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Physical Review

Some Properties of Correlation Functions of Irreducible Tensor Operators

Paul S. Hubbard

Published Apr 5, 1969• 123 citations
Publisher

Abstract

Properties of quantum-mechanical time-dependent correlation functions of pairs of irreducible tensor operators are derived from the transformation properties of the tensor operators under rotations, the assumed rotational invariance of the Hamiltonian of the system, and the Wigner-Eckart theorem. It is shown that many of the possible correlation functions are zero, and that the nonzero correlation functions can be expressed in terms of reduced matrix elements. Some examples of irreducible tensor operators which occur in applications are given, and their reduced matrix elements are calculated.

DOI
10.1103/physrev.180.319

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Properties of quantum-mechanical time-dependent correlation functions of pairs of irreducible tensor operators are derived from the transformation properties of the tensor operators under rotations, the assumed rotational invariance of the Hamiltonian of the system, and the Wigner-Eckart theorem. It is shown that many of the possible correlation functions are zero, and that the nonzero correlation functions can be expressed in terms of reduced matrix elements. Some examples of irreducible tensor operators which occur in applications are given, and their reduced matrix elements are calculated.
As in classical mechanics, rotation in quantum mechanics is a transformation which deals with angular momentum. The difference with classical mechanics comes from the fact that angular momentum is a vector operator and not a usual vector and its components do not commute. As for any transformation in quantum mechanics, to each rotation we can associate an operator which acts in state space. The expression of this operator depends on whether the rotation is passive, that is we do a rotation of the coordinate axes and the physical system is left unchanged, or active, in which case the coordinate axes are unchanged and the rotation is performed on the physical system. In the first part (Chaps. 1 and 2) of this book, details concerning both aspects are given. Following the definition of the geometrical transformation associated with the most general rotation, we give the expression of the rotation operator for specific cases. Transformation laws for scalar fields, vector fields and spinor fields are given as well as transformation laws for scalar operators, vector operators and more generally, for operators of any rank. The second part (Chaps. 3 and 4) deals with angular momentum algebra. We define the coupling coefficients of 2, 3 and 4 angular momenta as well as the recoupling coefficients. The definition of the irreductible tensor operator, which is a generalisation of scalar and vector operators, is given as well as the Wigner-Eckart theorem. The application of this theorem to more complex cases is studied.
Abstract It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J+, J0, J−} of LASU(2). The Hamiltonian of the system is proportional to J0, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.
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.