The Classical Limit

Pimon Ajanapon

Instead of regarding the classical limit as the limit ℏ→0, an alternative view based on the physical interpretation of the elements of the density matrix is proposed. The derivation of the classical limit of quantum mechanics is carried out in two stages. First, the statistical classical limit is derived, and then, with an appropriate initial condition, the deterministic classical limit is obtained. The derivation hinges on the use of the Feynman path-integral formulation of quantum mechanics in terms of the density matrix. It is shown that deterministic (Hamilton’s or Newton’s) classical mehanics is not the classical limit of Schrödinger’s wave mechanics. The classical limit of Schrödinger’s wave mechanics is only statistical and corresponds to the classical Liouville equation. In order to obtain the deterministic classical limit, it is necessary to start out initially with a quantum mechanical mixture.

R. Shankar

William Hughes Miller

Using a classical phase space integral to approximate certain quantum mechanical operations, the classical limit of the Fredholm determinant for a general multichannel scattering system is derived. This statistical form of the classical limit of scattering is found to give exactly the same ``classical S matrix'' for the case of potential scattering as does a dynamical treatment of the classical limit. For inelastic scattering, however, the classical S matrix which results from this Fredholm determinant is only an approximation to that obtained by exact classical dynamics. Reasons for this failure of the statistical version of the classical limit for inelastic scattering are discussed.

R. Shankar

John Adair Barker, C. H. J. Johnson

The behavior of the quantum-mechanical phase shifts for scattering by a potential well is examined in the classical limit. The phase shifts tend to zero in the classical limit for all cases in which the corresponding classical trajectory does not pass through the well. This disproves by counterexample the result of Curtiss and Powers which indicated that there should be a contribution to the phase shifts even in the classical limit from classically inaccessible but energetically possible regions between two inner classical turning points. The error in the latter result appears to arise from the neglect of bound states of positive energy which exist in the classical limit (and only in that limit).