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Physical Review A

Quantization of a particle with a force quadratic in the velocity

Catherine Stuckens, Donald H. Kobe

Published Nov 1, 1986• 21 citations
Publisher

Abstract

A particle with a nonconservative force quadratic in the velocity is described by an appropriate Lagrangian. The particle can be canonically quantized, but ambiguities concerning the ordering of operators arise. These ambiguities are eliminated if the Feynman path-integral procedure is used to quantize the particle.
DOI
10.1103/physreva.34.3565

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For a free falling particle moving in a media which has quadratic velocity force effect on the particle, two equivalent constants of motion, with units of energy, two Lagrangians, and two Hamiltonians are deduced. These quantities describe the dynamics of the same classical system. However, their quantization and the associated statistical mechanics (for an ensemble of particles) describe two completely different quantum and statistical systems. This is shown at first order in the dissipative parameter.
Some recent works on the path-integral formulation of nonconservative forces quadratic in velocity are examined critically. It is argued that the ambiguity resulting from ordering the operators in the Lagrangian and the Hamiltonian for this force is more serious than it is usually believed, and that one can construct a number of Hermitian Hamiltonians all satisfying Ehrenfest's theorem, but forming a set of noncommuting operators. A similar problem is encountered in the ordering of the Lagrangian for the path-integral approach. This is in addition to a number of well-known difficulties such as violation of the position-velocity uncertainty, and the differences between the invariances of the equation of motion, and the Lagrangian. An alternative way is suggested in which the damping force is replaced by a purely time-dependent term which produces the same classical motion, thus avoiding the above-mentioned difficulties. Using this formulation, wave functions and time-dependent energies are found for a quadratically damped harmonic oscillator, and for a particle moving in a dissipative medium when the friction is proportional to an arbitrary power of velocity.
It was an interested and important problem to give the description of quantization, i.e. of passing from classical physical systems to the corresponding quantum ones, from the moment that quantum mechanics came into existence. In the end we succeeded in giving it as follows: Let L be a Lagrangian function. Then the Hamiltonian function H is defined through the Legendre transformation of L. The Hamiltonian operator ( ) at time in quantum mechanics is defined from H. It should be noted that ( ) has ordering ambiguities (cf. [14]). Let be a probability amplitude at time . Then its temporal evolution can be given by the solution of the Schrodinger equation