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

International Journal of Quantum Chemistry

Ground and excited states of one‐dimensional self‐interacting nonlinear oscillators through time‐dependent quantum mechanics

Amita Wadehra, Amlan K. Roy, Bimalendu Deb

Published Jan 1, 2003• 17 citations
Publisher

Abstract

Abstract Energy eigenvalues, probability densities, and 〈x2r〉 values (r = 1 to 6) of one‐dimensional self‐interacting nonlinear oscillators have been obtained by evolving the time‐dependent Schrödinger equation in imaginary time, coupled with the minimization of energy expectation values. For excited states, the orthogonality constraint with lower states is maintained. Probability density plots for the ground and first three excited states are presented. A comparison of energy eigenvalues and probability densities plots is also made between oscillators with and without self‐interaction. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

DOI
10.1002/qua.10472

Related papers

Time-dependent quantum-mechanical calculation of ground and excited states of anharmonic and double-well oscillators

Amlan K. Roy, Neetu Gupta, Bimalendu Deb

Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.

Physical Review A•Dec 14, 2001•42 citations

On the time‐dependent solutions of the Schrödinger's equation. II. The one‐mode field perturbed harmonic oscillator

A. Palma, M. Villa, L. Sandoval

Abstract The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

International Journal of Quantum Chemistry•Oct 27, 2011•3 citations

Quantum confinement in 1D systems through an imaginary-time evolution method

Amlan K. Roy

Quantum confinement is studied by numerically solving time-dependent (TD) Schrödinger equation (SE). An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence, a simple accurate and reliable method is proposed for confinement in quantum systems.

arXiv:1801.06478•Modern Physics Letters A•Sep 23, 2015•11 citations

Trial introduction of a complex potential function for perturbation that causes quantum mechanical transition in a one‐dimensional harmonic oscillator

Mitsuru Yamada

Abstract Some results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V (x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum Chemistry•Jan 1, 2004•0 citations

Quantum Harmonic Oscillators in One and Two Dimensions

Guanghan Xia

The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.

Highlights in Science Engineering and Technology•Aug 21, 2023•0 citations

On the time‐dependent solutions of the Schrödinger equation. I. The linear time‐dependent potential

A. Palma, M. Villa, L. Sandoval

Abstract The solution of the Schrödinger equation for the linear time‐dependent potential has recently been the subject of several publications. In this report, we show this is one of the few systems that leads to a solvable Lie algebra. In fact, we consider a more general problem where the linear time‐dependent potential is only a particular case. The solution is found by using the well‐known theorem of Wei‐Norman. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

International Journal of Quantum Chemistry•Jul 8, 2010•7 citations
Abstract Energy eigenvalues, probability densities, and 〈x2r〉 values (r = 1 to 6) of one‐dimensional self‐interacting nonlinear oscillators have been obtained by evolving the time‐dependent Schrödinger equation in imaginary time, coupled with the minimization of energy expectation values. For excited states, the orthogonality constraint with lower states is maintained. Probability density plots for the ground and first three excited states are presented. A comparison of energy eigenvalues and probability densities plots is also made between oscillators with and without self‐interaction. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.
Abstract The solution of the time‐dependent Schrödinger's equation for a perturbed harmonic oscillator is obtained using a solvable Lie algebra. We choose a harmonic oscillator interacting with a one‐mode field, where the perturbation happens to be periodic in time. This leads to one of the simplest Floquet problems. Using the Wei–Norman theorem, the Floquet wave function is obtained as well as the semiclassical Floquet shift in the energy. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
Quantum confinement is studied by numerically solving time-dependent (TD) Schrödinger equation (SE). An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence, a simple accurate and reliable method is proposed for confinement in quantum systems.
Abstract Some results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V (x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.