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A simple expression is derived for the terms in the Baker-Campbell-Hausdorff series. One formulation of the result involves a finite number of operations with matrices of rational numbers. Generalizations are discussed.

Four-atom systems may soon be subject to state-to-state reactive scattering calculations and understanding body frames and their singularities will be an important part of this effort. This paper examines body frames in four-atom systems, building on a geometrical analysis of the nine-dimensional configuration space and the six-dimensional internal space. Kinematic rotations are an important tool in this analysis. A central role is played by the ''kinetic cube,'' the space of all asymmetric top shapes related by kinematic rotations. The singularities, multiple branches, and connectivity of the principal axis frame are examined in detail and related to the topology of the kinetic cube. The principal axis frame has singularities on all symmetric top shapes, both oblate and prolate, of both chiralities. A version of the Eckart frame, however, has singularities only on prolate symmetric top shapes of one chirality. Frame singularities are inevitable in the four-body problem and no other frame has a smaller singular set than the Eckart frame.

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order 2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in , starting with 0 , with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.

Asymptotic expressions for Clebsch-Gordan coefficients are derived from an exact integral representation. Both the classically allowed and forbidden regions are analyzed. Higher-order approximations are calculated. These give, for example, six digit accuracy when the quantum numbers are in the hundreds.

General formulas are given for the masses and spring constants of one-dimensional finite chains with linear dispersion relations, examples of which were given by Herrmann and Schmälzle in 1981 in their discussion of a well-known collision apparatus. The mathematical similarity to the problem of a Boson in a constant magnetic field is shown. The explicit formulas make a study of the continuum limit possible. This is shown to be related to the system of uniform rods studied by Bayman in 1976. Examples are given of chains with quadratic dispersion relations. Resonances that give singularities in the interaction time are discovered in certain chains of elastic spheres.