Quantization and classical limit of a linearly damped particle, a van der Pol system and a Duffing system (Q2755162)

In the Liouville formulation of classical mechanics the state of a given system is specified by the probability density \(F(z_1,\ldots,z_{N},t)\) and the dynamics is given by the deterministic Liouville equation NEWLINE\[NEWLINE{\partial F\over\partial t}+\sum\limits_{i=1}^{N}K_{i}{\partial F\over\partial z_{i}} =-F\sum\limits_{i=1}^{N}{\partial K_{i}\over\partial z_{i}}NEWLINE\]NEWLINE generated by the dynamical system \(dz_{i}/dt=K_{i}(z_1,\ldots,z_{N},t), i=1,\ldots,N\). Using the Wigner representation the author obtains: (i) the quantization of linearly damped particle with Newton equation \(\dot p=-\partial V/\partial q-\beta p, \dot q=p/m\), where \(q\) is the position, \(p\) is the linear momentum, \(m\) is the mass; (ii) the quantization of the van der Pol system \(\dot p=-\partial V/\partial q-\beta p(1-q^2), \dot q=p/m\); (iii) the quantization of the Duffing system \(\dot p=-\partial V/\partial q-\beta p-\kappa q^3, \dot q=p/m\). The classical limit \(\hbar\to 0\) in the quantum equations of motion is evaluated.