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Journal of Mathematical Physics

Classification of Three-Particle States According to SU3

Alex J. Dragt

Published Apr 1, 1965• 160 citations
Publisher

Abstract

It is shown that the set of states for three noninteracting particles can be put into a one-to-one correspondence with the set of irreducible representations of SU3. This classification leads to three-particle angular momentum states which treat all particles on an equal footing. The states exhibit the maximum localization compatible with a given total energy, momentum, and angular momentum. Only nonrelativistic particles are treated.

DOI
10.1063/1.1704306

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It is shown that the set of states for three noninteracting particles can be put into a one-to-one correspondence with the set of irreducible representations of SU3. This classification leads to three-particle angular momentum states which treat all particles on an equal footing. The states exhibit the maximum localization compatible with a given total energy, momentum, and angular momentum. Only nonrelativistic particles are treated.
The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.
We introduce here a new coupling scheme for three angular momenta. It relies on the properties of an operator which depends ``democratically'' upon the three individual angular momenta; this is in fact their mixed product. This operator, the total angular momentum and one of its components form a complete set of commuting observables. In the case where the three individual angular momenta are equal, the eigenstates of this set possess remarkable symmetry properties with respect to the permutation group S3.
When a classical particle moves in a central field the energy and the three components of angular momentum are all constants of the motion. The motion is an orbit whose orientation in space is fixed. To discuss the corresponding situation in quantum mechanics we must examine the commutation properties of the associated operators: If they all commute, then the quantities which they represent may all simultaneously be constants of the motion; if not, we must find as many mutually commuting operators as we can and obtain a maximal set of constants of the motion.
Abstract This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.
The angular momentum operators define a set of irreducible tensors which are unique except for a normalization constant. The normalization is conveniently defined in terms of statistical tensors which describe oriented states. The properties of the tensors discussed here include: (1) the trace of products of components of such tensors, (2) symmetry properties of the traces, and (3) the expansion of products of components of these tensors into a sum of irreducible tensors. The corresponding expansion of commutators and anticommutators of these components is also discussed briefly.
We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.