scholarly journals Chaotic dynamics of a free particle interacting linearly with a harmonic oscillator

2005 ◽  
Vol 208 (1-2) ◽  
pp. 96-114 ◽  
Author(s):  
Stephan De Bièvre ◽  
Paul E. Parris ◽  
Alex Silvius
Keyword(s):  
Harmonic Oscillator ◽  
Chaotic Dynamics ◽  
Free Particle

A note on summation over Feynman histories

1958 ◽  
Vol 54 (2) ◽  
pp. 302-304 ◽  
Author(s):  
C. W. Kilmister
Keyword(s):  
Harmonic Oscillator ◽  
Free Particle

ABSTRACTIt is pointed out that the free particle, the harmonic oscillator and certain other systems are such that the Feynman summation over histories is independent of the class of histories summed over. These examples are therefore useless as tests of methods of defining and carrying out the summation.

scholarly journals SOME ASPECTS OF QUANTUM MECHANICS AND FIELD THEORY IN A LORENTZ INVARIANT NONCOMMUTATIVE SPACE

2013 ◽  
Vol 28 (07) ◽  
pp. 1350017 ◽  
Author(s):  
EVERTON M. C. ABREU ◽  
M. J. NEVES
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Spectral Representation ◽  
Free Particle ◽  
Transition Amplitude ◽  
Noncommutative Space ◽  
Moyal Product ◽  
Massive Scalar Field ◽  
Scalar Action

We obtained the Feynman propagators for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher–Fredenhagen–Roberts–Amorim (DFRA) NC background that can be considered as an alternative framework for the NC space–time of the early universe. The operators' formalism was revisited and we applied its properties to obtain an NC transition amplitude representation. Two examples of DFRA's systems were discussed, namely, the NC free particle and NC harmonic oscillator. The spectral representation of the propagator gave us the NC wave function and energy spectrum. We calculated the partition function of the NC harmonic oscillator and the distribution function. Besides, the extension to NC DFRA quantum field theory is straightforward and we used it in a massive scalar field. We had written the scalar action with self-interaction ϕ4 using the Weyl–Moyal product to obtain the propagator and vertex of this model needed to perturbation theory. It is important to emphasize from the outset, that the formalism demonstrated here will not be constructed by introducing an NC parameter in the system, as usual. It will be generated naturally from an already existing NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.

scholarly journals Harmonic oscillator Brownian motion: Langevin approach revisited

2021 ◽  
Vol 18 (1) ◽  
pp. 97
Author(s):  
O. Contreras-Vergara ◽  
N. Lucero-Azuara ◽  
N. Sánchez-Salas ◽  
J. I. Jiménez-Aquino
Keyword(s):  
Harmonic Oscillator ◽  
Free Particle ◽  
Gaussian White Noise ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Brownian Movement ◽  
Deterministic Equation ◽  
Langevin Approach ◽  
The Mean ◽  
Movement Problem

The original strategy applied by Langevin to Brownian movement problem is used to solve the case of a free particle under a harmonic potential. Such straightforward strategy consists in separating the noise termin the Langevin equation in order to solve a deterministic equation associated with the Mean Square Displacement (MSD). In this work, to achieve our goal we first calculate the variance for the stochastic harmonic oscillator and then the MSD appears immediately. We study the problem in the damped and lightly damped cases and show that, for times greater than the relaxation time, Langevin's original strategy is quite consistent with the exact theoretical solutions reported by Chandrasekhar and Lemons, these latter obtained using the statistical properties of a Gaussian white noise. Our results for the MSDs are compared  with the exact theoretical solutions as well as with the numerical simulation.

scholarly journals Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

2018 ◽  
Vol 33 (16) ◽  
pp. 1850091 ◽  
Author(s):  
Kh. P. Gnatenko ◽  
O. V. Shyiko
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Rotational Symmetry ◽  
Free Particle ◽  
Second Order ◽  
Noncommutative Phase Space ◽  
Noncommutative Algebra ◽  
Ordinary Space ◽  
Canonical Type ◽  
Rotationally Invariant

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to the well-known noncommutative algebra of canonical type. In the noncommutative phase space, rotational symmetry influence of noncommutativity on the spectrum of free particle and the spectrum of harmonic oscillator is studied up to the second-order in the parameters of noncommutativity. We find that because of momentum noncommutativity, the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.

Some quantum mechanical solutions in nonrelativistic anti-Snyder framework

2017 ◽  
Vol 32 (27) ◽  
pp. 1750170 ◽  
Author(s):  
Homa Shababi ◽  
Won Sang Chung
Keyword(s):  
Neutron Star ◽  
Harmonic Oscillator ◽  
Free Particle ◽  
Three Dimensions ◽  
One Dimension ◽  
Quantum Mechanical ◽  
Momentum Representation ◽  
Eigenvalues And Eigenfunctions

In this paper, we investigate nonrelativistic anti-Snyder model in momentum representation and obtain quantum mechanical eigenvalues and eigenfunctions. Using this framework, first, in one dimension, we study a particle in a box and the harmonic oscillator problems. Then, for more investigations, in three dimensions, the quantum mechanical eigenvalues and eigenfunctions of a free particle problem and the radius of the neutron star are obtained.

Summation over Feynman histories: the free particle and the harmonic oscillator

1957 ◽  
Vol 53 (1) ◽  
pp. 199-205 ◽  
Author(s):  
H. Davies
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Free Particle ◽  
Phase Factor ◽  
Arbitrary Phase

ABSTRACTUsing a particular parametrization of paths, the free particle and the harmonic oscillator are treated by the Feynman method of summation over histories of the system. The propagators obtained are, apart from an arbitrary phase factor, those of conventional quantum mechanics.

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