Independent Eigenstates of Angular Momentum in a Quantum N-body System

The global rotational degrees of freedom in the Schr\"{o}dinger equation for an $N$-body system are completely separated from the internal ones. After removing the motion of center of mass, we find a complete set of $(2\ell+1)$ independent base functions with the angular momentum $\ell$. These are homogeneous polynomials in the components of the coordinate vectors and the solutions of the Laplace equation, where the Euler angles do not appear explicitly. Any function with given angular momentum and given parity in the system can be expanded with respect to the base functions, where the coefficients are the functions of the internal variables. With the right choice of the base functions and the internal variables, we explicitly establish the equations for those functions. Only (3N-6) internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.Comment: 24 pages, no figure, one Table, RevTex, Will be published in Phys. Rev. A 64, 0421xx (Oct. 2001

Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed

Geometric Phase, Curvature, and Extrapotentials in Constrained Quantum Systems

We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the external curvature of the constraint manifold, the (Riemannian) curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is an example of geometric phase, closely related to that originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the external curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.Comment: 27 pages with 1 figure, submitted to Phys. Rev.

Exact Classical Formulation of the 3-dimensional Atom-diatom Reactive Collision Problem in Terms of Natural Coordinates

The classical three-dimensional atom-diatom reactive collision problem is formulated without any approximation in terms of coordinates close to but similar than the natural collision coordinates (NCC) introduced by Marcus (1968). The exact equations of motion are derived. In this context, a step-by-step and rather vivid description of the reaction, as it proceeds, is made possible.Anglai

All-particle Hamiltonians for Polyatomic-molecules .1. Body-fixed Frames and Coordinates - Classical Treatment

With a view to quantization, exact classical expressions of Hamiltonians are derived for a molecular system made up of nu-electrons and N nuclei, by applying a body-fixed (BF) matrix procedure recently introduced. Two cases are considered, depending on whether the BF frame origin is at the centre of mass of the nuclei or at the total centre of mass. The arrangement of the nuclei is described by 3N - 6 internal coordinates, and the 3-nu-electron BF Cartesian coordinates are used. All terms which contribute to the total energy are identified. The similarities and the differences between the two cases are emphasized, particularly with regard to the consequences of (i) neglecting the electron/total system mass ratio, (ii) the assumption of infinitely slow nuclear motion and the fact that (iii) there can be either no distinction between the body-fixed and the space-fixed frames, i.e. no overall rotation, or not, i.e. no total angular momentum is considered. The separability of the Hamiltonians is discussed

Overall Rotation Along a Reaction-path

It is shown that, along a reaction path defined as a line in an atomic Cartesian component configuration space (whatever the actual definition of this line, provided that it joins continuously the reagent and the product representative points through all intermediate stationary points), overall (i.e. tumbling) rotation of the body-fixed frame of reference linked with the molecular system (the only relevant definition of 'molecular rotation' for a very deformable molecular system) necessarily takes place, a point generally overlooked and even sometimes denied

Principal-axis Hyperspherical Description of N-particle Systems - Classical Treatment

Principal-axis hyperspherical coordinates (made up of one hyperradius and 3N - 7 angles as internal coordinates) and three Eulerian angles as external (rotational) coordinates are defined from the Eckart coordinate system [Phys. Rev. 46, 383 (1934)]. They can describe any N-particle system. The exact classical Hamiltonian of the system in terms of these coordinates is derived, and it is remarkably simple. The quantization of this Hamiltonian is deferred for future study