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Malaysian Journal of Science

CONSTRUCTING QUANTUM ANGULAR MOMENTUM L3 IN SPECIFIC DIRECTION BY USING U(1) GROUP

Ahmad Aqwa Rosli, Nurisya Mohd Shah

Published Sep 30, 2024• 0 citations
Publisher

Abstract

The purpose of this work is to investigate the mathematical structure of finite quantum angular momentum in a specific direction L3 which can be constructed from the representation of the U(1) group. The angular momentum eigenstate is invariant under Abelian rotation symmetry. The character group of U(1) is constructed to show that there exists an additive unitary operator for the angular momentum eigenstate, and the rotation eigenstate is invariant under it. The Weyl relation is proved by showing that the angle and angular momentum L3 are the canonical conjugate pair of observables.

DOI
10.22452/mjs.vol43no3.7

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The purpose of this work is to investigate the mathematical structure of finite quantum angular momentum in a specific direction L3 which can be constructed from the representation of the U(1) group. The angular momentum eigenstate is invariant under Abelian rotation symmetry. The character group of U(1) is constructed to show that there exists an additive unitary operator for the angular momentum eigenstate, and the rotation eigenstate is invariant under it. The Weyl relation is proved by showing that the angle and angular momentum L3 are the canonical conjugate pair of observables.
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).
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.
The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU (2). A single formula for the bases is obtained from a polar decomposition of SU (2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum jjj and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3)\text{SO}(3)SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large jjj, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.