Transformation of phase-space volumes

We consider here the relation between volume elements in phase space in particular, the relation between dqdp and dQdP, where dq = dq dqn refers to Cartesian coordinates in a laboratory fixed coordinate system, dQ = dQ dQn refers to normal-mode coordinates, and p and P are the associated generalized conjugate momenta. 

For coordinate transformations we generally have the following relation between the volume elements dq = J dQ, where J is the absolute value of the Jacobian J, which is given by the determinant 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

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