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arXiv:1907.00108 · Journal of Mathematical Physics

On a center-of-mass system of coordinates for symmetric classical and quantum many-body problems

Érik Amorim

Published Nov 1, 2019• 1 citation
arXivHTMLPDF

Abstract

In the context of classical or quantum many-body problems involving identical bodies, a linear change of coordinates can be constructed with the properties that it includes the center-of-mass as one of the new coordinates and preserves the inherent permutation symmetry of both the Hamiltonian and the admissible states. This has advantages over the usual system of Jacobi coordinates in the study of many-body problems for which permutation symmetry of the bodies plays an important role. This paper contains the details of the construction of this system and the proof that these properties uniquely determine it up to trivial modifications. Examples of applications to both classical and quantum problems are explored, including a generalization to problems involving groups of different species of bodies.

DOI
10.1063/1.5119131

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In the context of classical or quantum many-body problems involving identical bodies, a linear change of coordinates can be constructed with the properties that it includes the center-of-mass as one of the new coordinates and preserves the inherent permutation symmetry of both the Hamiltonian and the admissible states. This has advantages over the usual system of Jacobi coordinates in the study of many-body problems for which permutation symmetry of the bodies plays an important role. This paper contains the details of the construction of this system and the proof that these properties uniquely determine it up to trivial modifications. Examples of applications to both classical and quantum problems are explored, including a generalization to problems involving groups of different species of bodies.
The most general transformations of coordinates that allow for the exact separation of the kinetic energy operator of a quantum many-body system into total center of mass kinetic energy and internal kinetic energy are found and discussed. It is found (i) that the suitable transformations, depending on the number of particles, have a certain number of free parameters and this allows for the generalization of the Jacobi coordinates to a much larger class of coordinates with the same properties and (ii) that there is a new, uncommon, additive group structure hidden in the transformation matrices that is connected to certain geometric properties of the set of coordinates.
The permutation symmetry of Newtonian stationary-state functions – such as the total momentum and the Hamiltonian – is crucial for correct state counting in classical statistical mechanics, but plays virtually no role for classical many-body systems. In contrast, it yields novel effects for quantum many-body systems of equal/identical particles. In order to avoid the misunderstandings Jaynes (1992) has rightly stressed, basic notions, such as equal and identical particles, (in)distinguishability and exchangeability (the really relevant notion) are discussed first within classical mechanics. Then, they are applied to the weight (limiting) functions. The permutation symmetry of the wave functions follow largely from that of the weight functions. The different effects for bosons and fermions are mentioned and illustrated. For the generalization towards anyons and the fractional quantum Hall effect, some ideas for further exploration are proposed. Entanglement is treated as a consequence of the conservation laws, in particular, for the angular momentum.
A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.
Abstract A system of 4 point masses (without spin) is considered. The potential energy is assumed to be a function of the distances between the particles only. It is possible to express the kinetic energy relative to the center of mass as the kinetic energy of 3 "reduced" masses. The different possibilities for that procedure are discussed for systems with an arbitrary number of particles. The positions of the 3 reduced masses are described by 9 coordinates. 6 of them, the "internal" coordinates, are needed to determine the internal structure of the system, and 3, the "external" coordinates, for its orientation in space. The classical Hamiltonian in such a set of coordinates is derived. In a first step the Hamiltonian is expressed in terms of linear and angular momenta. Then these quantities are determined as functions of the coordinates and the conjugate momenta by infinitesimal trans-formations. From the classical Hamiltonian we get the quantum mechanical one by the well-known translation rule. It can be expressed in terms of the linear and angular momentum operators quite analogous to the classical case. For the solution of the time-independent Schrödinger equation the rotational symmetry of the system is used. The wave function is expanded in terms of the eigenfunctions of the total angular momentum. This leads to a system of coupled differential equations for the expansion-coefficients which depend on the 6 internal coordinates only. The further symmetries of the system (reflection and particle exchange) appear in symmetry properties of these functions. Differences and similarities in the solution theory for the 4-body-problem with respect to the 3-body-problem are discussed.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.
A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.