On the Completeness of the Classical Limit of Quantum Mechanics

Thomas A. Ryckman

Quantum mechanics originated with a paper of 24-year-old Werner Heisenberg in early July 1925 seeking "to establish a quantum-theoretical mechanics based entirely on relations between quantities that are observable in principle". Heisenberg argues that experimental conditions limit simultaneous use of both kinematic and dynamic concepts in the description of an atomic object, complementarity more broadly enjoined an essential limitation in application of classical concepts to the quantum domain. The quantum-to-classical transition argument then targets the purported universality of quantum mechanics, the claim that descriptions of "real states" of macroscopic systems can be recovered as the classical limit of the more fundamental underlying quantum mechanical ones. The looming problem in quantum-to-classical transition is wave-particle duality and in particular the principle of superposition. As quantum mechanics holds that all possible physical information about the state of the individual system is represented by Ψ, the theory must be incomplete. Einstein suggest quantum orthodoxy holds a completely different sense of quantum mechanical completeness.

Valia Allori, Nino Zanghı̀

Blanchard Ph, Cini M, Maurizio Serva

The description of physical systems obtained in the classical limit of quantum mechanics is not consistent with the one given by classical statistical mechanics. In qantum mechanics the complete description of a physical system is given by specifying the wave function, which evolves completely deterministically. Probability comes into play only when a measurement is made. Classical statistical mechanics, on the contrary, concerns systems whose description is not complete and where probability enters as a consequence of a lack of knowledge. We show that Nelson's stochastic mechanics, whose predictions are identical with those of quantum mechanics, provides nevertheless a unified description of the deterministic and random aspects of quantum mechanics which eliminates the inconsistency outlined above and gives a simple and satisfactory solution to the measurement problem.

Jorge Bernal, A. Martín-Ruiz, J. C. García-Melgarejo

Donald H. Kobe

William Hughes Miller

A classical model for the interaction of molecular systems with infrared radiation is presented. It differs from the usual ’’semiclassical theory of radiation and matter’’ in that the molecular system is treated by classical rather than quantum mechanics, and the radiation field is described, as in Jaynes’ neoclassical theory, as classical (mechanical) harmonic oscillators rather than as a classical field (i.e., via Maxwell’s equations). The classical Hamiltonian for the composite system—molecules, radiation, and their interaction—is thus that of a completely mechanical system, and its classical dynamics is determined by computing the classical trajectories of the system. Quantum mechanical interference and tunneling effects can be built into the description within the framework of the classical S-matrix theory. Even within the strictly classical limit of the model, it is shown that all dynamical effects in the interaction of radiation and matter are obtained; in the perturbative limit, for example, it is shown that absorption and induced emission, and even spontaneous emission, which is often difficult to obtain in other classical or semiclassical models, appear in a completely straightforward manner.