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

Acta Phys. Polon.

ON THE APPROXIMATE APPLICABILITY OF THE SCHRODINGER EQUATION TO NON- ISOLATED SYSTEMS

B. Sredniava, J. Weyssenhoff

Published Feb 1, 1963• 0 citations
OpenAlex

Abstract

Problems in which Schrodinger's equation with timedependent hamiltonian is applied to one system should rather be considered as representing cases of two systems, the second acting appreciably on the first but not inversely; the systems as a whole is then considered to be isolated and, consequently, subject to treatment by means of a Schrodinger equation with time-independent hamiltonian. The composition and decomposition of systems is dealt with by the method of statistical operators of v. Neumann. Both methods of dealing with the problem are shown to lead in good approximation to the same results, but only for times fulfilling a given inequality. (auth)

Related papers

ON THE QUANTUM-MECHANICAL DESCRIPTION OF NON-ISOLATED SYSTEMS

K. Zalewski

The Schrodinger equation with a time-dependent hamiltonian is sometimes applied to the study of non-isolated systems. It is shown that this is correct (with a proper choice of the hamiltonian) if and only if the state of the system is initially pure and remains so with increasing time. Further the problem of describing a non-isolated system by the density matrix of a pure state is discussed. (auth)

Acta Phys. Polon.•Nov 1, 1963•0 citations

THE SCHRÖDINGER PICTURE

Mircea S. Rogalski, Stuart B. Palmer

We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.

CRC Press eBooks•Jan 8, 2014•0 citations

SOLUTION OF THE TIME-INDEPENDENT SCHRÖDINGER EQUATION FOR THE ROSEN–MORSE POTENTIAL BY USING THE GALERKIN METHOD

I Gusti Agung Widagda, Nengah Artawan, Ni Luh Putu Trisnawati +2 more

This study presents a numerical solution to the time-independent Schrödinger equation (TISE) for the Rosen-Morse potential using the Galerkin method. The Rosen-Morse potential, commonly used in atomic and molecular physics, has known analytical solutions under certain conditions. By transforming the TISE into a Jacobi differential equation, the analytical wave function and energy levels can be derived. However, analytical solutions are limited to ideal cases, highlighting the need for numerical methods in more general scenarios. The Galerkin method is implemented by expanding the wave function using Sine basis functions and projecting the TISE onto this basis. The resulting eigenvalue problem is solved by constructing the Hamiltonian matrix from kinetic and potential energy operators. Numerical results from the Galerkin method are compared with analytical solutions using graphical analysis, percentage error (% error), and statistical tests, including the Mann-Whitney U test. The results demonstrate that the probability densities obtained using the Galerkin method closely approximate the analytical solution. This is visually evident from the overlapping of probability density plots from both methods. The percentage error of the probability densities is below 1 %, entirely. Furthermore, the Mann–Whitney U test yields a p-value less than 0.05, indicating that the differences between the two sets of probability densities are statistically insignificant at the 95% confidence level. These findings highlight the Galerkin method’s effectiveness and accuracy as a robust numerical tool for solving the TISE with the Rosen-Morse potential.

Indonesian Physical Review•Sep 30, 2025•0 citations

A comparison of solutions to the time dependent Schrödinger equation

Chi-Ming Chang

Various methods providing approximate solutions to the time dependent Schrödinger equation proposed in a previous paper are compared using two examples for which the exact analytic solutions are known. Results show that the new method proposed in the previous paper is better than the traditional method for time dependent parameters.

The Journal of Chemical Physics•Oct 15, 1977•0 citations

A minimum principle for von Neumann’s equation

Sidney Golden

A minimum principle is described which has von Neumann’s equation of motion of the statistical operator as its ultimate solution. When the statistical operator is initially restricted to be a projection of trace unity, the minimum principle has Schrödinger’s time−dependent equation as its ultimate solution. As a result, the present minimum principle is shown to be a generalization of one first introduced by McLachlan, with a modification of the Dirac−Frenkel variational principle as a necessary consequence. In physical terms, the quantity that is minimized is shown to be the mean−squared deviation of an effective time−dependent Hamiltonian, which can be attributed to a time−dependent statistical operator initially chosen to describe a system, from the actual time−dependent Hamiltonian of the system. Examples of approximate solutions to von Neumann’s equation and Schrödinger’s equation that are obtained by use of the minimum principle confirm this identification.

Journal of Mathematical Physics•Jan 1, 1975•7 citations

The Schrödinger Equation in Three–Dimensions

Şaki̇r Erkoç

Consider an isolated two–body system (see Figure 7.1 ). The Hamiltonian representing the system is

CRC Press eBooks•Jul 28, 2006•0 citations
Problems in which Schrodinger's equation with timedependent hamiltonian is applied to one system should rather be considered as representing cases of two systems, the second acting appreciably on the first but not inversely; the systems as a whole is then considered to be isolated and, consequently, subject to treatment by means of a Schrodinger equation with time-independent hamiltonian. The composition and decomposition of systems is dealt with by the method of statistical operators of v. Neumann. Both methods of dealing with the problem are shown to lead in good approximation to the same results, but only for times fulfilling a given inequality. (auth)
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
This study presents a numerical solution to the time-independent Schrödinger equation (TISE) for the Rosen-Morse potential using the Galerkin method. The Rosen-Morse potential, commonly used in atomic and molecular physics, has known analytical solutions under certain conditions. By transforming the TISE into a Jacobi differential equation, the analytical wave function and energy levels can be derived. However, analytical solutions are limited to ideal cases, highlighting the need for numerical methods in more general scenarios. The Galerkin method is implemented by expanding the wave function using Sine basis functions and projecting the TISE onto this basis. The resulting eigenvalue problem is solved by constructing the Hamiltonian matrix from kinetic and potential energy operators. Numerical results from the Galerkin method are compared with analytical solutions using graphical analysis, percentage error (% error), and statistical tests, including the Mann-Whitney U test. The results demonstrate that the probability densities obtained using the Galerkin method closely approximate the analytical solution. This is visually evident from the overlapping of probability density plots from both methods. The percentage error of the probability densities is below 1 %, entirely. Furthermore, the Mann–Whitney U test yields a p-value less than 0.05, indicating that the differences between the two sets of probability densities are statistically insignificant at the 95% confidence level. These findings highlight the Galerkin method’s effectiveness and accuracy as a robust numerical tool for solving the TISE with the Rosen-Morse potential.
A minimum principle is described which has von Neumann’s equation of motion of the statistical operator as its ultimate solution. When the statistical operator is initially restricted to be a projection of trace unity, the minimum principle has Schrödinger’s time−dependent equation as its ultimate solution. As a result, the present minimum principle is shown to be a generalization of one first introduced by McLachlan, with a modification of the Dirac−Frenkel variational principle as a necessary consequence. In physical terms, the quantity that is minimized is shown to be the mean−squared deviation of an effective time−dependent Hamiltonian, which can be attributed to a time−dependent statistical operator initially chosen to describe a system, from the actual time−dependent Hamiltonian of the system. Examples of approximate solutions to von Neumann’s equation and Schrödinger’s equation that are obtained by use of the minimum principle confirm this identification.