Quantum Phase-space Representation Research Articles

Real wavefunction from Generalised Hamiltonian Schrodinger Equation in quantum phase space via HOA (Heaviside Operational Ansatz): exact analytical results

This note reports the decomposition of a complex wavefunction into real and imaginary parts via the Heaviside Operational Ansatz (HOA) in the Quantum Phase Space Representation; this for a Complex Schrodinger Equation (CSE) generated by a given Generalised Hamiltonian. Consequently, the CSE is re-cast as a two-component Real Schrodinger Equation (RSE) system that couples the real and imaginary parts of the complex wavefunction from the original CSE. By way of the HOA, the resulting real and imaginary parts of the complex wavefunction may be expressed in terms of each other: the ‘upshot’ being the de-coupling of the two-component system into a separate new RSE for the real part and a new separate RSE for the imaginary part, of the original CSE complex wavefunction. It follows that the new de-coupled RSE for the real (respectively imaginary) part of the original complex wavefunction is ‘informationally-equivalent’ to the new de-coupled RSE for the imaginary (respectively real) part of the original complex wavefunction. Moreover, each of these RSEs for the aforementioned real wavefunctions is also ‘informationally-equivalent’ to the original CSE and it’s complex wavefunction. What’s more, consequent of HOA, the real wavefunctions each have exact analytical expression in quadrature (integral form).

The one-dimensional Hulthén potential in the quantum phase space representation

AbstractThe analytic expression of the Wigner function for bound eigenstates of the Hulthén potential in quantum phase space is obtained and presented by plotting this function for a few quantum states. In addition, the correct marginal distributions of the Wigner function in spherical coordinates are determined analytically.

Wigner function for the orientation state

We introduce a quantum phase space representation for the orientation state of extended quantum objects, using the Euler angles and their conjugate momenta as phase space coordinates. It exhibits the same properties as the standard Wigner function and thus provides an intuitive framework for discussing quantum effects and semiclassical approximations in the rotational motion. Examples illustrating the viability of this quasi-probability distribution include the phase space description of a molecular alignment effect.

HOA (Heaviside Operational Ansatz) revisited: recent remarks on novel exact solution methodologies in wavefunction analysis

A viable methodology for the exact analytical solution of the multiparticle Schrodinger and Dirac equations has long been considered a holy grail of theoretical chemistry. Since a benchmark work by Torres-Vega and Frederick in the 1990s, the QPSR (Quantum Phase Space Representation) has been explored as an alternate method for solving various physical systems. Recently, the present author has developed an exact analytical symbolic solution scheme for broad classes of differential equations utilizing the HOA (Heaviside Operational Ansatz). An application of the scheme to chemical systems was initially presented in Journal of Mathematical Chemistry (Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/r(I) radial potential in quantum phase space. Journal of Mathematical Chemistry, 2009; 45(1):129–140). It is believed that the coupling of HOA with QPSR represents not only a fundamental breakthrough in theoretical physical chemistry, but it is promising as a basis for exact solution algorithms that would have tremendous impact on the capabilities of computational chemistry/physics. The novel methods allow the exact determination of the momentum [and configuration] space wavefunction from the QPSR wavefunction by way of a Fourier transform. In this note some remarks, examples and further directions, concerning HOA as a tool to solve and provide analytical insight into solutions of dynamical systems occurring in, but not limited to Mathematical Chemistry, are also posited.

Repulsive Nonlinear Schrödinger Equation and Bose-Einstein Condensate in Phase Space

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of repulsive nonlinear Schrödinger equation are solved, which models the dilute-gas Bose-Einstein condensate. The eigenfunctions in position and momentum spaces can be obtained through the “Fourier-like” projection transformation from the phase-space eigenfunctions. It shows that the wave-mechanics method in the phase-space representation could be extended to the nonlinear Schrödinger equations. The research provides the foundation for the approximate calculation in future.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

Attractive Nonlinear Schrödinger Equation and Bose-Einstein Condensate in Phase Space

In this paper, we solve the rigorous solutions of attractive nonlinear Schrödinger equation which models the Bose-Einstein condensate, within the framework of the quantum phase space representation established by Torres-Vega and Frederick. By means of the “Fourier-like” projection transformation, we obtain the eigenfunctions in position and momentum spaces from the phase space eigenfunctions. As an example, we discuss the eigenfunction with a hypersecant part.

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.

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector |q⟩λ,ν of linear combination of the coordinate and momentum operator, (λQ + νP), where λ, ν are real numbers, and |q⟩λν is complete, then the projector |q⟩λν⟨q| must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.

Moyal phase-space analysis of nonlinear optical Kerr media

Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.

Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise $${\frac{1}{r_{ij}}}$$ radial potential in quantum phase space

This brief note presents the exact analytical solution of an example class of dynamical systems, realized by differential equations. These problems occur in quantum mechanics and their generalizations in theoretical physics, but they particularly find application to chemical systems. Addressed is the exact analytical solution of the radial Schrodinger equation for the nonrelativistic N-particle system with pairwise \({\frac{1}{r_{ij}}}\) radial potential dynamics in the quantum phase space representation (QPSR) (Torres-Vega, Frederick, J. Chem. Phys. 98 3103, 1993). Applying the formal symbolic solution scheme Heaviside Operational Ansatz (heretofore, HOA) put forward in (Electron. J. Theor. Phys. 1 , 10–16, 2004), reduces solution of the radial Schrodinger equation to exact quadratures.

Wave mechanics in quantum phase space: hydrogen atom

Abstract The rigorous solutions of the stationary Schrodinger equation for hydrogen atom are solved with the wave-mechanics method within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The Fourier-like projection transformations of wave function from the phase space to position and momentum spaces are extended to three-dimensional systems. The eigenfunctions in general position and momentum spaces could be obtained through the transformations from eigenfunction in the phase space. *Supported by National Natural Science Foundation of China (Grant No. 20273008) and Excellent Talent Foundation of Beijing (Grant No. 2005ID0502209)

Rigorous Solutions for the One-Dimensional Hartmann Potential in the Quantum Phase-Space Representation

The rigorous solutions of the Schrodinger equation with the one-dimensional Hartmann potential for a particle are solved and discussed within the framework of the quantum phase space representation established by Torres-Vega and Frederick. For a simple example, the uncertainty principle for the quantum probability density functions is revealed in phase space representation.

Relationship between the Wigner function and the probability density function in quantum phase space representation

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

Exact solutions for nonlinear Schr dinger equation in phase space: applications to Bose–Einstein condensate

The stationary-state nonlinear Schrodinger equation, which models the dilute-gas Bose-Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the 'Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.

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.

One-dimensional hydrogen atom in quantum phase-space representation: rigorous solutions

The rigorous solutions of the Schrödinger equations for the one-dimensional hydrogen atom are solved within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The eigenfunctions in position and momentum spaces can be obtained through the `Fourier-like' projection transformation from the phase-space eigenfunctions.

Rigorous solutions of a particle in δ potential fields in phase space

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrodinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.

Rigorous solutions of diatomic molecule oscillator with empirical potential function in phase space

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrödinger equation of the diatomic molecule oscillator with an empirical potential function are solved and discussed, and the Heisenberg uncertainty principle is interpreted in this physical system.