Embedding of velocity-power forces into a class of energy-dependent potentials and their quantization

Abstract

Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force ${\mathit{F}}_{\mathit{n}}$=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials ${\mathit{V}}_{\mathit{n}}$(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that ${\mathit{F}}_{\mathit{n}}$ acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces ${\mathit{F}}_{\mathit{dn}}$=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.