scholarly journals Quantum Physics in Phase Space: an Analysis of Simple Pendulum

2018 ◽  
Vol 1 (2) ◽  
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Symplectic Structure ◽  
Unitary Representations ◽  
Galilei Group ◽  
Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

Analytical solution for quantum quartic oscillator in phase space

2020 ◽  
Vol 35 (20) ◽  
pp. 2050100
Author(s):  
A. X. Martins ◽  
T. M. R. Filho ◽  
R. G. G. Amorim ◽  
R. A. S. Paiva ◽  
G. Petronilo ◽  
...  
Keyword(s):  
Phase Space ◽  
Analytical Solution ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Schrodinger Equation ◽  
Unitary Representations ◽  
Algebraic Methods ◽  
Quantum Oscillator ◽  
Galilei Group ◽  
Lie Methods

In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.

SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

2013 ◽  
Vol 28 (05n06) ◽  
pp. 1350013 ◽  
Author(s):  
R. G. G. AMORIM ◽  
M. C. B. FERNANDES ◽  
F. C. KHANNA ◽  
A. E. SANTANA ◽  
J. D. M. VIANNA
Keyword(s):  
Quantum Mechanics ◽  
Group Theory ◽  
Phase Space ◽  
Wigner Function ◽  
Physical Theory ◽  
Star Product ◽  
Unitary Representations ◽  
Theory Approach ◽  
Galilei Group ◽  
Symplectic Representations

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

scholarly journals Classical and Quantum Physics in Selected Economic Models

2011 ◽  
Vol 3 (1) ◽  
pp. 7-20 ◽  
Author(s):  
Ewa Drabik
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Financial Markets ◽  
Uncertainty Principle ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Economic Models ◽  
Heisenberg Uncertainty Principle ◽  
Wave Particle Duality ◽  
Heisenberg Uncertainty

Classical and Quantum Physics in Selected Economic ModelsA growing number of economic phenomena are nowadays described with methods known in physics. The most frequently applied physical theories by economists are: (1) the universal gravitation law and (2) the first and second law of thermodynamics. Physical principles can also be applied to the theory of financial markets. Financial markets are composed of individual participants who may be seen to interact as particles in a physical system. This approach proposes a financial market model known as a minority game model in which securities and money are allocated on the basis of price fluctuations, and where selling is best option when the vast majority of investors tend to purchase goods or services, and vice versa. The players who end up being on the minority side win.The above applications of physical methods in economics are deeply rooted in classical physics. However, this paper aims to introduce the basic concepts of quantum mechanics to the process of economic phenomena modelling. Quantum mechanics is a theory describing the behaviour of microscopic objects and is grounded on the principle of wave-particle duality. It is assumed that quantum-scale objects at the same time exhibit both wave-like and particle-like properties. The key role in quantum mechanics is played by: (1) the Schrödinger equation describing the probability amplitude for the particle to be found in a given position and at a given time, and as (2) the Heisenberg uncertainty principle stating that certain pairs of physical properties cannot be economic applications of the Schrödinger equation as well as the Heisenberg uncertainty principle. We also try to describe the English auction by means the quantum mechanics methods.

On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation

2010 ◽  
Vol 76 (3-4) ◽  
pp. 645-653 ◽  
Author(s):  
RENATO FEDELE ◽  
SERGIO DE NICOLA ◽  
DUSAN JOVANOVIĆ ◽  
DAN GRECU ◽  
ANCA VISINESCU
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Von Neumann ◽  
Nonlinear Schrödinger ◽  
Von Neumann Equation

AbstractThe Wigner transformation is used to define the quasidistribution (Wigner function) associated with the wave function of the cylindrical nonlinear Schrödinger equation (CNLSE) in a way similar to that of the standard nonlinear Schrödinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the ‘cylindrical nonlinear von Neumann equation’ (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the ‘cylindrical Wigner function’ and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically and evaluated numerically.

The formulation of quantum mechanics in terms of phase space functions

1964 ◽  
Vol 60 (3) ◽  
pp. 581-586 ◽  
Author(s):  
D. B. Fairlie
Keyword(s):  
Wave Function ◽  
Distribution Function ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Time Dependence ◽  
Schrodinger Equation ◽  
Average Energy ◽  
Weighted Mean

AbstractA relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.

scholarly journals COMMUTATOR ANOMALY IN NONCOMMUTATIVE QUANTUM MECHANICS

2006 ◽  
Vol 21 (39) ◽  
pp. 2971-2976 ◽  
Author(s):  
SAYIPJAMAL DULAT ◽  
KANG LI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
Noncommutative Phase Space ◽  
Noncommutative Quantum Mechanics ◽  
Physical Observable ◽  
Noncommutative Quantum

In this paper, the Schrödinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space–space and space–momentum as well as momentum–momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.

scholarly journals Size-effect at finite temperature in a quark–antiquark effective model in phase space

2021 ◽  
pp. 2150121
Author(s):  
G. X. A. Petronilo ◽  
R. G. G. Amorim ◽  
S. C. Ulhoa ◽  
A. F. Santos ◽  
A. E. Santana ◽  
...  
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Size Effect ◽  
Wigner Function ◽  
Ground State ◽  
Finite Temperature ◽  
Size Effects ◽  
Schrodinger Equation ◽  
Effective Model ◽  
Linear Potential

A quark–antiquark effective model is studied in a toroidal topology at finite temperature. The model is described by a Schrödinger equation with linear potential which is embedded in a torus. The following aspects are analyzed: (i) the nonclassicality structure using the Wigner function formalism; (ii) finite temperature and size-effects are studied by a generalization of Thermofield Dynamics written in phase space; (iii) in order to include the spin of the quark, Pauli-like Schrödinger equation is used; (iv) analysis of the size-effect is considered to observe the fluctuation in the ground state. The size effect goes to zero at zero, finite and high temperatures. The results emphasize that the spin is a central aspect for this quark–antiquark effective model.

Accidental Degeneracy of the Hydrogen Atom and its Non-accidental Solution in Parabolic Coordinates

2021 ◽  
Author(s):  
P.C. Deshmukh ◽  
Aarthi Ganesan ◽  
Sourav Banerjee ◽  
Ankur Mandal
Keyword(s):  
Quantum Mechanics ◽  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Dynamical Symmetry ◽  
Coordinate Systems ◽  
Casimir Operators

The degeneracy associated with dynamical symmetry of a potential can be identified in quantum mechanics, by solving the Schrödinger equation analytically, using the method of separation of variables in at least two different coordinate systems, and in classical mechanics by solving the Hamilton-Jacobi equation. In the present pedagogical article, the notion of separability and superintegrability of a potential, with profound implications is discussed. In an earlier tutorial paper, we had addressed the n<sup>2</sup>-fold degeneracy of the hydrogen atom using the Casimir operators corresponding to the SO(4) symmetry of the 1/r potential. The present paper is a sequel to it, in which we solve the Schrödinger equation for the hydrogen atom using separation of variables in the parabolic coordinate systems. In doing so, we take the opportunity to revisit some excellent classical works on symmetry and degeneracy in classical and quantum physics, if only to draw attention to these insightful studies which unfortunately miss even a mention in most undergraduate and even graduate level courses in quantum mechanics and atomic physics.

scholarly journals General Quantum Theory: I ---- Deducing Quantum Physics and Solving Two Origin Crisises of Wave-Particle Duality and the First Quantization

2021 ◽  
Author(s):  
C. Huang ◽  
Yong-Chang Huang ◽  
Jia-Min Song
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Statistical Mechanics ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Time Derivative ◽  
Square Root ◽  
Classical Statistical Mechanics ◽  
Wave Particle Duality ◽  
Solid Bases

Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality, derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics. Quantum mechanics being just a generalization theory of the complex square root of classical statistical mechanics is both new physics and revolutionary discovery, which are affecting people&rsquo;s deep philosophical thinking for modern physics development, solve all the crisises of quantum mechanics, quantum information and so on, and make quantum mechanics have scientific solid bases being checked and both no basic axiom presumption and no all the quantum strange incomprehensible properties, because classical statistical mechanics and the complex square root of classical statistical mechanics have the scientific solid bases being checked. In addition, this paper discovers the reason no taking the time derivative of space coordinates in Schrȍdinger equation. Therefore, this paper gives solution to the crisis of the first quantization origin, and mainly deduces quantum physics no all the quantum current strange incomprehensible properties.

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