Free-Particle Schrödinger Equation: Wave Packets

2017 ◽  
pp. 53-67
Author(s):  
Paul R. Berman
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Packets ◽  
Free Particle

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.

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→∞.

Integro-differential Schrödinger equation and description of unstable particle

2014 ◽  
Vol 28 (30) ◽  
pp. 1450234
Author(s):  
Kwok Sau Fa
Keyword(s):  
Schrödinger Equation ◽  
Diffusion Equation ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Radioactive Decay ◽  
Total Probability ◽  
Fractional Schrödinger Equations ◽  
The Times ◽  
And Diffusion ◽  
Ordinary Quantum

The description of a particle in the quantum system is probabilistic. In the ordinary quantum mechanics the total probability of finding the particle is conserved, i.e. the probability is normalized for all the times. To find a non-constant total probability an imaginary term should be added to the potential energy which is not physical. Recently, generalizations of the ordinary Schrödinger equation have been proposed by using the Feynman path integral and analogy between the Schrödinger equation and diffusion equation. In this work, an integro-differential Schrödinger equation is proposed by using analogy between the Schrödinger equation and diffusion equation. The equation is obtained from the continuous time random walk model with diverging jump length variance and generic waiting time probability density. The equation generalizes the ordinary and fractional Schrödinger equations. One can show that the integro-differential Schrödinger equation can describe a non-constant total probability for a free particle, and it includes the exponential decay which is fundamental for the description of radioactive decay.

The Schrödinger equation: total interaction as perturbation

1982 ◽  
Vol 380 (1779) ◽  
pp. 489-494 ◽  
Keyword(s):  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Eigenvalue Problems ◽  
Free Particle ◽  
Analytic Approximation ◽  
Binding Interaction ◽  
General Applicability ◽  
Total Interaction

The quantization of the energy spectrum of a free particle in the presence of a binding interaction is reconsidered here. These considerations form the basis of a simple analytic approximation of general applicability to eigenvalue problems.

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