Abstract
The relations between various treatments of the classical linearly damped harmonic oscillator and its quantization 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 shown how y 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 thetime-dependent Caldirola-Kanai Hamiltonian. Canonical quantization of either formulation entails a violation of Heisenberg's 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 complex symplectic formulation by a simple canonical transformation. The fundamental commutator is still in error. Therefore it is demonstrated how, either separating the dual oscillators according to a modified version of Bopp's original treatment or reducing classical complex 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. The dissipation induces additional fluctuations. The same density operator equation is shown to arise in quantum optics in the weak coupling limit. Next, for weak friction. Hasse's pure state condition is used to derive an equivalent nonlinear but normconserving frictional Schrödinger equation. It involves a particular non-Hermitian Finally, this formalism is used to make contact with Kostin's fluid dynamical Schrödinger-Langevin equation.
Published Version
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