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Technical Physics

On the quantization of solid body rotation

В. А. Антонов, Б. П. Кондратьев

Published Aug 1, 2006• 1 citation
Publisher

Abstract

A general method for solving the problem of quantization of a top is proposed that allows for finding the Hamiltonian eigenfunctions in the form of polynomials of various degree n in the Cartesian coordinates with the use of the Lamé functions. All three coordinates x, y , and z are equivalently involved in computations, as well as their relations with ellipsoidal coordinates, which makes the computations symmetric.

DOI
10.1134/s1063784206080020

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A general method for solving the problem of quantization of a top is proposed that allows for finding the Hamiltonian eigenfunctions in the form of polynomials of various degree n in the Cartesian coordinates with the use of the Lamé functions. All three coordinates x, y , and z are equivalently involved in computations, as well as their relations with ellipsoidal coordinates, which makes the computations symmetric.
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We study the quantization of three-dimensional many-body systems in rotating coordinate frames defined implicitly by frame conditions. We carry out the elimination of orientational degrees of freedom in general, giving the Hamiltonian for the NNN-particle system in a broad class of body frames in terms of frame conditions and internal coordinates. We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature.
The dynamical system of a point particle constrained on a torus is quantized à la Dirac with two kinds of coordinate systems respectively; the Cartesian and toric coordinate systems. In the Cartesian coordinate system, it is difficult to express momentum operators in coordinate representation owing to the complication in structure of the commutation relations between canonical variables. In the toric coordinate system, the commutation relations have a simple form and their solutions in coordinate representation are easily obtained with, furthermore, two quantum Hamiltonians turning up. A problem comes out when the coordinate system is transformed, after quantization, from the Cartesian to the toric coordinate system.
The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.
We reconsider the problem of quantising a particle on the DDD-dimensional sphere. Adopting a Lagrangian method of reducing the degrees of freedom, the quantum Hamiltonian is found to be the usual Schrödinger operator without any boundary term. The equivalence with the Dirac Hamiltonian approach is demonstrated, either in the cartesian or in the curvilinear basis. We also briefly comment on the path integral approach.