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Physica Scripta

Quantization of Dissipative Systems with Friction Linear in Velocity

U. Das, Subrata K. Ghosh, Pranab Sarkar, B. Talukdar

Published Jan 1, 2005• 6 citations
Publisher

Abstract

It is shown that a dissipative system with friction linear in velocity can be analytically represented by two different Lagrangians which are harmless at the classical level. But they lead to different quantum mechanical systems. This is explicitly demonstrated by using both canonical quantization procedure as well as path integral method.

DOI
10.1238/physica.regular.071a00235

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Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0$ for a function V(q)V(q)V(q) is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.
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New method of quantization is presented. It is based on classical Newton–Lagrange equations of motion (representing the fundamental physical law of mechanics) rather than on their traditional Lagrangian and/or Hamiltonian precursors. It is shown that classical dynamics is governed by canonical two-form Ω, which embodies kinetic energy and forces acting within the system. New type of variational principle employing differential two-form Ω and "umbilical strings" is introduced. The Feynman path integral over histories of the system is then rearranged to "umbilical world-sheet" functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, world-sheet approach reduces to the standard quantum mechanics. As an example Quantum Mechanics with friction is analyzed in detail.