The quantisation of quadratic friction

F. Negro, Angelo Tartaglia

Subrata K. Ghosh, Supriya Chatterjee, B. Talukdar

J. Geicke

In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.

Rainer W. Hasse

The author studies the motion of N-dimensional wave packets in quadratic potentials without friction and also under the influence of two different quantum frictional potentials. Differential equations for the coordinate and momentum uncertainties and for the correlations are presented and solutions for oscillatory motion and barrier penetration are given where possible. The barrier penetration probabilities are compared with each other.

Bo-Sture Skagerstam

Angelo Tartaglia

Two examples of the motion of an object through a medium against friction are discussed, showing that the equations of motion can be derived from a lagrangian, which is no longer the difference between the kinetic and potential energy of the system. The transposition of the same problems into the quantum world is also performed using the canonical formalism and the related difficulties are discussed.