Proof of the Wigner–Eckart Theorem

Chapter 6 applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and then unify them using the notion of a representation operator. We then use this framework to derive a generalized selection rule, from which the various quantum-mechanical selection rules can be derived, and we also discuss the Wigner–Eckart theorem. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the ‘Dirac bilinears’ of QED.

Tensor operators as the irreducible submodules corresponding to the adjoint representation of the quantum algebra Ŭq( su 2) are equipped with q-analogue of the Hilbert–Schmidt scalar product by using the Wigner–Eckart theorem. Then, it is used to show that the adjoint representation of the quantum algebra Ŭq( su 2) is a *-representation.

Addition of Angular Momenta— The Wigner—Eckart Theorem

The Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular non-compact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finite-dimensional and infinite-dimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

This chapter applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and use these to give a quick overview of the Wigner–Eckart theorem and selection rules. These latter subjects are then made precise using the notion of a representation operator. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the “Dirac bilinears” of QED.

The relativistic approach to electroweak properties of two-particle composite systems developed previously is generalized here to the case of nonzero spin. This approach is based on the instant form of relativistic Hamiltonian dynamics. A special mathematical technique is used for the parametrization of matrix elements of electroweak current operators in terms of form factors. The parametrization is a realization of the generalized Wigner--Eckart theorem on the Poincar\'e group, form factors are corresponding reduced matrix elements and they have the sense of distributions (generalized functions). The electroweak current matrix element satisfies the relativistic covariance conditions and in the case of electromagnetic current it also automatically satisfies the conservation law.

To honour the memory of Brian Garner Wybourne, an analysis is presented of three components of the spin-other-orbit interaction for f electrons using the kind of Lie groups he would have been familiar with. The components have been named z 6, z 8 and z 10. They all belong to the irreducible representation (IR) (30) of Racah’s group G2. Near the middle of the f shell it is often found that fewer independent blocks of numbers are needed to express their matrix elements than the Wigner–Eckart theorem, generalized to the IRs U of G2, would indicate. Each block corresponds to a given U and U ′, and possesses rows and columns labelled by the angular momenta L and L′. The number of independent blocks would be expected to be given by Racah’s multiplicity function c(UU ′ (30)); but near the middle of the shell the number c(UU ′ (20)) (or less) often suffices. For this to occur, z 8 and z 10 have to be replaced by linear combinations corresponding to IRs of the types (20)×(10) and (21)×(10) of the direct product group G2A×G2B, where A and B refer to electrons with their spins up (A) and spins down (B). A detailed example is provided by the IR (31) of G2, which occurs in the configurations f 5 through f 9. In addition, two antiHermitian operators (z a6 and z a7) that also belong to the IR (30) of G2 are discussed.

A derivation of a special case of the Wigner–Eckart theorem suitable for use in an undergraduate course in quantum mechanics is presented. This presentation is restricted to the wave mechanical formalism of quantum mechanics.

We deal here with the use of Wigner–Eckart type arguments to calculate the matrix elements of a hyperbolic vector operator [Formula: see text] by expressing them in terms of reduced matrix elements. In particular, we focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum [Formula: see text] whose components [Formula: see text], [Formula: see text], [Formula: see text] satisfy an SO(2,1) Lie algebra. We show that the commutation rules between the components of [Formula: see text] and [Formula: see text] can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner–Eckart theorem, we can calculate the matrix elements of [Formula: see text] within a representation where [Formula: see text] and [Formula: see text] are jointly diagonal.

CHAPTER 6 - TENSORS

An invariant functional, analog to the group integral associated with a Lie group, is defined for the graded Lie algebras. A sufficient condition for the vanishing of the group volume is given. Orthogonality relations of the matrix elements of the representations are obtained, and the Wigner–Eckart theorem is proved for a class of graded Lie algebras.

CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry

CHAPTER 22 - INTERACTION BETWEEN ATOMS AND RADIATION

We shall use the magnetic nuclear hyperfine structure calculation in one-electron atoms to illustrate how the Wigner—Eckart theorem can be exploited in actual calculations. So far, we have calculated the perturbed one-electron spectrum (in hydrogen or in alkali atoms) only under the assumption the nucleus (the proton in hydrogen or the alkali atomic nucleus) have a charge but no magnetic properties. Because the proton has a spin (with spin quantum number 1/2), it also has a magnetic moment, similarly for alkali atoms. The spin of the stable isotope of Na, e.g., is 3/2, and hence, it also has a nuclear magnetic moment. These nuclear magnetic moments will have a magnetic interaction with the spin magnetic moment of the electron. In addition, the motion of the electron relative to the nucleus will set up an effective magnetic field at the nucleus (proportional to the orbital angular momentum of the electron). The magnetic moment of the nucleus will also lead to a magnetic interaction caused by this field. These magnetic interactions of nuclear origin will give rise to the so-called hyperfine structure splitting of the nlsj states of the one-electron atoms.

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