A modified quantum-mechanical Boltzmann equation has been derived for the general case in which the molecules have degenerate internal states. This is an equation of the Boltzmann type for a quantity which is simultaneously a Wigner distribution function in molecular phase space, and a density matrix in internal state space. In particular, the nondiagonal terms of this density matrix have been included in the formalism, resulting in the collision term being modified from the usual Boltzmann expression. Thus the collisions are described in terms of combinations of the Lippmann-Schwinger scattering matrix rather than the collision cross section. For nondegenerate states the usual collision term is obtained again.

This paper considers certain simple and practically useful properties of Cartesian tensors in three-dimensional space which are irreducible under the three-dimensional rotation group. Ordinary tensor algebra is emphasized throughout and particular use is made of natural tensors having the least rank consistent with belonging to a particular irreducible representation of the rotation group. An arbitrary tensor of rank n may be reduced by first deriving from the tensor all its linearly independent tensors in natural form, and then by embedding these lower-rank tensors in the tensor space of rank n. An explicit reduction of third-rank tensors is given as well as a convenient specification of fourth-and fifth-rank isotropic tensors. A particular classification of the natural tensors is through a Cartesian parentage scheme, which is developed. Some applications of isotropic tensors are given.

A one-parameter solvable model for three bosons subject to ␦-function attractive interactions in one dimension with periodic boundary conditions is studied. The energy levels and wave functions are classified and given explicitly in terms of three momenta. In particular, eigenstates and eigenvalues are described as functions of the model parameter c. Some of the states are given in terms of complex momenta and represent dimer or trimer configurations for large negative c. The asymptotic behavior for small and large values of the parameter, and at thresholds between real and complex momenta, is provided. The properties of the potential energy are also discussed. ͓S1050-2947͑98͒08905-7͔PACS number͑s͒: 03.65. Ϫw, 21.45.ϩv, 12.39.Pn, 36.90.ϩf

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Classical and quantum dynamics are compared in phase space and at the hydrodynamic level for one dimensional motion. An internal potential is defined as a common object to both classical statistical mechanics and to quantum mechanics. It reduces to the quantum potential in the quantum pure state case, but its presence and roll in determining the dynamics of the system is equally valid for classical flows and quantum mixed states. A numerical example is provided to illustrate the extent of the similarity between both mechanics in a scattering process where classical and quantum transmittances are alike.