Quantum/Classical Phase Space Analogies

Abstract

The phase space is a space endowed with a tensor, which gives it the geometric structure, is called a symplectic form, and a real-valued function on this space called the Hamiltonian. In classical mechanics, the points in the phase space represent dynamically possible states of a physical system while the Hamiltonian determines a class of curves in the phase space that can be viewed as the dynamically possible histories of the system. A unique dynamical trajectory passes through each point in phase space. For time-independent problems the phase space is spanned by (r, p), where r is the position vector and p its associated momentum in the Hamiltonian sense. For time-dependent problems a generalized phase space (r, t, p, w) should be considered, where w is the temporal frequency. However, in many situations, only the evolution in the temporal phase space (t, w) is of interest. Throughout this chapter we refer mainly to the spatial phase space (r, p). The phase space treatment of quantum or classical systems does not provide additional information about the state of the system that cannot be retrieved otherwise, but often offers new insights into many aspects of the phenomena under study.