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:hep-th/9306060 · Physical Review A

Separation of variables and exactly soluble time-dependent potentials in quantum mechanics

Costas John Efthimiou, Donald Spector

Published Apr 1, 1994• 27 citations
arXivHTMLPDF

Abstract

We use separation of variables as a tool to identify and to analyze exactly soluble time-dependent quantum-mechanical potentials. By considering the most general possible time-dependent redefinition of the spatial coordinate, as well as general transformations on the wave functions, we show that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models. We consider the formal structure underlying our findings, and the relationship between our results and other work on time-dependent potentials. As an application of our methods, we apply our results to the calculation of propagators.

DOI
10.1103/physreva.49.2301

Related papers

Time-Dependent Perturbation Theory

Shabnam Siddiqui

In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).

CRC Press eBooks•Oct 10, 2018•1 citation

Exactly Solvable Time-Dependent Problems: Potentials of Monotonously Decreasing Function of Time

Tae Jun Park

We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.

Bulletin of the Korean Chemical Society•Dec 20, 2002•11 citations

Exact solution to a general quantum mechanical problem with time-dependent boundary conditions

Rafael Almeida, Daniel A. Morales

Exact analytical solutions are found for the quantum mechanical problem of a particle subject to a time-dependent potential with time-dependent boundary conditions. The method of solution employs time-dependent invariants, rescaling of space and time variables along with an unitary transformation. Several applications are discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 827–833, 1997

International Journal of Quantum Chemistry•Jan 1, 1997•4 citations

General Solution and Quantum Phases of the Time-Dependent Linear Potential

Mai-Lin Liang, Zhiguo Zhang, Kai-Sheng Zhong

The exact general solution of the Schrödinger equation for the time-dependent linear potential (TLP) is obtained in momentum representation. The wave functions in coordinate representation are derived from that in momentum representation by the Fourier transform. Meanwhile, the dynamical and geometrical phases for the unbounded states of TLP are calculated.

Physica Scripta•Jan 1, 2004•3 citations

Saturated transitions in exactly soluble models of two-state curve crossing with time-dependent potentials

Vladimir Yurovsky, A. Ben‐Reuven

Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional $(\ensuremath{\sim}{t}^{\ensuremath{-}1})$ or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.

Physical Review A•Dec 1, 1999•19 citations

Separation of Variables

Steven M. Pascal

The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.

CRC Press eBooks•Jul 12, 2023•0 citations
We use separation of variables as a tool to identify and to analyze exactly soluble time-dependent quantum-mechanical potentials. By considering the most general possible time-dependent redefinition of the spatial coordinate, as well as general transformations on the wave functions, we show that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models. We consider the formal structure underlying our findings, and the relationship between our results and other work on time-dependent potentials. As an application of our methods, we apply our results to the calculation of propagators.
In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).
We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.
Two exactly soluble two-state curve crossing models, with potential matrices formed by a linear combination of time-independent terms and (I) inversely proportional $(\ensuremath{\sim}{t}^{\ensuremath{-}1})$ or (II) exponential time-dependent terms, are considered here. It is shown that the two models are related to each other by a simple transformation of the time variable. These models can be further transformed to a simpler one (the ``reduced'' model), in which the two crossing potentials are a constant (horizontal) one and an inversely proportional time-dependent one, and the interaction coupling them is time-independent. Analysis of the exact solution of the reduced model discloses that the nonadiabatic transition probability saturates at increasingly large coupling strengths without vanishing (in contradiction of the linear Landau-Zener theory and various adiabatic theories). Similar saturation can also occur in more general models, within certain ranges of the potential parameters involved.
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.