular systems, namely, in a quartic, the pendulum, and the kicked harmonic oscillator potentials. Regardless of the specific potential, there are common features in the dynamics, some of which are revealed when very small parts of the quantum phase-space densities are analyzed.

On coherent-state representations of quantum mechanics: Wave mechanics in phase space

In this article we argue that the state-vector phase-space representation recently proposed by Torres-Vega and co-workers [introduced in J. Chem. Phys. 98, 3103 (1993)] coincides with the totality of coherent-state representations for the Heisenberg-Weyl group. This fact leads to ambiguities when one wants to solve the stationary Schrödinger equation in phase space and we devise two schemes for the removal of these ambiguities. The physical interpretation of the phase-space wave functions is discussed and a procedure for computing expectation values as integrals over phase space is presented. Our formal points are illustrated by two examples.

On quantum mechanical phase-space wave functions

An approach to quantum mechanics based on the notion of a phase-space wave function is proposed within the Weyl–Wigner–Moyal representation. It is shown that the Schrödinger equation for the phase-space wave function is equivalent to the quantum Liouville equation for the Wigner distribution function. The relationship to the recent results by Torres-Vega and Frederick [J. Chem. Phys. 98, 3103 (1993)] is also discussed.

Chebyshev scheme for the propagation of quantum wave functions in phase space

The propagation scheme based on the Chebyshev polynomial expansion of evolution operators is adapted for use in the propagation of quantum mechanical wave functions in phase space. This shows, once more, that one can do true quantum mechanics in phase space, in a similar way as it is done in abstract and coordinate or momentum spaces.

Numerical method for the propagation of quantum-mechanical wave functions in phase space.

A recently proposed phase-space representation for quantum mechanics provides a formal means for investigating the correspondence between classical and quantum mechanics. To explore the implications of this new representation, we introduce a numerical method for propagating an initial wave function directly in phase space. We illustrate the usefulness of the proposed method by analyzing two simple quantum-mechanical systems in phase space.