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arXiv:physics/0010049

Generalized Radial Equations in a Quantum N-Body Problem

Zhong-Qi Ma, Duan, Bing, Xiaoyan Gu

Published Oct 20, 2000• 0 citations
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Abstract

We demonstrate how to separate the rotational degrees of freedom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular momentum ($\ell$) and the parity in the system can be expanded with respect to $(2\ell+1)$ base-functions, where the coefficients are the functions of the internal variables. We establish explicitly the equations for those functions, called the generalized radial equations, which are $(2\ell+1)$ coupled partial differential equations containing only $(3N-6)$ internal variables.

DOI
10.48550/arxiv.physics/0010049

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We demonstrate how to separate the rotational degrees of freedom in a quantum N-body problem completely from the internal ones. It is shown that any common eigenfunction of the total orbital angular momentum (ℓ\ellℓ) and the parity in the system can be expanded with respect to (2ℓ+1)(2\ell+1)(2ℓ+1) base-functions, where the coefficients are the functions of the internal variables. We establish explicitly the equations for those functions, called the generalized radial equations, which are (2ℓ+1)(2\ell+1)(2ℓ+1) coupled partial differential equations containing only (3N−6)(3N-6)(3N−6) internal variables.
The global rotational degrees of freedom in the Schr\"odinger equation for an N-body system are completely separated from the internal ones. After removing the motion of the center of mass, we find a complete set of (2l+1)(2\mathcal{l}+1)(2l+1) independent base functions with angular momentum l.\mathcal{l}.l. 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 $(3N\ensuremath{-}6)$ internal variables are involved both in the functions and in the equations. The permutation symmetry of the wave functions for identical particles is discussed.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
Schrödinger's equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, R9, naturally equipped with Jacobi's kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger's equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger's equation to corresponding PDEs solely defined on triangular parameters--i.e., at the level of R6/SO(3)--has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),R6) which enable us to obtain such a reduction of Schrödinger's equation of three-body systems to PDEs solely defined on triangular parameters.
In quantum mechanics, as well as in classical mechanics, the number of variables involved in the equations of motion may be reduced by utilizing the constants of motion of the system. For a set of N particles moving in a field-free space there are six integrals if N is greater than two. These correspond to the laws of conservation of linear and of angular momentum. It is the purpose of this paper to eliminate the coordinates of rotation and of translation of the total system from the Schrodinger equation and to obtain a new set of differential equations involving only 3N-6 variables.1
A general theory of quantum mechanical constants of motion of arbitrary order in the momenta is developed. Because of rotational invariance the theory assumes an elegant tensorial form. The fundamental equations for the nth-order constants of motion in m variables are set up in Cartesian coordinates. The first two equations of the set do not contain the potential function and their solutions are shown to be polynomials of degree n and n-1 with coefficients satisfying certain symmetry relations. The next two equations contain the first and the second derivatives of the potential function. From these equations all unknown functions are eliminated in the special cases n=3 and n=4. The elimination gives partial differential equations for the potential function, of second and third order for n=3, and third and fourth order for n=4. The possibility of extending the results to higher values of n is indicated.