Eckart Theorem Research Articles

Interpreting the Wigner–Eckart Theorem

The Wigner–Eckart theorem is central to the application of symmetry principles throughout atomic, molecular, and nuclear physics. Nevertheless, the theorem has a puzzling feature: it is dispensable for solving problems within these domains, since elementary methods suffice. To account for the significance of the theorem, I first contrast it with an elementary approach to calculating matrix elements. Next, I consider three broad strategies for interpreting the theorem: conventionalism, fundamentalism, and conceptualism. I argue that the conventionalist framework is unnecessarily pragmatic, while the fundamentalist framework requires more ontological commitments than necessary. Conceptualism avoids both defects, accounting for the theorem’s significance in terms of how it epistemically restructures the calculation of matrix elements. Specifically, the Wigner–Eckart theorem modularizes and unifies matrix element problems, thereby changing what we need to know to solve them.

Magnetic Ordering in Systems of Identical Particles with an Arbitrary Spin

The Wigner–Eckart theorem is used for considering the collective effects related to ordering spins in systems of identical particles in ferro- and antiferromagnetic electronic systems, as well as magnetic effects that occur in high spin systems. The Hamiltonian, written in the spin representation in the form obtained by Heisenberg, Dirac, and van Vleck used to describe spin ordering in systems of particles with spin ½, is not appropriate for a description of particle systems with a spin different from ½. “High” spin particles in the spin representation need other forms of the Hamiltonian of the exchange interaction in the spin representation. The Hamiltonian for high-spin particles has been developed from the first principles, and the effects of magnetic ordering in systems of identical particles with arbitrary spin are considered in this review. An effect of giant negative magnetoresistance in the Indium antimonide has been interpreted from the exchange contribution of a high spin heavy holes point of view.

Wigner–Eckart theorem and the false EDM of 199Hg

In neutron electric dipole moment (EDM) experiments, 199Hg is used as a comagnetometer. The comagnetometer suffers from a false EDM arising in leading order from a gradient ∂Bz∕∂z in the magnetic field. Our work concerns higher-order multipole corrections to the false EDM of 199Hg. We show that for spherical traps, all higher-order multipoles are identically zero. We further show that for the usual cylindrical traps used in EDM experiments, selection of quasi-spherical dimensions for the trap can reduce the higher-order contributions. The results indicate that trap geometry is an important consideration beyond leading order for experiments desiring to control this systematic effect.

Matrix elements of the proton–neutron symplectic model

The tensor properties of the [Formula: see text] algebra generators are determined in respect to the reduction chain [Formula: see text], which defines a shell-model coupling scheme of the proton–neutron symplectic model (PNSM). They are further used to calculate the matrix elements of the basic [Formula: see text] operators of the PNSM in the space of fully symmetric representations in the [Formula: see text]-coupled basis using a generalized Wigner–Eckart theorem. The obtained results allow further the matrix elements of any physical operator of interest, such as the relevant transition operators or the collective potential, to be calculated. As an illustration, the matrix elements of the basic irreducible tensor terms which appear in the [Formula: see text] decomposition of the long-range full major-shell mixing proton–neutron quadrupole–quadrupole interaction are presented.

From the Wigner–Eckart theorem to the Hilbert–Schmidt scalar product for an adjoint representation of the quantum algebra Up,q(su2)

Irreducible tensor operators as the irreducible submodules of an adjoint representation of the two-parametric quantum ∗-algebra [Formula: see text] are constructed by using its Jordan–Schwinger formulation on two independent [Formula: see text]-oscillator ∗-algebras. All [Formula: see text]-submodules are equipped with an appropriate Hilbert–Schmidt scalar product with the help of the Wigner–Eckart theorem. We show that with respect to this scalar product, not only the bases of all irreducible submodules of the adjoint representation are orthonormal, but also the adjoint representation is a ∗-representation.

Right SUq(2)- and left SUq−1(2)-invariances of the q-Hilbert–Schmidt Scalar products for an adjoint representation of the quantum algebra Ŭq(su2)

The Jordan–Schwinger realization is used to construct tensor operators as the even and odd dimensional irreducible submodules of an adjoint representation of the quantum algebra U ̆ q ( s u 2 ) . All U ̆ q ( s u 2 ) -submodules are equipped with the so-called left and right q -Hilbert–Schmidt scalar products by using the Wigner–Eckart theorem. The bases of all irreducible submodules of the adjoint representation are orthonormal with respect to the left q -Hilbert–Schmidt scalar product, and are orthogonal, but not normalized, with respect to the right one. Consequently, only with respect to the left q -Hilbert–Schmidt scalar product, the adjoint representation of the quantum algebra U ̆ q ( s u 2 ) on the tensor operators is a ∗ -representation. We show that both left and right q -Hilbert–Schmidt scalar products are right S U q ( 2 ) -invariant and left S U q − 1 ( 2 ) -invariant. Moreover, every irreducible submodule of the adjoint representation of the quantum algebra U ̆ q ( s u 2 ) as an associative algebra with unit, is a left quantum space for O ( S U q − 1 ( 2 ) ) and a right quantum space for O ( S U q ( 2 ) ) . Finally, it is shown that there is a natural compatibility between the coproducts and the Haar measures of the quantum groups O ( S U q − 1 ( 2 ) ) and O ( S U q ( 2 ) ) and the definitions of the left and right q -Hilbert–Schmidt scalar products on the tensor operators of the Hopf algebra U ̆ q ( s u 2 ) .

Spin-adapted matrix product states and operators.

Matrix product states (MPSs) and matrix product operators (MPOs) allow an alternative formulation of the density matrix renormalization group algorithm introduced by White. Here, we describe how non-abelian spin symmetry can be exploited in MPSs and MPOs by virtue of the Wigner-Eckart theorem at the example of the spin-adapted quantum chemical Hamiltonian operator.

Excitation of high orbital angular momentum Rydberg states with Laguerre–Gauss beams

We consider the excitation of Rydberg states through photons carrying an intrinsic orbital angular momentum degree of freedom. Laguerre–Gauss modes, with a helical wave-front structure, correspond to such a set of laser beams, which carry units of orbital angular momentum in their propagation direction, with ℓ0 the winding number. We demonstrate that, in a proper geometry setting, this orbital angular momentum can be transferred to the internal degrees of freedom of the atoms, thus violating the standard dipole selection rules. Higher orbital angular momentum states become accessible through a single photon excitation process. We investigate how the spacial structure of the Laguerre–Gauss beam affects the radial coupling strength, assuming the simplest case of hydrogen-like wavefunctions. Finally we discuss a generalization of the angular momentum coupling, in order to include the effects of the fine and hyperfine splitting, in the context of the Wigner–Eckart theorem.

Scalar product for the tensor operators of the quantum algebra Ŭq(su2) by the Wigner–Eckart theorem

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.

Wigner–Eckart theorem for the non-compact algebra 𝔰𝔩(2, ℝ)

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.

Non-abelian symmetries in tensor networks: A quantum symmetry space approach

A general framework for non-abelian symmetries is presented for matrix-product and tensor-network states in the presence of well-defined orthonormal local as well as effective basis sets. The two crucial ingredients, the Clebsch–Gordan algebra for multiplet spaces as well as the Wigner–Eckart theorem for operators, are accounted for in a natural, well-organized, and computationally straightforward way. The unifying tensor-representation for quantum symmetry spaces, dubbed QSpace, is particularly suitable to deal with standard renormalization group algorithms such as the numerical renormalization group (NRG), the density matrix renormalization group (DMRG), or also more general tensor networks such as the multi-scale entanglement renormalization ansatz (MERA). In this paper, the focus is on the application of the non-abelian framework within the NRG. A detailed analysis is presented for a fully screened spin- 3/2 three-channel Anderson impurity model in the presence of conservation of total spin, particle–hole symmetry, and SU(3) channel symmetry. The same system is analyzed using several alternative symmetry scenarios based on combinations of U(1)charge, SU(2)spin, SU(2)charge, SU(3)channel, as well as the enveloping symplectic Sp(6) symmetry. These are compared in detail, including their respective dramatic gain in numerical efficiency. In the Appendix, finally, an extensive introduction to non-abelian symmetries is given for practical applications, together with simple self-contained numerical procedures to obtain Clebsch–Gordan coefficients and irreducible operators sets. The resulting QSpace tensors can deal with any set of abelian symmetries together with arbitrary non-abelian symmetries with compact, i.e. finite-dimensional, semi-simple Lie algebras.

A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach

The Proper Generalized Decomposition (PGD) is a methodology initially proposed for the solution of partial differential equations (PDE) defined in tensor product spaces. It consists in constructing a separated representation of the solution of a given PDE. In this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting. In particular, we introduce a generalization of Eckart and Young theorem which allows to prove the convergence of the so-called progressive PGD for a large class of linear problems defined in tensor product Hilbert spaces.

Field-Induced Multipole States of Sm-Based Filled Skutterudites

We discuss microscopic aspects of multipole properties of Sm-based filled skutterudite compounds under a magnetic field by analyzing a seven-orbital Anderson model with the use of a numerical renormalization group method. In order to determine the multipole state, we evaluate the susceptibility for the multipole operator defined in the form of one-body spin-charge density by the Wigner–Eckart theorem. We show our numerical results on entropy, specific heat, and multipole susceptibility of the seven-orbital Anderson model for the case of five f electrons at an impurity site under an applied magnetic field. Here we consider three directions of the magnetic field as [0, 0, 1], [1, 1, 0], and [1, 1, 1]. For all directions, dipoles and higher-rank multipoles with the same Γ 4u symmetry are induced, but depending on the direction of the magnetic field, characteristic multipoles are found to be additionally induced. For the [1, 1, 1] direction, we observe that Γ 2u octupole moment is induced, while for [0, 0, 1]...

Loop approach to lattice gauge theories

We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space H L using the prepotential formulation of ( d + 1 ) -dimensional SU ( 2 ) lattice gauge theory. The resulting orthonormal and complete loop basis, explicitly constructed in terms of the d ( 2 d − 1 ) prepotential intertwining operators, is used to transcribe the gauge dynamics directly in H L without any redundant gauge and loop degrees of freedom. Using generalized Wigner–Eckart theorem and Biedenharn–Elliot identity in H L , we show that the above loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by real and symmetric 3 n j coefficients of the second kind (e.g., n = 6 , 10 for d = 2 , 3 respectively). The corresponding “ribbon diagrams” representing SU(2) loop dynamics are constructed. The prepotential techniques are trivially extended to include fundamental matter fields leading to a description in terms of loops and strings. The SU( N) gauge group is briefly discussed.

On loop states in loop-quantum gravity

We explicitly construct and characterize all possible independent loop states in (3 + 1)-dimensional loop-quantum gravity by regulating it on a 3D regular lattice in the Hamiltonian formalism. These loop states, characterized by the (dual) angular momentum quantum numbers, describe SU(2) rigid rotators on the links of the lattice. The loop states are constructed using the Schwinger bosons which are harmonic oscillators in the fundamental (spin half) representation of SU(2). Using the generalized Wigner–Eckart theorem, we compute the matrix elements of the volume operator in the loop basis. Some simple loop eigenstates of the volume operator are explicitly constructed.

WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

We prove Wigner–Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representations is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated. A construction of some elements of the center of Ut(su(2)) is given.

Analytic atomic wave functions of NMCSCF quality: Applications

AbstractConcise analytic, nonorthogonal but selectively orthogonalizable, modified Laguerre type atomic orbitals are developed, providing conciseness, clear physical interpretation, and near‐equivalent accuracy with numerical multi‐configuration self‐consistent field, to atomic configuration interaction calculations. By analyzing the general Eckart theorem, which introduces an energy augmentation to the variational criterion for excited states, securing against variational collapse, we use them to calculate several excited atomic states known to be in danger of suffering from variational collapse. We also use them to calculate highly excited states, applied to computation of recently measured VUV spectra of He states near the second ionization threshold. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Construction of form factors of composite systems by a generalized Wigner-Eckart theorem for the Poincaré group

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.

Simplified components of the spin-other-orbit interaction near the middle of the atomic f shell

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.

Wigner–Eckart Theorem in the Inductive Spaces

By use of the modified Wigner group projectors the reduced matrix element of Wigner-Eckart theorem is found to be an ordinary matrix element of the invariants built of the involved states and operator. This is used to derive the general form of Wigner-Eckart theorem for the inductive spaces and to propose a symmetry based procedure of the matrix elements calculations.