Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators

2002 ◽  
Vol 43 (11) ◽  
pp. 5223-5253 ◽  
Author(s):  
S. Benenti ◽  
C. Chanu ◽  
G. Rastelli
Keyword(s):  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
First Integrals ◽  
Symmetry Operators ◽  
Hamilton Jacobi Equation

Coherent states, Schrödinger equation, and the Hamilton–Jacobi equation

1995 ◽  
Vol 73 (7-8) ◽  
pp. 478-483
Author(s):  
Rachad M. Shoucri
Keyword(s):  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Hidden Variables ◽  
Classical Equation ◽  
Equation Of Motion ◽  
Phase Variable ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

The self-adjoint form of the classical equation of motion of the harmonic oscillator is used to derive a Hamiltonian-like equation and the Schrödinger equation in quantum mechanics. A phase variable ϕ(t) instead of time t is used as an independent variable. It is shown that the Hamilton–Jacobi solution in this case is identical with the solution obtained from the Schrödinger equation without the need to introduce the idea of hidden variables or quantum potential.

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 The Equivalence Postulate of Quantum Mechanics, Dark Energy, and the Intrinsic Curvature of Elementary Particles

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Alon E. Faraggi
Keyword(s):  
Quantum Mechanics ◽  
Dark Energy ◽  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
Axiomatic Approach ◽  
Quantum Potential ◽  
Curvature Term ◽  
Cocycle Condition ◽  
Hamilton Jacobi Equation

The equivalence postulate of quantum mechanics offers an axiomatic approach to quantum field theories and quantum gravity. The equivalence hypothesis can be viewed as adaptation of the classical Hamilton-Jacobi formalism to quantum mechanics. The construction reveals two key identities that underlie the formalism in Euclidean or Minkowski spaces. The first is a cocycle condition, which is invariant underD-dimensional Möbius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of theD-dimensional quantum Hamilton-Jacobi equation. In this approach, the solutions of the associated Schrödinger equation are used to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property of the construction is that the two solutions of the corresponding Schrödinger equation must be retained. The quantum potential, which arises in the formalism, can be interpreted as a curvature term. The author proposes that the quantum potential, which is always nontrivial and is an intrinsic energy term characterising a particle, can be interpreted as dark energy. Numerical estimates of its magnitude show that it is extremely suppressed. In the multiparticle case the quantum potential, as well as the mass, is cumulative.

scholarly journals On the Asymptotics of the Schrödinger Equation Solutions and the Euler Model of an Ideal Fluid

2021 ◽  
Author(s):  
Владислав Хаблов
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Ideal Fluid ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
Continuity Equation ◽  
Rotation Group ◽  
Potential Flows ◽  
Euler Model ◽  
Hamilton Jacobi Equation

In this paper we analyze the asymptotics of the Schrödinger equation solutions with respect to a small parameter ~. It is well known, that short- waveasymptoticstosolutionsofthisequationleadstothepairofequations— the Hamilton–Jacobi equation for the phase and the continuity equation. These equations coincide with the ones for the potential flows of an ideal fluid. The wave function is invariant with respect to the complex plane rotations group, and the asymptotics is constructed as a point-dependent action of this group on some function that is found by solving the transfer equation. It is shown in the paper, that if the Heisenberg group is used instead of the rotation group, then the limit of the equations solutions with ~ tending to zero leads to the equations for vortex flows of an ideal fluid in a potential field of forces. If the original Schrödinger equation is nonlinear, then equations for barotropic processes in an ideal fluid are obtained.

A NEW METHODOLOGY FOR THE CONSTRUCTION OF OPTIMIZED RUNGE–KUTTA–NYSTRÖM METHODS

2011 ◽  
Vol 22 (06) ◽  
pp. 623-634 ◽  
Author(s):  
D. F. PAPADOPOULOS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
First Integrals ◽  
New Method ◽  
Phase Lag ◽  
Nyström Methods ◽  
Algebraic Order ◽  
Runge Kutta ◽  
First Time ◽  
Zero Phase

In this paper, a new Runge–Kutta–Nyström method of fourth algebraic order is developed. The new method has zero phase-lag, zero amplification error and zero first integrals of the previous properties. Numerical results indicate that the new method is very efficient for solving numerically the Schrödinger equation. We note that for the first time in the literature we use the requirement of vanishing the first integrals of phase-lag and amplification error in the construction of efficient methods for the numerical solution of the Schrödinger equation.

scholarly journals QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD Uz(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION

1998 ◽  
Vol 13 (16) ◽  
pp. 1241-1252 ◽  
Author(s):  
ANGEL BALLESTEROS ◽  
FRANCISCO J. HERRANZ ◽  
PREETI PARASHAR
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Discretization ◽  
Quantum Deformation ◽  
Standard Quantum ◽  
R Matrix ◽  
Two Photon ◽  
Order Deformation ◽  
Symmetry Operators ◽  
Boson Representation

The non-standard quantum deformation of the (trivially) extended sl (2,ℝ) algebra is used to construct a new quantum deformation of the two-photon algebra h6 and its associated quantum universal R-matrix. A deformed one-boson representation for this algebra is deduced and applied to construct a first-order deformation of the differential equation that generates the two-photon algebra eigenstates in quantum optics. On the other hand, the isomorphism between h6 and the (1+1) Schrödinger algebra leads to a new quantum deformation for the latter for which a differential-difference realization is presented. From it, a time discretization of the heat-Schrödinger equation is obtained and the quantum Schrödinger generators are shown to be symmetry operators.

scholarly journals Emmy Noether and Linear Evolution Equations

10.14311/1811 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
P. G. L. Leach
Keyword(s):  
Variational Principle ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
First Integrals ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Linear Evolution ◽  
Action Integral ◽  
Linear Evolution Equations

Noether’s Theorem relates the Action Integral of a Lagrangian with symmetries which leave it invariant and the first integrals consequent upon the variational principle and the existence of the symmetries. These each have an equivalent in the Schrödinger Equation corresponding to the Lagrangian and by extension to linear evolution equations in general. The implications of these connections are investigated.

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