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

Chapman & Hall/CRC eBooks

BOUND STATES OF SIMPLE SYSTEMS

Vishnu S. Mathur, Surendra Singh

Published Dec 12, 2008• 0 citations
Publisher

Abstract

We shall now apply the time-independent Schro¨dinger equation to study the bound states of simple systems. 1. A free particle in a box with sharp boundaries. 2. Particle moving in a one-dimensional harmonic potential well (a simple harmonic oscillator).
DOI
10.1201/b17176-8

Related papers

Exactly Solvable Systems I: Bound States

S. Rajasekar, R. Velusamy

This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.

CRC Press eBooks•Sep 29, 2022•0 citations

PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS

Vishnu S. Mathur, Surendra Singh

Among the simplest applications of Schro¨dinger ’s time-independent equation for stationary states are the problems pertaining to the motion of a particle in the presence of potential steps. In the present chapter we shall consider the following problems: (a) Motion of a particle across a potential step of finite height and infinite extent.

Chapman & Hall/CRC eBooks•Dec 12, 2008•0 citations

Effective Schrodinger equation for one-dimensional systems with rapidly oscillating boundary conditions

N. P. Tretyakov, Павел Евгеньевич Голосов, Anton Kochanov

The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.

International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT)•Nov 1, 2018•1 citation

Energy and Quantum Dynamics

Gennaro Auletta, Shang-Yung Wang

In this chapter we shall deal with quantum dynamics and in particular with the basic equation that governs quantum dynamics, the Schro¨dinger equation. We shall consider two of the most important model systems in quantum mechanics: the free particle and the harmonic oscillator. Moreover, we shall deal with the density matrix formalism and composite systems. Another crucial concept that will be introduced here is that of entanglement, a particular manifestation of quantum features.

•Jun 20, 2014•0 citations

Wave Mechanics and Its Simple Applications

Ajay Sharma

The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.

CRC Press eBooks•Jun 18, 2021•0 citations

Three-dimensional problems

Kenichi Konishi, Giampiero Paffuti

Abstract In this chapter several simple three-dimensional systems are studied: the free particles, the potential well or barrier, the harmonic oscillator, and the hydrogen atom.

•Mar 5, 2009•0 citations
This chapter deals with the study of bound states of some simple one-dimensional systems. Study of bound systems and their bound states is very important because atoms, molecules and other fundamental matters are bound systems and their characteristics are understood by quantum mechanics. The chapter considers a one-dimensional linear harmonic oscillator, an anharmonic oscillator, a particle in a box, Morse oscillator, Poschl–Teller potentials and pendulum system. The time evolution of stationary state eigenfunctions is easy to obtain. The linear harmonic oscillator (LHO) is a fundamental system solved exactly in both classical and quantum mechanics. It is used in the modelling of subnuclear quarks to cosmic systems. The connection between the quantized energy eigenvalue spectrum of a bound state and the classical motions of the corresponding classical system has become increasingly interesting as the ability to experimentally probe the quantum-classical interface has dramatically improved.
The method of the effective Schrodinger equation applied to the one-dimensional problem of a moving particle in a potential box with a potential function and high-frequency time-dependent boundary conditions is investigated. The method of converting the initial equation written for a system with time-dependent boundary conditions to an equation describing a system in a constant region is briefly presented. The resulting system can be considered as a particle in a potential box, located in the field of some effective potential, different from the initial one. Next we consider a particular case of wall motion, with the subsequent construction of the effective Schrodinger equation. Rapid oscillations of the potential box as a whole (i.e., without changing its width) lead to an additional term in the effective equation proportional to the second derivative of the potential. This, in particular, can lead to the formation of potential wells at potential peaks in which bound states of a particle are possible. In the considered case, it was assumed that the oscillation amplitudes are small in comparison with the characteristic dimensions of the system.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.