Cambridge University Press eBooks

The Theory of Turns Adapted from Hamilton

Lawrence C. Biedenharn, James D. Louck, Peter A. Carruthers

Published Dec 28, 1984• 1 citation

Abstract

An Alternative Approach to Rotations In the preceding chapters, we have discussed the standard view of the quantum theory of angular momentum. The present chapter is concerned with developing a new view of the subject starting from the beginning with a fresh look at rotations based on a viewpoint adapted primarily from Hamilton [1], but also partly from Klein and Sommerfeld [2] and from Wigner [3] (see Note 1). This new survey will be rewarded by the concept of a novel geometric entity – the turn – which in its quantal version (Chapter 5) will be realized by an elementary operator structure. In discussing this alternative viewpoint, we are primarily interested in motivation and in suggesting concepts. Accordingly, we shall proceed, at first, intuitively. When reduced to fundamentals, the concept of an elementary (point) particle in physics (as discussed in Chapter 1) becomes synonymous with the fundamental symmetries assumed to characterize physical space. We have seen that, from the homogeneity and isotropy assumed for space, one induces the symmetries of spatial displacements (translations) and rotations; that is, one assumes that space is Euclidean and three-dimensional, having the isometry group, E (3), of rotations and translations. It is quite easy to characterize translations, since three-dimensional translations necessarily (Artin [4]) form an abelian (commutative) group. For physics, the translation generator is the (Hermitian) momentum operator p , and the associated operator generating finite displacements is realized by U ( a ) = exp(– i a · p / ħ ). This operator displaces the system by the (numerical) vector a .

DOI: 10.1017/cbo9780511759888.007

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Some Applications to Physical Problems

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Cambridge University Press eBooksDec 28, 19844 citations

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The Boson Calculus Applied to the Theory of Turns

Lawrence C. Biedenharn, James D. Louck, Peter A. Carruthers

Introduction The space of turns was shown in the preceding chapter to be the carrier space of the defining (fundamental) representation of the group SU (2) * SU (2) – the group of the symmetric top. We shall show in the present chapter how the complete representation theory of this structure can be obtained from a mapping (the generalized Jordan map) that maps the generic turn into an operator (the matrix boson operator) acting in a Hilbert space. By means of this construction we shall obtain a unified presentation of the angular momentum states, the rotation matrices, and the Wigner coefficients as well. An interesting aspect of this technique is that it leads automatically to a view of the Wigner coefficients as a form of discretized rotation matrix (Section 8 below), which in turn implies a relationship between the Wigner coefficients and Jacobi polynomials (Gel'fand et al . [1, Supplement III]). The key idea behind this unified presentation – the Jordan mapping of the turn into a matrix boson – is of more general validity than the application made here. To put this structure in a larger context we first develop, in some detail, the techniques of the boson calculus . It is hoped that this digression does not break the thread of the development too badly. Excursus on the Boson Calculus The harmonic oscillator . The boson calculus originated in the treatment of a basic physical problem, the quantum mechanics of the linear harmonic oscillator.

Cambridge University Press eBooksDec 28, 19841 citation

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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).

Cambridge University Press eBooksDec 28, 19840 citations