A Guide to Rotations in Quantum Mechanics

Abstract

To lay a foundation for the study and use of rotation operators in graduate quantum mechanics and in research, a thorough discussion is presented of rotations in Euclidean three space (R 3 ) and of their effect on kets in the Hilbert space of a single particle. The Wigner D-matrices are obtained and used to rotate spherical harmonics. An extensive ready-reference appendix of the properties of these matrices, expressed in a consistent notation, is provided. Careful attention is paid throughout to various conventions (e.g. active versus passive viewpoints) that are used in the literature. 466 466 466 469 470 473 473 473 474 474 475 475 475 476 476 478 478 480 480 480 481 482 496 0004-9506/87/040465$02.00 Rotation operators, their matrix representations, and their effect on quantum states are an essential part of the quantum mechanics of microscopic systems. In graduate-level treatments of this topic (cf. Gottfried 1966; Shankar 1980;Messiah 1966;Schiff 1968;Merzbacher 1970; Sakurai 1985;Bohm 1979;Cohen-Tannoudji et at. 1977) the introduction of the rotation operator in Hilbert space and the Euler angle parametrization of rotations leads in short order to the Wigner rotation matrices.* These matrices, in turn, are widely used in a variety of applications (cf. Weissbluth 1979).In our experience, newcomers to the theory of angular momentum, be they students or practicing physicists, are frequently confused by differences in the conventions and viewpoints used by various authors and by the lack of clarity in many introductory texts. Moreover, many have difficulty relating their intuitive (classical) notion of a rotation to the quantum mechanics of rotations in Hilbert spaceprimarily, we think, because their training in rotations in ~3 is poor. Although a few recent papers have dealt with rotations in Euclidean three space (~3) (Leubner 1980) and with rotation matrices (Bayha 1984), few have addressed the rotation operators per se (see, however, Wolf 1969).Our objec1:ive is to lay a foundation for the study of rotations in quantum mechanics and to provide a compendium of essential results of the theory of angular momentum. We begin in § II by discussing geometrical rotations in ~3 and the corresponding rotation matrices, both for a general rotation and in the Euler angle parametrization. (Goldstein 1980). \Ve also discuss the difference between active and passive rotation conventions -a notorious source of confusion.There follows in § III an introduction to rotations in Hilbert space. This leads to the \Vigner rotation matrices, which are illustrated by rotating the familiar spherical harmonics. We have not included proofs of the properties of these matrices, since such proofs can be found in many texts and treatises on angular momentum (Brink and Satchler 1962;Rose 1957;Biedenharn and Louck 1981;Normand 1980;Silver 1976;Cushing 1975;Wybourne 1970).t To compensate for this lack, we have listed the important properties of the Wigner matrices in a "ready-reference appendix." The mater...

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