Quantum equations of motion and the Liouville equation

Abstract

Equations of motion for explicitly time-dependent operators in the Heisenberg and Schrödinger pictures, respectively, are reviewed. A simple transformation reduces the related equation in the Heisenberg picture to closed form. An algorithm is introduced for the classical limit which, in either picture, returns the classical equation of motion for dynamical functions. Applications of this algorithm to the equation of motion for the density matrix reduces it to the classical Liouville equation. A property related to this algorithm is established for the commutator of any two analytic functions of finite polynomials of conjugate variables. Thus, if$\hat F$ and Ĝ are two such functions and$\hat F^\prime$ and Ĝ′ contain an arbitrary permutation of variables, then

$$[\hat F,\hat G] = [\hat F^\prime ,\hat G^\prime ] + \hbar ^2 \hat{A}$$

where  is a remainder commutator. In the classical limit [$\hat F$, Ĝ] is invariant to such permutation.