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Reviews of Modern Physics

Tensor equations of motion for the excitations of rotationally invariant or charge-independent systems

DAVID J. ROWE, C. Ngo-Trong

Published Apr 1, 1975• 61 citations
Publisher

Abstract

A tensor equations-of-motion formalism for the excitation of a many-body system is presented, following the same general approach as the uncoupled formalism presented in a previous article. It is designed to take explicit account of the geometrical constraints imposed on excitation operators by the requirements that stationary states should have good angular momentum and, for a charge-independent nuclear Hamiltonian, good isospin. These developments are particularly relevant for application to molecular, atomic, and nuclear systems. By recognizing the geometrical constraints and exploiting the invariance properties of the excitation operators, the equations of motion become more readily applicable to nonscalar systems; i.e. to systems with nonzero angular momentum and/or isospin.

DOI
10.1103/revmodphys.47.471

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A tensor equations-of-motion formalism for the excitation of a many-body system is presented, following the same general approach as the uncoupled formalism presented in a previous article. It is designed to take explicit account of the geometrical constraints imposed on excitation operators by the requirements that stationary states should have good angular momentum and, for a charge-independent nuclear Hamiltonian, good isospin. These developments are particularly relevant for application to molecular, atomic, and nuclear systems. By recognizing the geometrical constraints and exploiting the invariance properties of the excitation operators, the equations of motion become more readily applicable to nonscalar systems; i.e. to systems with nonzero angular momentum and/or isospin.
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We develop a new formalism to treat nuclear many-body systems using the bare nucleon–nucleon interaction. It has become evident that the tensor interaction plays an important role in nuclear many-body systems due to the role of the pion in strongly interacting systems. We take the antisymmetrized molecular dynamics (AMD) as a basic framework and add a tensor correlation operator acting on the AMD wave function using the concept of the tensor-optimized shell model. We demonstrate a systematical and straightforward formulation utilizing the Gaussian integration and differentiation method and the antisymmetrization technique to calculate all the matrix elements of the many-body Hamiltonian. We can include the three-body interaction naturally and calculate the matrix elements systematically in the progressive order of the tensor correlation operator. We call the new formalism “tensor-optimized antisymmetrized molecular dynamics”.
We discuss the rotational excitations of highly quantum many-body systems (for example, polyatomic molecules or microdroplets of helium). For a general system, the states Fί, where and ί is a rotationally invariant ground or vibrational state, are shown to be eigenfunctions of L 2 and Lz , with eigenvalues L(L+1)ħ 2 and Lħ (for even L). These wavefunctions preserve the translational invariance and the permutation and inversion symmetries of the N-particle state ί. For harmonic pair interactions, the f = 1 wavefunctions are shown to be exact eigenstates of the N-body hamiltonian. For large N, the states Fί(f=1) represent surface oscillations of the type first proposed by Bohr. An inequality for the rotational excitation energy is obtained variationally; it depends on two, three, and four-particle correlations. Other translationally invariant angular momentum eigenfunctions are also discussed.
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.
This paper develops a gauge-independent symmetry theory of non-relativistic quantum mechanical systems, in line with that previously considered in the context of classical mechanics. We first discuss at length the motivation for adopting the view that the invariance of a system K under a physical symmetry operation .<I' should be taken to mean invariance of the equation of motion of K under a certain gauge-independent unitary transformation U c( .<1'). The formal development of the theory is then carried through, and some detailed examples are presented. In particular, corresponding to every direction along which a system K happens to be translation invariant there exists a gauge-independent generator of translations which leaves K invariant but which is not, in general, a component of either the canonical or the kinetic momentum. The connexion between such invariant generators of translations and the so-called magnetic translation operators is referred to.