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Chinese Physics Letters

Wavefunction of Coupled Quantum-Classical Oscillator

He Wei-Zhong, Xu Liu-Su, Zou Feng-wu

Published Apr 24, 2003• 3 citations
Publisher

Abstract

Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wavefunction of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.

DOI
10.1088/0256-307x/20/5/302

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Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wavefunction of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.
The relation between the dynamical properties of a coupled quasiparticle-oscillator system in the mixed quantum-classical and fully quantized descriptions is investigated. The system is considered as a model for applying a stepwise quantization. Features of the nonlinear dynamics in the mixed description such as the presence of a separatrix structure or regular and chaotic motion are shown to be reflected in the evolu- tion of the quantum state vector of the fully quantized system. In particular, it is demonstrated how wave packets propagate along the separatrix structure of the mixed description, and that chaotic dynamics leads to a strongly entangled quantum state vector. Special emphasis is given to viewing the system from a dyn- amical Born-Oppenheimer approximation defining integrable reference oscillators, and elucidating the role of the nonadiabatic couplings which complement this approximation into a rigorous quantization scheme.
The nature of the adiabatic approximation is investigated. This approximation is at the center of the Born-Oppenheimer approximation, commonly employed in the construction of molecular potentials. First, general behaviors of the approximation and some implications of these results are discussed. We show that the ground-state energy, calculated using the adiabatic approximation, will be lower than the calculated energy when no approximations are made. Next, a numerical and analytical application of the adiabatic approximation to systems comprising two coupled harmonic oscillators is described. These systems are used to demonstrate the results derived in the first part of the paper and to investigate how the accuracy of the approximation depends on the frequencies of the oscillators and the coupling strength.
Using the method of adiabatic invariants and the Born-Oppenheimer approximation, we have calculated the ground-state wave function of a pair of coupled oscillators in so-called semiquantum chaos. Some interesting characteristics, e.g., the similarities and differences between the wave functions in the regular and chaotic states have been found. Time-correlation functions of the wave functions and their Fourier spectra in two states have also been investigated. The sensitivity of the wave function in the chaotic state to the initial conditions has been identified.
The Born-Oppenheimer adiabatic approximation is used to describe the dynamic realization of wave-function collapse in quantum measurement. In the adiabatic limit, it is shown that the wave function of the total system formed by the measured quantum system plus the measuring apparatus can be factorized as an entangled state with correlation between adiabatic quantum states and quasiclassical motion configurations of the large system. When the apparatus effectively behaves as a classical object, this adiabatic entanglement leads to the wave-function collapse, which creates an ideal quantum measurement process.
The dynamics of systems consisting of coupled quantum–classical oscillators is numerically investigated. It is shown that, under certain conditions, the quantum oscillator exhibits chaos. When the mass of the classical oscillator increases, the chaos will be suppressed; if the energy of the system and/or the coupling strength between the two oscillators increases, chaotic behaviour of the system appears. This result will be helpful to understand the probability of the emergence of quantum chaos and may be applied to explain the spectra of complex atoms qualitatively.