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Journal of Anshan Normal University

The Commutation Relation of Angular Momentum and Rotational Symmetry

Zhao Yan

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The commutation relation of angular momentum is deduced under the base of the rotational symmetry in system.According to the relation,a united definition equation is defined.The definition equation applies to not only orbital angular momentum but also spin angular momentum.

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The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
The association and difference between general angular momentum and orbital angular momentum are discussed. Their association lies in the identity of local property (Lie algebra) between group SU (2) and SO (3) (isomorphism), while their difference is their distinctive integral property. The minimal unit ?/2 of the general angular momentum is determined by the local properties of the groups, namely the commutation relation of Lie algebra (or commutor of angular momentum). However, the minimal unit ? of orbital angular momentum depends upon the integral property of group SO (3).
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.