Point Canonical Transformation Research Articles

Exact solutions of the Schrödinger equation with position-dependent mass for some Hermitian and non-Hermitian potentials

We study the exact solutions of the Schrödinger equation with position-dependent mass by using the method of point canonical transformations. Given a position-dependent mass distribution, we construct the exactly solvable target potentials by choosing the Rosen–Morse-type and Scarf-type potentials as the reference potentials. The energy spectra of the bound states and corresponding wavefunctions for the target potentials are given explicitly.

Mapping of the position-dependent mass Schrödinger equation under point canonical transformation

In this paper, the three-dimensional radial position-dependent mass Schrodinger equation is exactly solved through mapping this wave equation into the constant mass Schrodinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.

Quantized normal matrices: some exact results and collective field formulation

We formulate and study a class of U ( N ) -invariant quantum mechanical models of large normal matrices with arbitrary rotation-invariant matrix potentials. We concentrate on the U ( N ) singlet sector of these models. In the particular case of quadratic matrix potential, the singlet sector can be mapped by a similarity transformation onto the two-dimensional Calogero–Marchioro–Sutherland model at specific couplings. For this quadratic case we were able to solve the N-body Schrödinger equation and obtain infinite sets of singlet eigenstates of the matrix model with given total angular momentum. Our main object in this paper is to study the singlet sector in the collective field formalism, in the large- N limit. We obtain in this framework the ground state eigenvalue distribution and ground state energy for an arbitrary potential, and outline briefly the way to compute bona-fide quantum phase transitions in this class of models. As explicit examples, we analyze the models with quadratic and quartic potentials. In the quartic case, we also touch upon the disk-annulus quantum phase transition. In order to make our presentation self-contained, we also discuss, in a manner which is somewhat complementary to standard expositions, the theory of point canonical transformations in quantum mechanics for systems whose configuration space is endowed with non-Euclidean metric, which is the basis for constructing the collective field theory.

Mapping of the five-parameter exponential-type potential model into trigonometric-type potentials

We choose the five-parameter exponential-type potential model as input, and construct five trigonometric-type potentials via point canonical transformations. Their energy spectra and wavefunctions are obtained in a unified manner by using the expressions for the energy spectra and wavefunctions of the five-parameter exponential-type potential model.

The extended Thomas–Fermi kinetic energy density functional with position-dependent effective mass in one dimension

The point canonical transformations map the Schrodinger equation with constant mass to a wave equation with a position-dependent effective mass. Using such a technique we derive, for a one-dimensional inhomogeneous system of noninteracting fermions with density ρ(x) and spatially dependent effective mass distribution m(x), the semiclassical kinetic energy density functional τ(ρ) in the so-called extended Thomas–Fermi model up to order 2. For a given position-dependent mass, we compare numerically the total semiclassical kinetic energy with its exact quantum mechanical counterpart. The qualitative agreement is excellent.

Exact solutions of the position-dependent mass Schrödinger equation in D dimensions

In this Letter, we employ the point canonical transformation to solve the D-dimensional position-dependent effective mass Schrödinger equation with physical potentials. By mapping this wave equation into the well-known exactly solvable D-dimensional Schrödinger equation with constant mass, for a given spatial dependent mass distribution, the exact bound state solutions including the energy spectrum and corresponding wave functions are derived. As examples, the cases of the harmonic oscillator and Coulomb potential are considered.

Point Canonical Transformation for Solving Five-Parameter Exponential-Type Potential**The project supported by National Natural Science Foundation of China under Grant No. 49905001, the Sichuan Youth Science and Technology Foundation, and the Scientific Research Foundation of Chengdu University of Information Technology (CRF200211)

Using the approach of mapping of shape invariant potentials under point canonical transformations, the energy spectra and wave functions are most easily determined for the bound states of the five-parameter exponential-type potential with a little extra effort.

Nonrelativistic Green's Function for Systems with Position-Dependent Mass

Given a spatially dependent mass we obtain the two-point Green's function for exactly solvable nonrelativistic problems. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation. The one-dimensional oscillator class is considered and examples are given for several mass distributions.

EXACT SOLUTIONS OF DIRAC AND SCHRÖDINGER EQUATIONS FOR A LARGE CLASS OF POWER-LAW POTENTIALS AT ZERO ENERGY

We obtain exact solutions of Dirac equation for radial power-law relativistic potentials at rest mass energies. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrödinger equation at zero energy carrying representations of SO(2,1) Lie algebra. The latter are obtained by point canonical transformations of the exactly solvable problem of the three dimensional oscillator. The wave function solutions are written in terms of the confluent hypergeometric functions and almost always square integrable. For most cases these solutions support bound states at zero energy. Some exceptional unbounded states are normalizable for nonzero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.

Solutions of the nonrelativistic wave equation with position-dependent effective mass

Given a spatially dependent mass distribution we obtain potential functions for exactly solvable nonrelativistic problems. The energy spectrum of the bound states and their wavefunctions are written down explicitly. This is accomplished by mapping the wave equation for these systems into well-known exactly solvable Schrodinger equations with constant mass using point canonical transformation. The Oscillator, Coulomb, and Morse class of potentials are considered.

MAPPING OF NON-CENTRAL POTENTIALS UNDER POINT CANONICAL TRANSFORMATIONS

Motivated by the observation that all known exactly solvable shape invariant central potentials are inter-related via point canonical transformations, we develop an algebraic framework to show that a similar mapping procedure also exists between a class of non-central potentials. As an illustrative example, we discuss the inter-relation between the generalized Coulomb and oscillator systems.

Erratum: Graded extension of so(2,1) Lie algebra and the search for exact solutions of the Dirac equation by point canonical transformations [Phys. Rev. A65, 042109 (2002)

Erratum: Graded extension of so(2,1) Lie algebra and the search for exact solutions of the Dirac equation by point canonical transformations [Phys. Rev. A<b>65</b>, 042109 (2002)

Graded extension of so(2,1) Lie algebra and the search for exact solutions of the Dirac equation by point canonical transformations

SO(2,1) is the symmetry algebra for a class of three-parameter problems that includes the oscillator, Coulomb and Morse potentials as well as other problems at zero energy. All of the potentials in this class can be mapped into the oscillator potential by point canonical transformations. We call this class the "oscillator class". A nontrivial graded extension of SO(2,1) is defined and its realization by two-dimensional matrices of differential operators acting in spinor space is given. It turns out that this graded algebra is the supersymmetry algebra for a class of relativistic potentials that includes the Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in fact, the relativistic extension of the oscillator class. A new point canonical transformation, which is compatible with the relativistic problem, is formulated. It maps all of these relativistic potentials into the Dirac-Oscillator potential.

Supersymmetry and shape invariance of theeffective screened potential

This paper discusses the supersymmetry and shape invariance (SI) of theeffective screened potential. It is shownthat the effective screened potential (ESP) has SI and belongs to thefirst class of shape-invariant potentials, thence the energylevels of this potential are obtained. Furthermore, by using themethod of point canonical transformation we find that the ESPbelongs to the same subclass as Pöschl-Teller potential 1; the bound-state spectra and the eigenfunctions of this potentialare obtained. The results obtained can readily be applied to thespecial case of the ESP, the famousHulthén potential.

Comprehensive analysis of conditionally exactly solvable models

We study a quantum mechanical potential introduced previously as a conditionally exactly solvable (CES) model. Besides an analysis following its original introduction in terms of the point canonical transformation, we also present an alternative supersymmetric construction of it. We demonstrate that from the three roots of the implicit cubic equation defining the bound-state energy eigenvalues, there is always only one that leads to a meaningful physical state. Finally we demonstrate that the present CES interaction is, in fact, an exactly solvable Natanzon-class potential.

Polynomial approach and point canonical transformations in the constructing of quasi-exactly solvable quantum mechanical potentials

Polynomial approach and point canonical transformations in the constructing of quasi-exactly solvable quantum mechanical potentials

On Variational Principles for Gravitating Perfect Fluids

The connection is established between two different action principles for perfect fluids in the context of general relativity. For one of these actions,S, the fluid four–velocity is expressed as a sum of products of scalar fields and their gradients (the velocity–potential representation). For the other action,S, the fluid four–velocity is proportional to the totally antisymmetric product of gradients of the fluid Lagrangian coordinates. The relationship betweenSandSis established by expressingSin Hamiltonian form and identifying certain canonical coordinates as ignorable. Elimination of these coordinates and their conjugates yields the actionS. The key step in the analysis is a point canonical transformation in which all tensor fields on space are expressed in terms of the Lagrangian coordinate system supplied by the fluid. The canonical transformation is of interest in its own right. It can be applied to any physical system that includes a material medium described by Lagrangian coordinates. The result is a Hamiltonian description of the system in which the momentum constraint is trivial.

Methods for generating quasi-exactly solvable potentials

We describe three different methods for generating quasi-exactly solvable potentials, for which a finite number of eigenstates are analytically known. The three methods are respectively based on (i) a polynomial ansatz for wave functions; (ii) point canonical transformations; (iii) supersymmetric quantum mechanics. The methods are rather general and give considerably richer results than those available in the current literature.

Quasiexactly solvable problems in the path-integral formalism.

Quasiexactly solvable problems are treated via the path-integral approach. The generalized Duru-Kleinert formalism involving a combination of a point-canonical transformation and a new-time transformation is applied to power-law and Morse-like quasiexactly solvable problems, and it is shown that these can be grouped into two families. A truncated kernel hypothesis is advanced, whereby the exactly solvable parts of the kernels of each problem within a given family can be transformed into one another. As an application, the ground-state energy eigenvalue and wave function of the N=0 generalized harmonic oscillator are derived from those of the N=0 generalized Coulomb problem.

BRST symmetry of field redefinitions

Field redefinitions are considered as a special case of BRST gauge fixings of more general field-enlarging transformations. At the two-loop level an extra local term in the action, the Gervais-Jevicki potential, is generated by point canonical transformations in the bosonic part of the theory. We give an example of how this additional term can be cancelled by the inclusion of the ghost fields. This is achieved through an anomalous Jacobian factor from the ghost measure.