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

Jurnal Penelitian Fisika dan Aplikasinya (JPFA)

Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators

James Mendoza Gallo, Bienvenido Masirin Butanas Jr.

Published Dec 31, 2019• 0 citations
Publisher

Abstract

The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.

DOI
10.26740/jpfa.v9n2.p92-104

Related papers

Quantum Propagator Dynamics of a Harmonic Oscillator in a Multimode Harmonic Oscillators Environment using White Noise Functional Analysis

Bienvenido Masirin Butanas Jr., Roland Cristopher F. Caballar

White noise analysis is used to derive the propagator of an open quantum system consisting of a harmonic oscillator which is coupled to an environment consisting of N multimode harmonic oscillators. The quantum propagators are obtained after solving for the normal modes of the system-environment interaction in order to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian is then used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator is then found to consist of a product of N simple harmonic oscillator propagators. N-2 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other 2 correspond to the non-degenerate normal mode frequencies. This method of deriving the propagators greatly simplifies the task of mathematically describing the dynamics of the open quantum system. The ease of use of this method suggests that, together with its inherent mathematical rigor, white noise analysis can be a powerful tool in analyzing the dynamics of an open quantum system.

arXiv:1703.04909•Mar 15, 2017•2 citations

Dynamics of coupled harmonic oscillators in an environment using white noise analysis

Bienvenido Masirin Butanas Jr.

White noise analysis has gained considerable attention in the past years in the analysis of quantum systems, polymer entanglement, transport, and diffusion phenomena. However, there is dearth on the application of the analysis into open quantum systems specifically coupled harmonic oscillators in an environment. In this work, white noise analysis was used to study the dy- namics of the propagator of an open quantum system consisting of coupled harmonic oscillators that are coupled to an environment consisting of N-2 multimode harmonic oscillators. The quantum propagators were obtained after solving for the normal modes of the system-environment interaction to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian was used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator was found to consist of the product of N simple harmonic oscillator propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other three correspond to the non-degenerate normal mode frequen- cies, two of these nondegeneracies correspond to complex normal modes. This approach of describing the dynamics of the open quantum system greatly simplifies the task mathematically.

AIP conference proceedings•Jan 1, 2020•2 citations

Normal mode propagator dynamics of a double harmonic oscillator in an environment using white noise analysis

Bienvenido Masirin Butanas Jr., Roland Cristopher F. Caballar

This paper presents an application of white noise analysis to derive the quantum propagator dynamics at the normal mode of an open quantum system consisting of double harmonic oscillators which are coupled to an environment consisting of N-2 finite harmonic oscillators. The propagators were obtained after finding the normal modes of the system-environment interactions. We show that the full quantum propagator is just a product of N individual propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies while the other three correspond to the non-degenerate normal mode frequencies.

Proceedings of the Samahang Pisika ng Pilipinas•Jun 7, 2017•1 citation

Coupled harmonic oscillator in a multimode harmonic oscillator bath: Derivation of quantum propagator and master equation using white noise analysis

Bienvenido Masirin Butanas Jr., Roland Cristopher F. Caballar

This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. The obtained propagator is then used to derive the time evolution equation for the density matrix describing the system. The method used to derive the propagator of the open quantum system considered in this paper shows promise for analyzing open quantum systems, due to its ease of use and mathematical rigor.

arXiv:1607.01906•Jul 7, 2016•1 citation

Derivation of the quantum propagator of a three body system modelled as tri-coupled harmonic oscillators using white noise path integration

Bienvenido Masirin Butanas Jr., Roland Cristopher F. Caballar

White noise path integration is applied successfully to derive the quantum propagator of a three body particle system consisting of three harmonic oscillators coupled to each other, or tri-coupled harmonic oscillators. It is shown that the full quantum propagator is a product of three simple harmonic oscillator propagators. These propagators were obtained after two successive transformations of the coordinates for the first two harmonic oscillators, and the coordinates for the coupling between the third and one of the first two harmonic oscillators in the system. Furthermore, when the third coordinate and coupling constants are set to zero the resulting propagator reduces to that of the propagator of the coupled harmonic oscillators.

Proceedings of the Samahang Pisika ng Pilipinas•Jun 7, 2017•1 citation

Derivation of quantum propagator for coupled harmonic oscillator with uniform electric field in a single harmonic oscillator environment using white noise functional approach

Norodin A. Rangaig, Caironesa T. Pada, Vernie C. Convicto

The use of white noise analysis is utilized in this study. We consider an open quantum system of coupled harmonic oscillator with uniform two dimensional electric field in a single harmonic oscillator environment. We obtain the quantum propagator for the said system by considering the Feynman path integral as a white noise functional. In addition, the resulting quantum propagator was consistent with the previous literature and also, this study gives more general form due to the inclusion of the electric field.

Journal of King Saud University - Science•Apr 14, 2018•6 citations
The ring model of the coupled oscillator has enormously studied from the perspective of quantum mechanics. The research efforts on this system contribute to fully grasp the concepts of energy transport, dissipation, among others, in mesoscopic and condensed matter systems. In this research, the dynamics of the quantum propagator for the ring of oscillators was analyzed anew. White noise analysis was applied to derive the quantum mechanical propagator for a ring of four harmonically coupled oscillators. The process was done after performing four successive coordinate transformations obtaining four separated Lagrangian of a one-dimensional harmonic oscillator. Then, the individual propagator was evaluated via white noise path integration where the full propagator is expressed as the product of the individual propagators. In particular, the frequencies of the first two propagators correspond to degenerate normal mode frequencies, while the other two correspond to non-degenerate normal mode frequencies. The full propagator was expressed in its symmetric form to extract the energy spectrum and the wave function.
White noise analysis is used to derive the propagator of an open quantum system consisting of a harmonic oscillator which is coupled to an environment consisting of N multimode harmonic oscillators. The quantum propagators are obtained after solving for the normal modes of the system-environment interaction in order to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian is then used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator is then found to consist of a product of N simple harmonic oscillator propagators. N-2 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other 2 correspond to the non-degenerate normal mode frequencies. This method of deriving the propagators greatly simplifies the task of mathematically describing the dynamics of the open quantum system. The ease of use of this method suggests that, together with its inherent mathematical rigor, white noise analysis can be a powerful tool in analyzing the dynamics of an open quantum system.
White noise analysis has gained considerable attention in the past years in the analysis of quantum systems, polymer entanglement, transport, and diffusion phenomena. However, there is dearth on the application of the analysis into open quantum systems specifically coupled harmonic oscillators in an environment. In this work, white noise analysis was used to study the dy- namics of the propagator of an open quantum system consisting of coupled harmonic oscillators that are coupled to an environment consisting of N-2 multimode harmonic oscillators. The quantum propagators were obtained after solving for the normal modes of the system-environment interaction to decouple the coordinates in the Lagrangian describing the dynamics of the system, the environment and their interaction with each other. The decoupled Lagrangian was used in the path integral corresponding to the propagator of the system, with the path integral evaluated using white noise analysis. The resulting propagator was found to consist of the product of N simple harmonic oscillator propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies of the system-environment interaction, while the other three correspond to the non-degenerate normal mode frequen- cies, two of these nondegeneracies correspond to complex normal modes. This approach of describing the dynamics of the open quantum system greatly simplifies the task mathematically.
This paper presents an application of white noise analysis to derive the quantum propagator dynamics at the normal mode of an open quantum system consisting of double harmonic oscillators which are coupled to an environment consisting of N-2 finite harmonic oscillators. The propagators were obtained after finding the normal modes of the system-environment interactions. We show that the full quantum propagator is just a product of N individual propagators. N-3 of these propagators correspond to the degenerate normal mode frequencies while the other three correspond to the non-degenerate normal mode frequencies.
This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. The obtained propagator is then used to derive the time evolution equation for the density matrix describing the system. The method used to derive the propagator of the open quantum system considered in this paper shows promise for analyzing open quantum systems, due to its ease of use and mathematical rigor.
White noise path integration is applied successfully to derive the quantum propagator of a three body particle system consisting of three harmonic oscillators coupled to each other, or tri-coupled harmonic oscillators. It is shown that the full quantum propagator is a product of three simple harmonic oscillator propagators. These propagators were obtained after two successive transformations of the coordinates for the first two harmonic oscillators, and the coordinates for the coupling between the third and one of the first two harmonic oscillators in the system. Furthermore, when the third coordinate and coupling constants are set to zero the resulting propagator reduces to that of the propagator of the coupled harmonic oscillators.
The use of white noise analysis is utilized in this study. We consider an open quantum system of coupled harmonic oscillator with uniform two dimensional electric field in a single harmonic oscillator environment. We obtain the quantum propagator for the said system by considering the Feynman path integral as a white noise functional. In addition, the resulting quantum propagator was consistent with the previous literature and also, this study gives more general form due to the inclusion of the electric field.