Matrix Elements Of Tensor Operators Research Articles

The multipole form factors and the influence of the weak interaction in e-- 6 Li elastic scattering at high energies

By using the formulas to calculate the reduced matrix elements of tensor operators, the author has completely set up the formulas for calculating the multipole form factors of nuclei with six nucleons. Applying the given formulas, the author has also determined all multipole form factors that are present in elastic scattering cross section when the nucleus is in ground state. Based on the calculated multipole form factors, the influence of weak interaction is investigated corresponding to different incident electron energies.

Programs for generating Clebsch–Gordan coefficients of SU(3) in SU(2) and SO(3) bases

Computer codes are developed to calculate Clebsch–Gordan coefficients of SU(3) in both SU(2)- and SO(3)-coupled bases. The efficiency of this code derives from the use of vector coherent state theory to evaluate the required coefficients directly without recursion relations. The approach extends to other compact semi-simple Lie groups. The codes are given in subroutine form so that users can incorporate the codes into other programs. Program summary Title of program: SU3CGVCS Catalogue identifier: ADTN Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTN Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: Persons requesting the program must sign the standard CPC non-profit use license Computers for which the program is designed and others on which it is operable: SGI Origin 2000, HP Apollo 9000, Sun, IBM SP, Pentium Operating systems under which the program has been tested: IRIX 6.5, HP UX 10.01, SunOS, AIX, Linux Programming language used: FORTRAN 77 Memory required to execute with typical data: On the HP system, it requires about 732 KBytes. Disk space used for output: 2100+2460 bytes No. of bits in a word: 32 bit integer and 64 bit floating point numbers. No. of processors used: 1 Has the code been vectorized: No No. of bytes in distributed program, including test data, etc.: 26 309 No. of lines in distributed program, including test data, etc.: 3969 Distribution format: tar gzip file Nature of physical problem: The group SU(3) and its Lie algebra su(3) have important applications, for example, in elementary particle physics, nuclear physics, and quantum optics [1–3]. The code presented is particularly relevant for the last two fields. Clebsch–Gordan (CG) coefficients are required whenever the symmetries of many-body systems are used for the evaluation of matrix elements of tensor operators. Moreover, the construction of CG coefficients for SU(3) serves as a nontrivial prototype for larger compact semi-simple Lie algebras and even for non semi-simple Lie algebras. It is the simplest Lie algebra to have multiplicity in its outer products and a non-canonical subalgebra, i.e., SO(3). Method of solution: Vector coherent state theory is first used to construct bases for the products of two irreducible representations (irreps) [4]. The bases are SU(2)-coupled so that SU(2)-reduced CG (or isoscalar factors) can be constructed naturally. The CG coefficients in the SO(3) bases are constructed subsequently from the overlaps between the SU(2) and SO(3) bases. Restriction on the complexity of the problem: The programs are limited by computer memory and the maximum size of variable arrays. As dimension overflow conditions are possible, they are flagged and can be fixed by following the directions given as part of the error message. Typical running time: The calculation time for a single SU(3) CG coefficient is very different for SU(2) and SO(3) bases. It varies between 7.3–54.1 ns in SGI Origin 2000, 0.81–5.48 ms in HP Apollo 9000, or 0.055–0.373 ms in Intel Pentium 4 for SU(2) bases while it is between 0.027–0.255 s in Intel Pentium 4 for SO(3) bases. Unusual features of the program: Intrinsic bit functions: and, or, and shift, called iand, ior, and ishft, respectively, in FORTRAN, are used for packing and unpacking the labels for the irreps. Intrinsic logical btest is used to test the bit for the phase factor.

Electric dipole moments of pendular molecules

Abstract External electric fields can halt the rotation of polar molecules and produce pendular states in which the molecular dipole μ is confined to librate over a limited angular range about the space-fixed direction of the external field. The pendular eigenfunctions are hybrids of field-free rotor states, (J,K,M) with indefinite (for K ≠ 0) or alternating (for K = 0) parities; this renders the parity of the hybrid wavefunctions indefinite for any values of the good quantum numbers K or M. As a result, the parity selection rule for the matrix elements of tensor operators in the pendular basis, notably the electric dipole moment, is lifted. We give the expectation values of the space-fixed dipole moment operator and of the orientation cosine of the figure axis and illustrate the lifting with field strength of the parity selection rule for prototype hybrid states of polar molecules. The symmetry properties of the pendular Hamiltonian are used to sort the eigenstates according to the sign of the product of...

Spectroscopy of Hexafluorides with an Odd Number of Electrons. A Tensorial Formalism Adapted to the Chain SU(2) ⊗ C I ⊃ OSh

A tensorial formalism adapted to the case of octahedral molecules or molecular ions with half-integer angular momenta is proposed as an extension of the usual formalism for the group chain O(3) ⊃ O h . We use the chain SU(2) ⊗ C I ⊃ O S h , where O S h is the octahedral group with its spinorial representations. Coupling coefficients (3 j − m, isoscalar factors, etc.) and formulas for the computation of matrix elements of tensor operators are derived for this chain. A deduction of coupling coefficients (3 C − σ, 6 C, .) and similar formulas is proposed for the non-simply reducible group O S itself.

Angular momentum and Heisenberg's correspondence principle. II. More complex matrix elements

For pt.I see ibid., vol.25, no.20, p.4045-57 (1992). Previous work on the correspondence principle limit of matrix elements of tensor operators is extended to more complex matrix elements. Limiting forms of several such matrix elements are obtained; each case yields an approximate form for an angular momentum coupling coefficient: two for the 6-j symbol and one for the 9-j symbol. These are similar to existing approximations but are more symmetrical in form and, as a result, more accurate.

Angular momentum and Heisenberg's correspondence principle

Heisenberg's form of the correspondence principle is applied to the matrix elements of tensor operators. A correspondence limit of the Wigner-Eckart theorem is obtained. It is shown that the Clebsch-Gordan coefficient tends in this limit to the product of a reduced rotation matrix element with a simple multiplicative factor. A similar, but not identical, expression has long been known as a formal limit obtaining when two of the three angular momenta involved become large. It is characteristic of correspondence principle expressions that the initial and final quantal states must be represented by some 'mean' classical orbit, the choice of which remains ambiguous. Here recursion relations are used to arrive at the optimal choice. In this way, a correspondence principle approximation to the Clebsch-Gordan coefficient is obtained which may be used with success far from the limit to which it nominally refers. This approximation is tested numerically against both exact and semiclassical computations.

Wigner-Eckart theorem for the quantum group Uq(n)

Tensor operators transforming under finite dimensional irreducible representations of the quantum group Uq(n) are defined. Using them tensor operators transforming under representations of the quantum algebra Uq(u(n)) are introduced. The Wigner-Eckart theorem on matrix elements of tensor operators defined is derived.

New relations and identities for generalized hypergeometric coefficients

Generalized hypergeometric coefficients 〈 p F q( a; b)¦λ〉 enter into the problem of constructing matrix elements of tensor operators in the unitary groups and are the expansion coefficients of a multivariable symmetric function generalization p F q( a; b; z), z = ( z 1, z 2,…, z t ), of the Gauss hypergeometric function in terms of the Schur functions e λ( z), where λ = (λ 1, λ 2,…, λ t ) is an arbitrary partition. As befits their group-theoretic origin, identities for these generalized hypergeometric coefficients characteristically involve series summed over the Littlewood-Richardson numbers g( μνλ). Identities that may be interpreted as generalizations of the Bailey, Saalschütz,… identities are developed in this paper. Of particular interest is an identity which develops in a natural way a group-theoretically defined expansion over new inhomogeneous symmetric functions. While the relations obtained here are essential for the development of the properties of tensor operators, they are also of interest from the viewpoint of special functions.

The method of spherical harmonics in the nuclear collective model

A method for constructing a complete set of spherical tensors (under the SO(3) group of physical rotations) which are functions of a l-rank tensor alpha lm constituting a set of dynamical variables for a l-pole nuclear collective mode is given. The tensors are classified according to the group chain SO(2l+1) contains/implies SO(3). They are linear combinations of the l-azimuthal spherical harmonics in the (2l+1)-dimensional space. These harmonics possess a definite magnetic quantum number. Matrix elements of tensor operators are calculated within the constructed basis. The cases of l=2 and 3 are discussed.

A simple procedure for the calculation of matrix elements of tensor operators: The transition probability matrix for an ABC spin system

A simple procedure for the calculation of matrix elements of tensor operators: The transition probability matrix for an ABC spin system

Reduced matrix elements of tensor operators

The matrix elements of non-scalar products of two irreducible double tensorial sets of operators, of any order, between arbitrarily L-S coupled, N-electron configurations are expressed as products of two recoupling coefficients and one-particle reduced matrix elements. The formulation developed here should be of wide application in various problems relating to atomic structure.

Matrix elements of tensor operators

The Wigner-Eckart theorem for matrix elements of tensor operators transforming according to a finite-dimensional representation of a non-compact semisimple Lie group is proved. The applications of this theorem are discussed.

Selection rules forG 2 vectors in the atomicf shell

In various tabulations of such spectroscopic coefficients as the matrix elements of tensor operators or fractional parentage coefficients, it is found that many entries are unexpectedly zero. A survey is made of all cases that occur in the atomicf shell and that involve the 7-dimensional vector representation of the groupG2. Direct explanations are given in terms of the group structure of the electronic configurations that comprise the shell. The techniques used depend on a splitting of the state space into “spin-up” and “spin-down” parts, and, for other cases, the extensive use of the methods of second quantization. TheF terms of the atomicf shell are found to split into three classes. The separation that this classification provides for the twoF terms belonging to the irreducible representation (31) ofG2 coincides with Racah's separation. An improved separation of theH states of (31) is described.

Reduced Matrix Elements of Tensor Operators

Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.

AN IDENTITY INVOLVING 9-j COEFFICIENTS

An identity involving 9-j coefficients which may be of some value in spectroscopic calculations is obtained. Two proofs of the identity are described. In one proof the recoupling of matrix elements of tensor operators is considered; in the other the general methods of Yutsis, Levinsonas, and Vanagas are used. Two applications of the identity to nuclear shell-model theory are described.

Tensor operators under semi-simple groups

ABSTRACTTensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.

The matrix elements of tensor operators for configurations of three equivalent electrons

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .