Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators
DOI:
https://doi.org/10.26740/jpfa.v9n2.p92-104Keywords:
white noise analysis, path integrals, coupled harmonic oscillatorsAbstract
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.
References
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https://doi.org/10.1002/qua.560350502.
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Zuniga J, Bastid A, and Requena A. Quantum Solution of Coupled Harmonic Oscillator Systems Beyond Normal Coordinates. Journal of Mathematical Chemistry. 2017; 55(10): 1964-1984. DOI:
https://doi.org/10.1007/s10910-017-0777-1.
Dolfo G and Vigue J. Damping of Coupled Harmonic Oscillators. European Journal of Physics. 2018; 39(2): 025005. DOI: https://doi.org/10.1088/1361-6404/aa9ec6.
Rodriguez SRK. Classical and Quantum Distinctions Between Weak and Strong Coupling. European Journal of Physics. 2016; 37(2): 025802. DOI:
https://doi.org/10.1088/0143-0807/37/2/025802.
Macedo DX and Guedes I. Time-Dependent Coupled Harmonic Oscillators: Classical and Quantum Solutions. International Journal of Modern Physics E. 2014; 23(09): 1450048. DOI: https://doi.org/10.1142/S0218301314500487.
Macedo DX and Guedes I. Time-dependent coupled harmonic oscillators. Journal of Mathematical Physics. 2012; 53: 052101. DOI: https://doi.org/10.1063/1.4709748.
McDermott RM and Redmount RH. Coupled Classical and Quantum Oscillators; 2004. Available from: https://arxiv.org/abs/quant-ph/0403184.
Dutra ADS. On the Quantum Mechanical Propagator for Driven Coupled Harmonic Oscillators. Journal of Physics A: Mathematical and General. 1992; 25(15), 4189. DOI: https://doi.org/10.1088/0305-4470/25/15/026.
Pabalay JR and Bornales JB. Coupled Harmonic Oscillators: A White Noise Functional Approach. Undergraduate Thesis. Unpublished. Iligan: Iligan Institute of Technology Mindanao State University; 2007.
Butanas-Jr BM and Caballar RCF. Coupled Harmonic Oscillator in A Multimode Harmonic Oscillator Bath: Derivation of Quantum Propagator and Master Equation Using White Noise Analysis; 2016. Available from: https://arxiv.org/abs/1607.01906.
Butanas-Jr BM and Caballar RCF. Quantum Propagator Dynamics of a Harmonic Oscillator in a Multimode Harmonic Oscillators Environment using White Noise Functional Analysis; 2017. Available from: https://arxiv.org/pdf/1703.04909.
Feynman RP. Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics. 1948; 20(2):367-387. DOI: https://doi.org/10.1103/RevModPhys.20.367.
Bernido CC and Bernido MVC. White Noise Analysis: Some Applications in Complex Systems, Biophysics and Quantum Mechanics. International Journal of Modern Physics B. 2012; 26(29): 1230014. DOI: https://doi.org/10.1142/S0217979212300149.
Bernido CC and Bernido MVC. Methods and Application of White Noise Analysis in Interdisciplinary Sciences; 2015. DOI: https://doi.org/10.1142/9789814569125_0006.
Bock W and Grothaus M. A White Noise Approach to Phase Space Feynman Path Integrals; 2010. Available from: http://arxiv.org/abs/1012.1125v1.
Streit L and Hida T. Generalized Brownian Functionals and the Feynman Integral. Stochastic Processes and their Applications. 1984; 16(1): 55-69. DOI: https://doi.org/10.1016/0304-4149(84)90175-3.
Butanas BM and Esguerra JP. White Noise Functional Approach to the Brownian Motion of the Free Particle. Proceedings of the Samahang Pisika ng Pilipinas 36. 2018. Available from: https://paperview.spp-online.org/proceedings/article/view/SPP-2018-PB-03.
Butanas BM and Caballar RCF. On the Derivation of Nakajima-Zwanzig Probability Density Function Via White Noise Analysis. AIP Conference Proceedings. 2017; 1871: 020006. DOI: https://doi.org/10.1063/1.4996516.
Butanas BM and Caballar RCF. Normal Mode Propagator Dynamics of a Double Harmonic Oscillator in an Environment Using White Noise Analysis. Proceedings of the Samahang Pisika ng Pilipinas 35. 2017. Available from: https://paperview.spp-online.org/proceedings/article/view/110.
Baybayon RN, Bornales JB, and Cubero RJ. A Hida-Streit Formulation Approach in Evaluating the Quantum Mechanical Propagator of a Particle Moving in a Constant Force Field with Constant Friction. Proceedings of the 12th SPVM National Physics Conference. 2019. Available from: https://www.researchgate.net/publication/330467899_A_Hida-Streit_Formulation_Approach_in_Evaluating_the_Quantum_Mechancal_Propagator_of_a_Particle_Moving_in_aConstant_Force_field_with_Constant_Friction.
Butanas BM and Caballar RCF. The Caldeira-Leggett Model in A Harmonic Potential: Derivation of Master Equation and Quantum Propagator Through A White Noise Functional Approach. Proceedings of the Samahang Pisika ng Pilipinas. 2016. Available from: https://paperview.spp-online.org/proceedings/article/view/355.
Ito K. Stochastic Processes and Their Applications. New York: Springer; 2008.
Gradshteyn IS and Ryzhik IM. Table of Integrals, Series, and Products. San Diego: Academic Press; 1980.
Mazzucchi S. Functional-Integral Solution for The Schrodinger Equation with Polynomial Potential: A White Noise Approach. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2011; 14(04): 675-688. DOI: https://doi.org/10.1142/S0219025711004572.
Gemao B and Bornales J. White Noise Path Integral Treatment of The Probability Distribution for The Area Enclosed by a Polymer Loop in Crossed Electric-Magnetic Fields. International Journal of Modern Physics: Conference Series. 2012; 17: 77-82. DOI: https://doi.org/10.1142/S2010194512007969.
Gemao B, Bornales J, and Loquero M. On Polymer Loop in a Gel Under External Fields: Analytical Approach using White Noise Analysis. IOP Conference Series: Materials Science and Engineering. 2015; 79: 012012. DOI: https://doi.org/10.1088/1757-899X/79/1/012012.
Aure RRL, Bernido CC, Carpio-Bernido MV, and Bacabac RG. Damped White Noise Diffusion with Memory for Diffusing Microprobes in Ageing Fibrin Gels. Biophysical Journal. 2019; 117(6): 1029-1036. DOI:
https://doi.org/10.1016/j.bpj.2019.08.014.
Lü X and Dai W. A White Noise Approach to Stochastic Partial Differential Equations Driven by the Fractional Lévy Noise. Advances in Difference Equations. 2018; 2018: 420. DOI:
https://doi.org/10.1186/s13662-018-1861-y.
Bock W, Grothaus M, and Jung S. The Feynman Integrand for the Charged Particle in A Constant Magnetic Field as White Noise Distribution. Communications on Stochastic Analysis. 2012; 6(4): 10. DOI: http://doi.org/10.31390/cosa.6.4.10.
Bock W, da Silva JL, and Suryawan HP. Local Times for Multifractional Brownian Motion in Higher Dimensions: A White Noise Approach. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2016; 19(04): 1650026. DOI: https://doi.org/10.1142/S0219025716500260.
Bock W. Hamiltonian Path Integrals in Momentum Space Representation via White Noise Techniques. Reports on Mathematical Physics. 2014; 73(1): 91-107. DOI: https://doi.org/10.1016/S0034-4877(14)60034-3.
Violanda RR, Bernido CC, and Carpio-Bernido MV. White Noise Functional Integral for Exponentially Decaying Memory: Nucleotide Distribution in Bacterial Genomes. Physica Scripta. 2019; 94(12): 125006. DOI: https://doi.org/10.1088/1402-4896/ab3739.
Bernido CC, Carpio-Bernido MV, and Escobido MGO. Modified Diffusion with Memory for Cyclone Track Fluctuations. Physics Letters A. 2014; 378(30-31): 2016-2019. DOI: https://doi.org/10.1016/j.physlets.2014.06.003.
Weiss U. Quantum Dissipative Systems. Singapore: World Scientific; 1999.
https://doi.org/10.1038/nature09721.
Makarov DN. Coupled Harmonic Oscillators and Their Quantum Entanglement. Physical Review E. 2018; 97: 042203. DOI: https://doi.org/10.1103/PhysRevE.97.042203.
Chakraborty S and Sarma AK. Entanglement Dynamics of Two Coupled Mechanical Oscillators in Modulated Optomechanics. Physical Review A. 2018; 97: 022336. DOI: https://doi.org/10.1103/PhysRevA.97.022336.
Delor M, Archer SA, Keane T, Meijer AJHM, Sazanvich IV, Greetham GM, Towrie M, and Weinstein JA. Directing the Path of Light-Induced Electron Transfer at a Molecular Fork using Vibrational Excitation. Nature Chemistry. 2017; 9: 1099-1104. DOI: https://doi.org/10.1038/nchem.2793.
Sasihithlu K. A Coupled Harmonic Oscillator Model to Describe the Near-Field Radiative Heat Transfer Between Nanoparticles and Planar Surfaces; 2018. Available from: https://arxiv.org/abs/1810.02548.
Romero E, Augulis A, Novoderezhkin VI, Ferritti M, Thieme J, Zigmantas D, and van Grondelle R. Quantum Coherence in Photosynthesis for Efficient Solar-Energy Conversion. Nature Physics. 2014; 10: 676-682. DOI: https://doi.org/10.1038/nphys3017.
Fuller FD, Pan J, Gelzinis A, Butkus V, Senlik S, Wilcox DE, Yocum CF, Valkunas L, Abravicius D and Ogilvie JP. Vibrionic Coherence in Oxygenic Photosynthesis. Nature Chemistry. 2014; 6: 706-711. DOI: https://doi.org/10.1038/nchem.2005.
Bhattacharya M and Shi H. Coupled Second-Quantized Oscillators. American Journal of Physics. 2013; 81: 267-273. DOI: https://doi.org/10.1119/1.4792696.
Hong-Yi F. New Unitary Transformation for The Three Coupled Oscillators. International Journal of Quantum Chemistry. 1989; 35(5): 585-592. DOI:
https://doi.org/10.1002/qua.560350502.
Hong-Yi F. Unitary Transformation of Four Harmonically Coupled Identical Oscillators. Physical Review A. 1990; 42: 4337. DOI: https://doi.org/10.1103/PhysRevA.42.4377.
Zuniga J, Bastid A, and Requena A. Quantum Solution of Coupled Harmonic Oscillator Systems Beyond Normal Coordinates. Journal of Mathematical Chemistry. 2017; 55(10): 1964-1984. DOI:
https://doi.org/10.1007/s10910-017-0777-1.
Dolfo G and Vigue J. Damping of Coupled Harmonic Oscillators. European Journal of Physics. 2018; 39(2): 025005. DOI: https://doi.org/10.1088/1361-6404/aa9ec6.
Rodriguez SRK. Classical and Quantum Distinctions Between Weak and Strong Coupling. European Journal of Physics. 2016; 37(2): 025802. DOI:
https://doi.org/10.1088/0143-0807/37/2/025802.
Macedo DX and Guedes I. Time-Dependent Coupled Harmonic Oscillators: Classical and Quantum Solutions. International Journal of Modern Physics E. 2014; 23(09): 1450048. DOI: https://doi.org/10.1142/S0218301314500487.
Macedo DX and Guedes I. Time-dependent coupled harmonic oscillators. Journal of Mathematical Physics. 2012; 53: 052101. DOI: https://doi.org/10.1063/1.4709748.
McDermott RM and Redmount RH. Coupled Classical and Quantum Oscillators; 2004. Available from: https://arxiv.org/abs/quant-ph/0403184.
Dutra ADS. On the Quantum Mechanical Propagator for Driven Coupled Harmonic Oscillators. Journal of Physics A: Mathematical and General. 1992; 25(15), 4189. DOI: https://doi.org/10.1088/0305-4470/25/15/026.
Pabalay JR and Bornales JB. Coupled Harmonic Oscillators: A White Noise Functional Approach. Undergraduate Thesis. Unpublished. Iligan: Iligan Institute of Technology Mindanao State University; 2007.
Butanas-Jr BM and Caballar RCF. Coupled Harmonic Oscillator in A Multimode Harmonic Oscillator Bath: Derivation of Quantum Propagator and Master Equation Using White Noise Analysis; 2016. Available from: https://arxiv.org/abs/1607.01906.
Butanas-Jr BM and Caballar RCF. Quantum Propagator Dynamics of a Harmonic Oscillator in a Multimode Harmonic Oscillators Environment using White Noise Functional Analysis; 2017. Available from: https://arxiv.org/pdf/1703.04909.
Feynman RP. Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics. 1948; 20(2):367-387. DOI: https://doi.org/10.1103/RevModPhys.20.367.
Bernido CC and Bernido MVC. White Noise Analysis: Some Applications in Complex Systems, Biophysics and Quantum Mechanics. International Journal of Modern Physics B. 2012; 26(29): 1230014. DOI: https://doi.org/10.1142/S0217979212300149.
Bernido CC and Bernido MVC. Methods and Application of White Noise Analysis in Interdisciplinary Sciences; 2015. DOI: https://doi.org/10.1142/9789814569125_0006.
Bock W and Grothaus M. A White Noise Approach to Phase Space Feynman Path Integrals; 2010. Available from: http://arxiv.org/abs/1012.1125v1.
Streit L and Hida T. Generalized Brownian Functionals and the Feynman Integral. Stochastic Processes and their Applications. 1984; 16(1): 55-69. DOI: https://doi.org/10.1016/0304-4149(84)90175-3.
Butanas BM and Esguerra JP. White Noise Functional Approach to the Brownian Motion of the Free Particle. Proceedings of the Samahang Pisika ng Pilipinas 36. 2018. Available from: https://paperview.spp-online.org/proceedings/article/view/SPP-2018-PB-03.
Butanas BM and Caballar RCF. On the Derivation of Nakajima-Zwanzig Probability Density Function Via White Noise Analysis. AIP Conference Proceedings. 2017; 1871: 020006. DOI: https://doi.org/10.1063/1.4996516.
Butanas BM and Caballar RCF. Normal Mode Propagator Dynamics of a Double Harmonic Oscillator in an Environment Using White Noise Analysis. Proceedings of the Samahang Pisika ng Pilipinas 35. 2017. Available from: https://paperview.spp-online.org/proceedings/article/view/110.
Baybayon RN, Bornales JB, and Cubero RJ. A Hida-Streit Formulation Approach in Evaluating the Quantum Mechanical Propagator of a Particle Moving in a Constant Force Field with Constant Friction. Proceedings of the 12th SPVM National Physics Conference. 2019. Available from: https://www.researchgate.net/publication/330467899_A_Hida-Streit_Formulation_Approach_in_Evaluating_the_Quantum_Mechancal_Propagator_of_a_Particle_Moving_in_aConstant_Force_field_with_Constant_Friction.
Butanas BM and Caballar RCF. The Caldeira-Leggett Model in A Harmonic Potential: Derivation of Master Equation and Quantum Propagator Through A White Noise Functional Approach. Proceedings of the Samahang Pisika ng Pilipinas. 2016. Available from: https://paperview.spp-online.org/proceedings/article/view/355.
Ito K. Stochastic Processes and Their Applications. New York: Springer; 2008.
Gradshteyn IS and Ryzhik IM. Table of Integrals, Series, and Products. San Diego: Academic Press; 1980.
Mazzucchi S. Functional-Integral Solution for The Schrodinger Equation with Polynomial Potential: A White Noise Approach. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2011; 14(04): 675-688. DOI: https://doi.org/10.1142/S0219025711004572.
Gemao B and Bornales J. White Noise Path Integral Treatment of The Probability Distribution for The Area Enclosed by a Polymer Loop in Crossed Electric-Magnetic Fields. International Journal of Modern Physics: Conference Series. 2012; 17: 77-82. DOI: https://doi.org/10.1142/S2010194512007969.
Gemao B, Bornales J, and Loquero M. On Polymer Loop in a Gel Under External Fields: Analytical Approach using White Noise Analysis. IOP Conference Series: Materials Science and Engineering. 2015; 79: 012012. DOI: https://doi.org/10.1088/1757-899X/79/1/012012.
Aure RRL, Bernido CC, Carpio-Bernido MV, and Bacabac RG. Damped White Noise Diffusion with Memory for Diffusing Microprobes in Ageing Fibrin Gels. Biophysical Journal. 2019; 117(6): 1029-1036. DOI:
https://doi.org/10.1016/j.bpj.2019.08.014.
Lü X and Dai W. A White Noise Approach to Stochastic Partial Differential Equations Driven by the Fractional Lévy Noise. Advances in Difference Equations. 2018; 2018: 420. DOI:
https://doi.org/10.1186/s13662-018-1861-y.
Bock W, Grothaus M, and Jung S. The Feynman Integrand for the Charged Particle in A Constant Magnetic Field as White Noise Distribution. Communications on Stochastic Analysis. 2012; 6(4): 10. DOI: http://doi.org/10.31390/cosa.6.4.10.
Bock W, da Silva JL, and Suryawan HP. Local Times for Multifractional Brownian Motion in Higher Dimensions: A White Noise Approach. Infinite Dimensional Analysis, Quantum Probability and Related Topics. 2016; 19(04): 1650026. DOI: https://doi.org/10.1142/S0219025716500260.
Bock W. Hamiltonian Path Integrals in Momentum Space Representation via White Noise Techniques. Reports on Mathematical Physics. 2014; 73(1): 91-107. DOI: https://doi.org/10.1016/S0034-4877(14)60034-3.
Violanda RR, Bernido CC, and Carpio-Bernido MV. White Noise Functional Integral for Exponentially Decaying Memory: Nucleotide Distribution in Bacterial Genomes. Physica Scripta. 2019; 94(12): 125006. DOI: https://doi.org/10.1088/1402-4896/ab3739.
Bernido CC, Carpio-Bernido MV, and Escobido MGO. Modified Diffusion with Memory for Cyclone Track Fluctuations. Physics Letters A. 2014; 378(30-31): 2016-2019. DOI: https://doi.org/10.1016/j.physlets.2014.06.003.
Weiss U. Quantum Dissipative Systems. Singapore: World Scientific; 1999.
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2019-12-31
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Gallo, J. M. and Butanas Jr., B. M. (2019) “Quantum Propagator Derivation for the Ring of Four Harmonically Coupled Oscillators”, Jurnal Penelitian Fisika dan Aplikasinya (JPFA), 9(2), pp. 92–104. doi: 10.26740/jpfa.v9n2.p92-104.
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