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CLASSICAL AND QUANTUM MECHANICS OF THE DAMPED HARMONIC OSCILLATOR
Description
The relations between various treatments of the classical linearly damped harmonic oscillator and its quantizatlon are investigated. In the course of a historical survey typical features of the problem are discussed on the basis of Havas’ classical Hamiltonian and the quantum mechanical Süssmann—Hasse—Albrecht models as coined by the München’Garching nuclear physics group. It is then shown how by imposing a restriction on the classical trajectories in order to connect the Hamiltonian with the energy, the time-independent Bateman—Morse—Feshbach—Bopp Hamiltonian leads to the time-dependent Caldirola—Kanai Hamiltonian. Canonical quantization of either formulation entails a violation of Heisenbergs principle. By means of a unified treatment of both the electrical and mechanical semi-infinite transmission line, this defect is related to the disregard of additional quantum fluctuations that are intrinsically connected with the dissipation. The difficulties of these models are discussed. Then it is proved that the Bateman dual Hamiltonian is connected to a recently developed comples svmplectic formulation by a simple canonical transformation. The undamental commutator is still in error. Therefore it is demonstrated how, either separating the dual oscillators according to a modified version of Bopps original treatment or reducing classical comples phase space by an integration over the mirror image subspace. a quantum continuity equation is obtained that leads to Dekker’s master equation following the usual operator algebra.
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