Analogy between classical and quantum mechanics

P. A. M. Dirac

Narasimhaiengar Mukunda

Jaume Masoliver A, Ana Ros Camacho

In this paper, we revise the main aspects of the ‘Hamiltonian analogy’: the fact that optical paths are completely analogous to mechanical trajectories. We follow Schrödinger's original idea and go beyond this analogy by changing over from the Hamilton's principal function S to the wavefunction Ψ. We thus travel from classical to quantum mechanics using optics as a guide. Unfortunately, and despite its mathematical beauty and simplicity, the connection between classical and quantum mechanics through optics is nowadays hardly known and mostly ignored in university education. The present work tries to fill this gap.

Qian Shang-Wu

By comparison between equations of motion of geometrical optics(GO)and that of classical statistical mechanics(CSM),we find that there should be an analogy between GO and CSM instead of GO and classical mechanics (CM).Furthermore,by comparison between the classical limit(CL)of quantum mechanics(QM)and CSM,we find that CL of QM is CSM not CM,hence we demonstrate that QM is a natural generalization of CSM instead of CM.

J. H. Irving, Robert W. Zwanzig

This paper is concerned with certain extensions of a formal technique devised by Wigner for handling problems in quantum-statistical mechanics, especially to problems in quantum-mechanical transport processes. The approach is to find the closest possible analogy between classical and quantum-statistical mechanics, so that the extensive work in classical statistical mechanics can be utilized. This analogy is attained with the Wigner distribution function, with which averages of dynamical variables in quantum mechanics may be calculated by integrations in phase space. We will first state some basic properties of distribution functions in classical statistical mechanics, and then state the corresponding properties of the density matrix in quantum mechanics. We will define and discuss the Wigner distribution function, show that it has the desired averaging properties, and obtain the analog of the Liouville equation satisfied by this function. We will derive the analog of the Liouville equation in reduced phase space, and then obtain the equations of hydro-dynamics from quantum-statistical mechanics.

Philip Yu