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:2506.21280

Quantum dynamics in confined pseudo-harmonic oscillator in a time-dependent moving

Akash Halder, Amlan K. Roy, Debraj Nath

Published Jun 26, 2025• 0 citations
arXivHTMLPDF

Abstract

In this work, we present analytical solution of Schrödinger equation of confined pseudoharmonic potential in presence of a moving boundary condition, for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of time-dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product h/2. The TD average energy is derived analytically in a closed form and the corresponding average force and average pressure are defined. Moreover, time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, H2, N2) is obtained. It is found to depend on internuclear distance ratio at two different time domains. The TD survival probability and average life-time of molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non-harmonic central potential in a TD moving boundary condition.

DOI
10.48550/arxiv.2506.21280

Related papers

Quantum Dynamics in Confined Pseudo‐Harmonic Oscillator in a Time‐Dependent Moving Boundary

Akash Halder, Amlan K. Roy, Debraj Nath

ABSTRACT In this work, we present an analytical solution of Schrödinger equation of confined pseudoharmonic potential in the presence of a moving boundary condition for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is the internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of the time‐dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product . The TD average energy is derived analytically in a closed form, and the corresponding average force and average pressure are defined. Moreover, the time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, , ) is obtained. It is found to depend on the internuclear distance ratio at two different time domains. The TD survival probability and average life‐time of a molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity, and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non‐harmonic central potential in a TD moving boundary condition.

Advanced Theory and Simulations•Dec 29, 2025•0 citations

Time-correlation function and average energy of molecules in presence of Deng-Fan potential in a moving boundary

Debraj Nath, Amlan K. Roy

Abstract For the Deng-Fan potential within a moving boundary condition, the time-dependent Schrödinger equation is considered analytically. The eigenvalue equation is solved by using a combination of Pekeris and Greene-Aldrich approximations. Various time-dependent quantities including density distribution function, auto-correlation function, disequilibrium, average energy, quantum similarity, quantum similarity index are obtained for selected eight diatomic molecules. The motion of the peak of soliton with moving boundary condition is investigated for ground states of some diatomic molecules along with the corresponding peak values.

Nonlinear Dynamics•Sep 16, 2022•3 citations

Exact Time-Dependent Wave Functions of a Confined Time-Dependent Harmonic Oscillator with Two Moving Boundaries

Chi‐Fai Lo

By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.

Communications in Theoretical Physics•Nov 1, 2009•2 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

Comparison between time-independent and time-dependent quantum systems in the context of energy, Heisenberg uncertainty, average energy, force, average force and thermodynamic quantities

Debraj Nath

Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.

arXiv:2110.05609•Oct 12, 2021•0 citations

Time-dependent Dunkl–Schrödinger equation with an angular-dependent potential

B. C. Lütfüoğlu, Amar Benchikha, B. Hamil +1 more

In this paper, we investigate the analytical solution of the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent mass and frequency, coupled with angular-dependent potential energy by utilizing the Dunkl derivatives. To obtain the solution, we employ the Lewis–Riesenfeld invariant methodology. Our approach broadens the scope of quantum mechanical analyses, offering exact solutions and new insights into dynamic quantum systems under varying conditions.

arXiv:2408.02021•Modern Physics Letters A•Feb 28, 2025•15 citations
In this work, we present analytical solution of Schrödinger equation of confined pseudoharmonic potential in presence of a moving boundary condition, for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of time-dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product h/2. The TD average energy is derived analytically in a closed form and the corresponding average force and average pressure are defined. Moreover, time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, H2, N2) is obtained. It is found to depend on internuclear distance ratio at two different time domains. The TD survival probability and average life-time of molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non-harmonic central potential in a TD moving boundary condition.
ABSTRACT In this work, we present an analytical solution of Schrödinger equation of confined pseudoharmonic potential in the presence of a moving boundary condition for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is the internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of the time‐dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product . The TD average energy is derived analytically in a closed form, and the corresponding average force and average pressure are defined. Moreover, the time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, , ) is obtained. It is found to depend on the internuclear distance ratio at two different time domains. The TD survival probability and average life‐time of a molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity, and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non‐harmonic central potential in a TD moving boundary condition.
Abstract For the Deng-Fan potential within a moving boundary condition, the time-dependent Schrödinger equation is considered analytically. The eigenvalue equation is solved by using a combination of Pekeris and Greene-Aldrich approximations. Various time-dependent quantities including density distribution function, auto-correlation function, disequilibrium, average energy, quantum similarity, quantum similarity index are obtained for selected eight diatomic molecules. The motion of the peak of soliton with moving boundary condition is investigated for ground states of some diatomic molecules along with the corresponding peak values.
By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.
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.
Exact solutions of time-dependent Schrödinger equation in presence of time-dependent potential is defined by point transformation and separation of variables. Energy and Heisenberg uncertainty relation are pursued for time-independent potential whereas average energy and Heisenberg uncertainty relation are defined for time-dependent potential. Forces acting on a fixed boundary wall as well as average force acting on moving boundary wall are presented along various trajectories. For high temperature, analytical forms of partition function and the corresponding thermodynamic quantities are derived following the Euler-Maclaurin summation formula over a finite as well as an infinite domain for accurate presentation. Three quantum systems are generated with the help of point transformation, separation of variables and super-symmetric quantum mechanics from one quantum system and the corresponding results are compared among all systems, where two of them are time-independent and another two are time-dependent.