scholarly journals The Magnetic Hooke-Newton Transmutation in Momentum Space

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 608
Author(s):  
De-Hone Lin
Keyword(s):  
Schrödinger Equation ◽  
Effective Potential ◽  
Geometric Structure ◽  
Momentum Space ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Interaction Force ◽  
Hyperbolic Surface ◽  
Linear Interaction ◽  
Angular Momentum Quantum Number

The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

scholarly journals SYMMETRY REDUCTION OF BROWNIAN MOTION AND QUANTUM CALOGERO–MOSER MODELS

2012 ◽  
Vol 13 (01) ◽  
pp. 1250007
Author(s):  
SIMON HOCHGERNER
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Symmetry Reduction ◽  
Reduction Scheme ◽  
Polar Actions ◽  
Stochastic Setting ◽  
Hamilton Jacobi Equation

Let Q be a Riemannian G-manifold. This paper is concerned with the symmetry reduction of Brownian motion in Q and ramifications thereof in a Hamiltonian context. Specializing to the case of polar actions, we discuss various versions of the stochastic Hamilton–Jacobi equation associated to the symmetry reduction of Brownian motion and observe some similarities to the Schrödinger equation of the quantum–free particle reduction as described by Feher and Pusztai [10]. As an application we use this reduction scheme to derive examples of quantum Calogero–Moser systems from a stochastic setting.

scholarly journals Geometrization of the Schrödinger equation: Application of the Maupertuis Principle to quantum mechanics

2014 ◽  
Vol 11 (08) ◽  
pp. 1450066 ◽  
Author(s):  
Antonia Karamatskou ◽  
Hagen Kleinert
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Particle Density ◽  
Free Particle ◽  
Beltrami Operator ◽  
Classical Particle ◽  
Geometric Form ◽  
Curved Space ◽  
Maupertuis Principle ◽  
Semiclassical Expansion

In its geometric form, the Maupertuis Principle states that the movement of a classical particle in an external potential V(x) can be understood as a free movement in a curved space with the metric gμν(x) = 2M[V(x) - E]δμν. We extend this principle to the quantum regime by showing that the wavefunction of the particle is governed by a Schrödinger equation of a free particle moving through curved space. The kinetic operator is the Weyl-invariant Laplace–Beltrami operator. On the basis of this observation, we calculate the semiclassical expansion of the particle density.

THE FERMION-FERMION EFFECTIVE ACTION FOR THE THREE-DIMENSIONAL MASSIVE THIRRING MODEL

1994 ◽  
Vol 09 (18) ◽  
pp. 3143-3151 ◽  
Author(s):  
R.F. RIBEIRO ◽  
E.R. BEZERRA DE MELLO
Keyword(s):  
Schrödinger Equation ◽  
Effective Potential ◽  
Effective Action ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
Bound State ◽  
Thirring Model ◽  
Coupling Constant ◽  
Massive Thirring Model

In this paper a nonrelativistic fermion-fermion effective potential for a three-dimensional massive Thirring model is obtained in a 1/N expansion. We show, by analyzing the Schrödinger equation in the presence of this potential, that the system presents a fermion-fermion bound state for a positive value of the coupling constant g.

scholarly journals Dirac Equation in the Field of a Charged Cosmic String and in the Field of a Point Charge with Z > 137

1991 ◽  
Vol 44 (6) ◽  
pp. 585
Author(s):  
TJ Allen ◽  
LJ Tassie
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Internal Structure ◽  
Effective Potential ◽  
Cosmic String ◽  
Point Charge ◽  
Schrodinger Equation ◽  
Cylindrical Coordinates ◽  
Line Charge ◽  
Radial Dirac Equation

In both spherical and cylindrical coordinates, the radial Dirac equation can be written in the form of a Schrodinger equation with an effective potential. It is shown that the difficulties at r -+ 0 for the Dirac equation in the field of a point charge for Z > 137 are the same as those for the Schrodinger equation with a l/r2 potential. The effective potential is used to show that similar difficulties do not arise for the field of a line charge, so allowing the consideration of the motion of electrons in the field of a charged superconducting cosmic string without considering the internal structure of the string.

scholarly journals On a Conjecture regarding Fisher Information

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Angelo Plastino ◽  
Guido Bellomo ◽  
Angel Ricardo Plastino
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Free Particle ◽  
Quantum Probability ◽  
Time Dependent ◽  
Theoretical Physics ◽  
One Dimension ◽  
Pure States ◽  
Time Dependent Solution

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.

New Bound States for Modified Vibrational-Rotational Structure of Supersingular Plus Coulomb Potential of the Schrödinger Equation in One-Electron Atoms

2017 ◽  
Vol 73 ◽  
pp. 31-45 ◽  
Author(s):  
Abdelmadjid Maireche
Keyword(s):  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Bound States ◽  
Star Product ◽  
Three Dimensional ◽  
Real Space ◽  
Rotational Structure ◽  
Angular Momentum Quantum Number ◽  
Quantum Symmetries

In this study, three-dimensional modified time-independent Schrödinger equation of modified vibrational-rotational structure of supersingular plus Coulomb (v.r.s.c) potential was solved using Boopp’s shift method instead to apply star product, in the framework of both noncommutativity three dimensional real space and phase (NC: 3D-RSP). We have obtained the explicit energy eigenvalues for ground state and first excited state for interactions in one-electron atoms. Furthermore, the obtained corrections of energies are depended on infinitesimal parameters (Θ,χ) and (θ,σ) which are induced by position-position and momentum-momentum noncommutativity, respectively, in addition to the discreet atomic quantum numbers: j=l±1/2,s=±1/2,l and the angular momentum quantum number m. We have also shown that, the usual states in ordinary three dimensional spaces for ordinary vibrational-rotational structure of supersingular plus Coulomb potential are canceled and has been replaced by new degenerated 2(2l+1) sub-states in the extended new quantum symmetries of (NC: 3D-RSP).

Sign in / Sign up

Export Citation Format

Share Document