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arXiv:physics/0108029 · Physical Review A

Independent eigenstates of angular momentum in a quantumN-body system

Xiaoyan Gu, Bin Duan, Zhong-Qi Ma

Published Sep 13, 2001• 12 citations
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Abstract

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 $(2\mathcal{l}+1)$ independent base functions with angular momentum $\mathcal{l}.$ 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.

DOI
10.1103/physreva.64.042108

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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)(3N\ensuremath{-}6)(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.
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 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.
Work on the reduction of independent variables in the Schr\"odinger equation for two-particle systems from six to three is extended to give a closed expression of the variational equation for states of arbitrary total angular momentum. The method consists of writing the Hamiltonian in six coordinates, by a suitable choice of transformation equations, and evaluating integrals involving real angular-momentum functions. An existing variational equation for P states follows as a special case of the present formalism.
Rigid body rotations are described using the Cayley Klein parametrization. On quantization, the wave functions are simply polynomials in the parameters. Also, half-integral as well as integral angular momenta are allowed. The Cayley Klein parametrization provides a simple representation of both the space fixed and body fixed angular momentum operators. The construction of the allowed energies and the associated wavefunctions basically involves only algebra; no reference needs be made to the Schrödinger partial differential equations of the Euler angle parametrization.
This chapter shows that the three operators are Hermitian and (in a rotationally invariant system) commute with the Hamiltonian; therefore they are observable constants of the motion. It finds out what their physical significance is. By knowing from classical mechanics, the constant of the motion arising from rotational invariance is the angular momentum. The angular momentum may split up into two parts: orbital and intrinsic. The orbital part can always be described by the dependence of a wave function, whereas the intrinsic part is by definition contained in the transformation property of the wave function. In the case of a single component (scalar), no intrinsic angular momentum exists. There are—apart from the vector addition and the scalar product—two other ways of combining two vectors: the direct sum and the direct product. Both occur in the theory of angular momentum, in fact whenever symmetry is involved.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.