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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Angular-momentum—angle commutation relations and minimum-uncertainty states

K. Yamada

Published Jun 15, 1982• 13 citations
Publisher

Abstract

We extend the canonical commutation relations (CCR) in quantum mechanics to the case where appropriate dynamical variables are angular momenta and angles. It is found that projection operators of the resultant Weyl algebra provide us with a new and powerful way of characterizing minimum-uncertainty states, including those obtained by Carruthers and Nieto. The uniqueness theorem of the Schr\"odinger representation remains valid for extended CCR in a simple case. Finally, a wide range of applicability of our method is suggested.

DOI
10.1103/physrevd.25.3256

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We extend the canonical commutation relations (CCR) in quantum mechanics to the case where appropriate dynamical variables are angular momenta and angles. It is found that projection operators of the resultant Weyl algebra provide us with a new and powerful way of characterizing minimum-uncertainty states, including those obtained by Carruthers and Nieto. The uniqueness theorem of the Schr"odinger representation remains valid for extended CCR in a simple case. Finally, a wide range of applicability of our method is suggested.
Abstract Realizing the canonical commutation relations (CCR) [N, Θ] = - i as N = - i d/dϑ and Θ to be the multiplication by ϑ on the Hilbert space of square integrable functions on [0, 2π], in the physical literature there seems to be some contradictions concerning the Heisenberg uncertainty principle ⟨ΔN⟨ ⟨ΔΘ⟨ ≥ 1/4. The difficulties may be overcome by a rigorous mathematical analysis of the domain of state vectors, for which Heisenberg's inequality is valid. It is shown that the exponentials exp {i t N} and exp{i sΘ} satisfy some commutation relations, which are not the Weyl relations. Finally, the present work aims at a better understanding of the phase and number operators in non-Fock representations.
We perform an analysis of a graphical representation for the addition of two angular momenta, focusing our attention on the angle δ between the xy components of two angular momenta. Then we propose a new complete set of commuting observables, which differ from the usual sets that are connected by the Clebsch–Gordan coefficients. This set shows that the angle δ can be a well-defined variable in quantum mechanics. An empirical analysis of the graphical representations of the angular momenta relations, which may include the angle δ, followed by quantum mechanical considerations, leads to the vanishing of certain quantum angular momentum commutators for specific states. Therefore, although the commutators are not null in general, the quantum addition of angular momenta may be represented using classical-like diagrams.
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.