Bateman's dual system revisited: quantization, geometric phase and relation with the ground-state energy of the linear harmonic oscillator

2001/02/28 by Massimo Blasone, Petr Jizba · 33 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced MEMS and NEMS Technologies #Algorithm #Computer science #Energy (signal processing) #Ground state #Harmonic #Harmonic oscillator #Mathematical analysis #Mathematics #Mechanical and Optical Resonators #Phase (matter) #Photonic and Optical Devices #Physics #Quantization (signal processing) #Quantum mechanics #Relation (database) #State (computer science) #hep-th #quant-ph

paper · pdf · doi:10.1016/j.aop.2004.01.008

published in Annals of Physics 312(2), 354-397 (Elsevier BV) · 24 pages, RevTeX, 4 figures. Fully revised version

arxiv created 2001/03/18 · openalex publication_date 2004/02/27 · arxiv updated 2014/11/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

By using the Feynman-Hibbs prescription for the evolution amplitude, we quantize the system of a damped harmonic oscillator coupled to its time-reversed image, known as Bateman's dual system. The time-dependent quantum states of such a system are constructed and discussed entirely in the framework of the classical theory. The corresponding geometric (Pancharatnam) phase is calculated and found to be directly related to the ground-state energy of the 1D linear harmonic oscillator to which the 2D system reduces under appropriate constraint.

Citations

• Geometry, Topology and Physics
• Periodic Orbits and Classical Quantization Conditions
• Periodic Hamiltonian and Berry’s phase in harmonic oscillators
• Appearance of Gauge Structure in Simple Dynamical Systems
• Classical and quantum mechanics of the damped harmonic oscillator
• Quantal phase factors accompanying adiabatic changes
• The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics
• Quantum mechanics as a statistical theory
• Quantum Theory: Concepts and Methods
• On the Quantum Correction For Thermodynamic Equilibrium