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

arXiv:0911.2144

Quantum Mechanics in the general quantum systems (VI): exact series solution of stationary Schrödinger's equation

An Min Wang

Published Nov 11, 2009• 0 citations
arXivHTMLPDF

Abstract

We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.

DOI
10.48550/arxiv.0911.2144

Related papers

Quantum mechanics in general quantum systems (I): Exact solution

An Min Wang

Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a $c$-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.

arXiv:quant-ph/0611216•Nov 21, 2006•0 citations

Exact solution and perturbation theory in a general quantum system

An Min Wang

By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schrödinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.

arXiv:quant-ph/0602055•Feb 5, 2006•0 citations

Quantum mechanics in general quantum systems (III): open system dynamics

An Min Wang

We investigate the exact solution, perturbation theory and master equation of open system dynamics based on our serial studies on quantum mechanics in general quantum systems [An Min Wang, quant-ph/0611216 and quant-ph/0611217]. In a system-environment separated representation, a general and explicit solution of open system dynamics is obtained, and it is an exact solution since it includes all order approximations of perturbation. In terms of the cut-off approximation of perturbation and our improved scheme of perturbation theory, the improved form of the perturbed solution of open systems absorbing the partial contributions from the high order even all order approximations is deduced. Moreover, only under the factorizing initial condition, the exact master equation including all order approximations is proposed. Correspondingly, the perturbed master equation and its improved form different from the existed master equation are given. In special, the Redfield master equation is derived out without using Born-Markov approximation. The solution of open system dynamics in the Milburn model is also gained. As examples, Zurek model of two-state open system and its extension with two transverse fields are studied.

arXiv:quant-ph/0601051•Jan 9, 2006•0 citations

Novel Numerical Method for the Solution of Schrödinger’s Equation: I. A Particle in an Interaction Potential of General Shape

M. Rieth, Wolfram Schommers, Sotirios Baskoutas

A new method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods have been discussed critically.

•Jul 9, 2015•1 citation

Development of the perturbation theory using polynomial solutions

Fethi Mohamed Maiz

The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.

Journal of Mathematical Physics•Jan 1, 2019•2 citations

EXACT NUMERICAL SOLUTION OF SCHRÖDINGER'S EQUATION FOR A PARTICLE IN AN INTERACTION POTENTIAL OF GENERAL SHAPE

M. Rieth, Wolfram Schommers, Sotirios Baskoutas

A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.

International Journal of Modern Physics B•Oct 30, 2002•45 citations
We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a ccc-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schrödinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.
We investigate the exact solution, perturbation theory and master equation of open system dynamics based on our serial studies on quantum mechanics in general quantum systems [An Min Wang, quant-ph/0611216 and quant-ph/0611217]. In a system-environment separated representation, a general and explicit solution of open system dynamics is obtained, and it is an exact solution since it includes all order approximations of perturbation. In terms of the cut-off approximation of perturbation and our improved scheme of perturbation theory, the improved form of the perturbed solution of open systems absorbing the partial contributions from the high order even all order approximations is deduced. Moreover, only under the factorizing initial condition, the exact master equation including all order approximations is proposed. Correspondingly, the perturbed master equation and its improved form different from the existed master equation are given. In special, the Redfield master equation is derived out without using Born-Markov approximation. The solution of open system dynamics in the Milburn model is also gained. As examples, Zurek model of two-state open system and its extension with two transverse fields are studied.
A new method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods have been discussed critically.
The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.
A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.