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arXiv:2306.17293

Coherent loop states and angular momentum

Bruce Bartlett, Nzaganya Nzaganya

Published Jun 29, 2023• 0 citations
arXivHTMLPDF

Abstract

We study Bohr-Sommerfeld states in the context of the irreducible representations of SU(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular momentum eigenstates used in physics, and give a self-contained proof in this setting of the formula of Bothwick, Paul and Uribe for the asymptotics of the inner product of arbitrary coherent loop states. As an application, we use these states to derive Littlejohn and Yu's geometric formula for the asymptotics of the Wigner matrix elements.

DOI
10.48550/arxiv.2306.17293

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Cambridge University Press eBooks•Dec 28, 1984•0 citations
We study Bohr-Sommerfeld states in the context of the irreducible representations of SU(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular momentum eigenstates used in physics, and give a self-contained proof in this setting of the formula of Bothwick, Paul and Uribe for the asymptotics of the inner product of arbitrary coherent loop states. As an application, we use these states to derive Littlejohn and Yu's geometric formula for the asymptotics of the Wigner matrix elements.
Rotor coherent state constructions are given for the Wigner supermultiplet SU(4) contains SU(2)xSU(2) and for the special irreducible representations (N0) of the SO(5) contains SO(3) contains SO(2) group chain in exact parallel with the rotor coherent state construction for the SU(3) contains SO(3) contains SO(2) case given by Rowe, LeBlanc,, and Repka. Matrix elements of the coherent state realizations of the group generators are given in all cases by very simple expressions in terms of angular momentum Wigner coefficients involving intrinsic projection labels K. The K-matrix technique of vector coherent state theory is used to effectively elevate these K labels to the status of good quantum numbers. Analytic expressions are given for the (K K*)-matrices for many of the more important irreducible representations.
Wigner representations of the rotational motion of a rotator or spherical top, as well as symmetrical and arbitrary tops are presented. The unique form of the transformations to these representations is derived using a set of natural requirements. As a particular case of these representations, a Wigner representation of angular momentum orientation is found. Relations between this representation and those of irreducible tensor operators and coherent states are established. For large angular momentum l >>1, rough equations for the Wigner functions are derived, and it is shown that they are similar to the well known equation for the Wigner function for translational motion.
THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
Using the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum we find the formula for the quantum Hamiltonian H = ia † i ij U † jl al for SU(2) rotation U , in this way we further specify the angular velocity , where is the Pauli matrix. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.
It has been the purpose of the monograph Angular Momentum in Quantum Physics (AMQP) to develop comprehensively those aspects of angular momentum theory that are required in carrying out research in modern physics and chemistry. The emphasis there was principally on physical concepts and the symmetry principles that underlie the general applicability of angular momentum theory to a broad area of physical phenomena. Mathematical techniques were introduced within the context of the physical concepts themselves. Physical applications of angular momentum theory are to a large extent applications of specific properties of Wigner or Racah functions (more generally of 3 n-j coefficients) or of the representation functions themselves. Such applications tend to emphasize the “numerical” aspects of these functions and thus obscure the relationships to other areas of mathematics. The purpose of the present monograph is to show by specific examples the many interrelations that exist between concepts originating in angular momentum theory and various areas of mathematics. It turns out that this is a substantial task—the diversity of the interrelations is far greater than might be thought. Even a brief acquaintance with the contents of AMQP will show that the concept that recurs again and again in physical applications is that of an irreducible tensor operator with respect to a group [in our case, SU (2), the quantal rotation group]. Indeed, it is this concept that carries the mathematical apparatus of the physical theory beyond the related but much simpler results of the Lie algebraic theory of the angular momentum itself and the implied representation theory of the group SU (2).