Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematics

Study on the Phenomena of Potential Well of Infinite Depth on the View Point of Natural Statistical Physics

Yoshifumi Ito

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

In this paper, we study the phenomena of potential well of infinite depth on the view point of natural statistical physics. The mathematical model of this physical system is the system of micro-particles moving periodically with constant velocity in the interval D = [−a, a], (a > 0). Thereby we obtain the structure of this physical system at the stationary state. Thus we clarify that this physical system is the composed state of the proper physical systems at the stationary state and the ratio of their composition is given by the sequence {|an|2}n=0 ∪ {|bn|2}∞n=1, where an, (n ≥ 0) and bn, (n ≥ 1) are the Fourier type coefficients of the initial state ψ ∈ L2. Then we obtain the energy expectation of the total physical system Ē =∞Σn=0 n2π2ℏ2/2ma2 (|an|2 + |bn|2), where m denotes the mass of one micro-particle.

Related papers

Study on the Phenomena of the System of Free Particles on the View Point of Natural Statistical Physics

Yoshifumi Ito

In this paper, we study the phenomena of the system of free particles on the view point of natural statistical physics. Thereby we derive the Schrödinger equation of this physical system and solve the initial value problem of this Schrödinger equation. Thereby we obtain the structure of this physical system at the stationary state. Thus we obtain the information of the natural statistical distribution of the position variable and the momentum variable of this physical system.

Journal of Mathematics•Jan 1, 2015•0 citations

Study on the Phenomena of Potential Well in the View Point of Natural Statistical Physics

Yoshifumi Ito

In this paper, we study on the phenomena of potential well in the view point of natural statistical physics. We derive the Schrodinger equation of potential well by virtue of my new theory of natural statistical physics. This derivation is very new and natural and reasonable in the physical sense. We solve the eigenvalue problem for the Schrodinger equation of potential well. Thereby we obtain the new and true eigenfunctions of the Schrodinger operator of the potential well. By virtue of my new theory, we can understand the phenomena of potential well naturally and reasonably in the physical sense. At last, we study the meaning of the impact force naturally and reasonably in the physical sense. As for these results, we refer to Ito [1] [4]. Especially, as for the phenomena of potential well, we refer to Ito [2], [4]. 2010 Mathematics Subject Classi cation. Primary 81Q05 ; Secondary 81Q99, 81V45, 82D35.

•Sep 1, 2012•0 citations

Further Discussions on One Dimensional Infinitely Deep Potential Well Model

Ce-Xiang Mei

One-dimensional infinitely deep potential well(ODIDPW) model is one kind of the fundamental problems in quantum mechanics.This paper firstly uses the coordinate-momentum uncertainty relation and the operator theory respectively to prove why the energy of particles in ODIDPW is always greater than zero,and then makes a further discussions on stationary state wavefunction,the quantization of physical quantity,energy level interval,zero-point energy,the coordinate-momentum uncertainty relation and the applicable condition of the ODIDPW model.

Journal of Xianyang Normal University•Jan 1, 2010•0 citations

Existence Conditions of Particle's Bound State in One-dimensional Unfinite Deep Potential Well

Le Wang

For a moving particle in an semi- infinite potential well of one dimension,a continuous and monotonous function is introduced by solving stationary Schr? dinger equation,then Existence conditions of one even more Particle's bound state are analyzed by means of the zero- point theorem of continuous function. The results show the number of bound state is connected with the width and depth of one – dimensional Infinite potential well. For a certain quality of particle,more bound states need much deeper and wider potential well.

Journal of Heze University•Jan 1, 2014•2 citations

Time evolutions of quantum mechanical states in a symmetric double-well potential

Peter Senn

The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.

American Journal of Physics•Mar 1, 1992•6 citations

Energetic Spectrum of a Particle in Three-dimensional Infinite Potential Square Well in Point of View of Number Theory and Bayesian Statistics

Pavel Jahoda, Jan Kracík, David Ulčák

Using results of number theory we develop an approximate statistical model of energy levels of particles in a three-dimensional infinite potential well depending on whether there is exactly one particle or more than one particles in the well. The model is used to perform a statistical inference about the number of particles in the well. The estimation procedure is developed within the Bayesian framework.

Advances in Electrical and Electronic Engineering•Jul 20, 2017•0 citations
In this paper, we study the phenomena of potential well of infinite depth on the view point of natural statistical physics. The mathematical model of this physical system is the system of micro-particles moving periodically with constant velocity in the interval D = [−a, a], (a > 0). Thereby we obtain the structure of this physical system at the stationary state. Thus we clarify that this physical system is the composed state of the proper physical systems at the stationary state and the ratio of their composition is given by the sequence {|an|2}n=0 ∪ {|bn|2}∞n=1, where an, (n ≥ 0) and bn, (n ≥ 1) are the Fourier type coefficients of the initial state ψ ∈ L2. Then we obtain the energy expectation of the total physical system Ē =∞Σn=0 n2π2ℏ2/2ma2 (|an|2 + |bn|2), where m denotes the mass of one micro-particle.
In this paper, we study on the phenomena of potential well in the view point of natural statistical physics. We derive the Schrodinger equation of potential well by virtue of my new theory of natural statistical physics. This derivation is very new and natural and reasonable in the physical sense. We solve the eigenvalue problem for the Schrodinger equation of potential well. Thereby we obtain the new and true eigenfunctions of the Schrodinger operator of the potential well. By virtue of my new theory, we can understand the phenomena of potential well naturally and reasonably in the physical sense. At last, we study the meaning of the impact force naturally and reasonably in the physical sense. As for these results, we refer to Ito [1] [4]. Especially, as for the phenomena of potential well, we refer to Ito [2], [4]. 2010 Mathematics Subject Classi cation. Primary 81Q05 ; Secondary 81Q99, 81V45, 82D35.
One-dimensional infinitely deep potential well(ODIDPW) model is one kind of the fundamental problems in quantum mechanics.This paper firstly uses the coordinate-momentum uncertainty relation and the operator theory respectively to prove why the energy of particles in ODIDPW is always greater than zero,and then makes a further discussions on stationary state wavefunction,the quantization of physical quantity,energy level interval,zero-point energy,the coordinate-momentum uncertainty relation and the applicable condition of the ODIDPW model.
For a moving particle in an semi- infinite potential well of one dimension,a continuous and monotonous function is introduced by solving stationary Schr? dinger equation,then Existence conditions of one even more Particle's bound state are analyzed by means of the zero- point theorem of continuous function. The results show the number of bound state is connected with the width and depth of one – dimensional Infinite potential well. For a certain quality of particle,more bound states need much deeper and wider potential well.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.