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Canadian Journal of Physics

Generalized Schwinger boson realizations and the oscillator-like coherent states of the rotation groups and the asymmetric top

P. Gulshani

Published Jul 1, 1979• 16 citations
Publisher

Abstract

Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.

DOI
10.1139/p79-139

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Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.
Using the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum we find the formula for the quantum Hamiltonian H = ia † i ij U † jl al for SU(2) rotation U , in this way we further specify the angular velocity , where is the Pauli matrix. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.
The Jordan–Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan–Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.
The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.
Wigner representations of the rotational motion of a rotator or spherical top, as well as symmetrical and arbitrary tops are presented. The unique form of the transformations to these representations is derived using a set of natural requirements. As a particular case of these representations, a Wigner representation of angular momentum orientation is found. Relations between this representation and those of irreducible tensor operators and coherent states are established. For large angular momentum l >>1, rough equations for the Wigner functions are derived, and it is shown that they are similar to the well known equation for the Wigner function for translational motion.