Newtonian trajectories and quantum waves in expanding force fields

Abstract

The authors study nonrelativistic particles and waves in N dimensions in a time-dependent potential V(r/l(t))/(l(t))2, which describes a force field that expands and weakens as the scale factor l increases. If l2 is a quadratic function of time, then, in a reference frame expanding with the system and employing clocks recalibrated to read a scaled time that depends on l(t), the classical and quantum evolutions can be described by a conservative Hamiltonian differing from the original one by an 'inertial' term quadratic in the position variables. The quantal 'expanding modes' form a complete set whose energies decrease and which carry current outwards from the centre of expansion. Non-equilibrium statistical ensembles can be constructed, expanding with the force field.