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Journal of Anhui Science and Technology University

One - dimensional Infinite Potential Well of Particles Moving Wigner Function

Yanhui Li

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.

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As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.
The bound states of a particle confined in a one-dimensional finite square well cannot be solved analytically, since the eigen-energies are determined by transcendental equations. Here, we numerically calculate the bound states and show their non-classical properties, using Wigner's quasi-probability distribution (also called the Wigner functions) in the phase space (x, p). In contrast to the infinite-well case, we find that the Wigner functions spread over the space dimension x, get squeezed along the momentum dimension p, and show negativity outside the well. Negativity in a Wigner function indicates non-classical properties of the bound states.
We study the Wigner function for a quantum system with a discrete, infinite-dimensional Hilbert space, such as a spinless particle moving on a one-dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase-space construction, propose a meaningful definition of the Wigner function in this case and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system, which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states and to their superpositions.
As a quasi-probability distribution function in phase-space and a special representation of the density matrix, Wigner function has great significance in physics. In this paper, first, Wigner function’s characteristic and calculation approach are introduced. Then, with Star-eigen equation we obtain the Wigner function for three-dimensional harmonic oscillator. In the end, we discuss the method describing the interaction between phonons and electrons (or photons) and obtain the equation that transition probability satisfies in phase space.
In this article,the qualities of Wigner function and the corresponding stationary perturbation theory are introduced and applied to one-dimensional infinite potential well and one-dimensional harmonic oscillator,and then the particular Wigner function of one-dimensional infinite potential well is specified and a special constriction effect in its pure state Wigner function is discovered,to which,simultaneously,a detailed and reasonable explanation is elaborated from the perspective of uncertainty principle.Ultimately,the amendment of Wigner function and energy of one-dimensional infinite potential well and one-dimensional harmonic oscillator under perturbation are calculated according to stationary phase space perturbation theory.
In this article,via numerical method,we have shown the wave function diagrams corresponding to the different analytical solutions of the infinitely deep square potential,and the reason which causes the different phase of the analytic solution has been discussed.The probability density distribution curve is given,through which we found that the probability density distribution of the analytical solutions are identical even though the phase of them are different.This conclusion reflects in one side that a phase factor difference of the wave functions can exist in quantum mechanics.