Geometry and symmetries of multi-particle systems

Abstract

Abstract. The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, by-passes all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d-dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter.We develop the systematics and technology of this approach, introducing the relevant mathematics tutorially, by analogy to the familiar theory of angular momentum in three dimensions. The angular basis functions so obtained are treated in a manifestly coordinate-free manner, thus serving as a flexible generalized basis for carrying out detailed studies of wave function evolution in multi-particle systems.

Scaling Properties of the Two-Dimensional Randomly Stirred Navier-Stokes Equation

“…[15]). The K JL 's are decaying harmonic functions in a one-to-one correspondence with the H JL 's specified by the so-called Kelvin transform [3]:…”

“…It is expedient to choose a representation of hyperspherical harmonics adapted to the group-subgroup chain adapted to SO(d n ) ⊃ SO(d) n−1 (see e.g. [11,15]). If we focus on the SO(d)-isotropic sector of C 4 as in [18] for permutation invariant states the representation is two-dimensional and all calculations can be performed explicitly [31].…”

A general rigorous quantum dynamics algorithm to calculate vibrational energy levels of pentaatomic molecules

“…That is, the resulting Hamiltonian is very compact with a partitioned structure, and has a common formulation except for the mass-dependent prefactors regardless of types of orthogonal vectors [11,13,25,34,36,49,[66][67][68]. Otherwise, if non-orthogonal coordinates [25, [69][70][71][72][73][74][75][76] are used, the kinetic energy operators obtained are rather complicated. The complicated Hamiltonian will make calculations difficult, especially for polyatomic molecules, although the basis size can be somewhat reduced for rigid or semi-rigid molecules.…”

Independent eigenstates of angular momentum in a quantumN-body system

“…Since the partial angular momenta are generally not conserved, one has to deal with, in principle, an infinite number of partial angular momentum states. This problem also occurs in the hyperspherical harmonic function method and its improved versions [17][18][19][20][21][22]. It causes unnecessary degeneracy of the hyperspherical harmonic states because, as Wigner proved, only (2ℓ + 1) base functions with angular momentum ℓ are involved in the calculation.…”

“…In the present paper we will separate the global rotational variables in the Schrödinger equation for an N-body system from the internal ones by a generalized method following the approach described above. In our approach, the number of base functions with the given angular momentum is finite, but that number in the hyperspherical harmonic function method and its improved versions [13,[17][18][19][20][21][22] is infinite due to the unconserved partial angular momenta. We also avoid the heavy differential calculus with respect to the Euler angles which is sometimes necessary for expressing kinetic energy operators.…”