Basis states for equivalent electrons. II. States for the Lie group Sp(4l + 2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1724-1728
Author(s):  
William R. Ross
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Orbital Angular Momentum ◽  
Lie Group ◽  
Total Spin ◽  
Irreducible Representations ◽  
Total Orbital Angular Momentum ◽  
Intermediate Group ◽  
Slater Basis

The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of the Lie group U(4l + 2). States which are eigenfunctions of the total spin and total orbital angular momentum form the basis for irreducible representations of SO(3) × SU(2). In this paper the intermediate group Sp(4l + 2) is studied. The basis states for irreducible representations of Sp(4l + 2) are expressed in terms of the Slater basis states.

Orbital angular momentum eigenfunctions for many-particle systems

1985 ◽  
Vol 63 (7) ◽  
pp. 1719-1722 ◽  
Author(s):  
John Avery
Keyword(s):  
Angular Momentum ◽  
Symmetric Group ◽  
Orbital Angular Momentum ◽  
Particle System ◽  
Total Spin ◽  
Particle Systems ◽  
Irreducible Representations ◽  
Angular Momentum Operator ◽  
Total Orbital Angular Momentum ◽  
Orbital Angular Momentum Operator

Methods are presented for constructing eigenfunctions of the total orbital angular momentum operator of a many-particle system without the use of the Clebsch–Gordan coefficients. One of the equations derived in this paper is analogous to Dirac's identity for total spin; and through this equation, a connection is established between eigenfunctions of L2 and irreducible representations of the symmetric group Sn.

Basis states for equivalent electrons. IV. States for the Lie group SO(2l + 1) × SU(2) using U(2l + 1) × SU(2) states

1981 ◽  
Vol 59 (3) ◽  
pp. 315-324
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Irreducible Representations ◽  
Intermediate Group ◽  
Slater Basis

In this paper we examine the irreducible representations of SO(2l + 1) × SU(2) that are present when we have equivalent electrons. Basis states for these irreducible representations are defined using the basis states for U(2l + 1) × SU(2). Using earlier results, where the U(2l + 1) × SU(2) states were defined using the U(4l + 2) states, we are able to express the SO(2l + 1) × SU(2) states in terms of the Slater basis states associated with the irreducible representation (1N) of U(4l + 2). The SO(2l + 1) × SU(2) states obtained in this paper using the intermediate group U(2l + 1) × SU(2) are compared with those obtained earlier using the intermediate group Sp(4l + 2).

Basis states for equivalent electrons. I. States for the Lie group U(2l + 1) × SU(2)

1980 ◽  
Vol 58 (12) ◽  
pp. 1718-1723
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Spin Operator ◽  
Total Spin ◽  
Slater Basis

The antisymmetric Slater basis states for N equivalent electrons form the basis for an irreducible representation of U(4l + 2). When we consider the subgroup U(2l + 1) × SU(2) we obtain states which are eigenstates of the total spin operator. The basis states for the irreducible representation of U(2l + 1) × SU(2) are expressed in terms of the Slater basis states. General expressions are obtained which can easily be applied regardless of the number of electrons, the value of l, or the irreducible representation that is considered.

Basis states for equivalent electrons. III. States for the Lie group SO(2l + 1) × SU(2) using Sp(4l + 2) states

1981 ◽  
Vol 59 (2) ◽  
pp. 207-212
Author(s):  
William R. Ross
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Unitary Group ◽  
Symplectic Group ◽  
Irreducible Representations ◽  
Slater Basis

The Lie group SO(2l + 1) × EU(2) is a subgroup of the symplectic group Sp(4l + 2), which in turn is a subgroup of the unitary group U(4l + 2). The Slater basis states for N equivalent electrons form the basis for the irreducible representation (1N) of U(4l + 2). The basis states for the irreducible representations of SO(2l + 1) × SU(2) are expressed in terms of the states for irreducible representations of Sp(4l + 2). The basis states for SO(2l + 1) × SU(2) are also expressed in terms of the Slater basis states.

scholarly journals Spherical HarmonicsYlm(θ,ϕ): Positive and Negative Integer Representations ofsu(1,1)forl-mandl+m

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
H. Fakhri
Keyword(s):  
Angular Momentum ◽  
Irreducible Representation ◽  
Lie Algebra ◽  
Spherical Harmonics ◽  
Compact Manifold ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Angular Momentum Operator ◽  
Quantum Numbers ◽  
Azimuthal Quantum Number

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.

Decoupling of measured spherical tensor components of resolved J states of collisionally excited (Ar)+

1996 ◽  
Vol 74 (11-12) ◽  
pp. 955-958 ◽  
Author(s):  
O. Yenen ◽  
B. W. Moudry ◽  
D. H. Jaecks
Keyword(s):  
Angular Momentum ◽  
Density Matrix ◽  
Nonlinear Equations ◽  
Total Spin ◽  
Reflection Symmetry ◽  
Density Matrices ◽  
Total Orbital Angular Momentum ◽  
Orientation Parameters ◽  
Single Set

We measured Stoke's parameters P1, P2, P3, and P4 of excited Ar+ in coincidence with the scattered He for the two-electron process[Formula: see text]From the measured Stoke's parameters of the 4610 Å (1 Å = 10−10 m) radiation of the 2F(J = 7/2) to the 2D(J = 5/2) transition and the 4590 Å radiation of the 2F(J = 5/2) to 2D(J = 3/2) transition we determined the alignment and orientation parameters of the upper states using the Fano–Macek formalism. These parameters were then converted to the appropriate spherical tensor components in the J representation to obtain, in general eight values of [Formula: see text]. Since these states are best described by LS coupling, we decoupled the spherical tensor components of the density matrix of each J state in terms of the spherical components of the density matrices of the total orbital angular momentum L and total spin 5. The unknowns in the expansion are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], with [Formula: see text] because of reflection symmetry. We used Mathematica to calculate the decoupling coefficients and solve these nonlinear equations. The decoupling also allows us to optically determine the octopole moments of the total orbital momentum provided that [Formula: see text] is nonzero. When [Formula: see text], the two sets of equations are linearly dependent. In this case, the determination of a single set of [Formula: see text] specifies [Formula: see text].

HARDY’S THEOREM FOR GABOR TRANSFORM

2018 ◽  
Vol 106 (2) ◽  
pp. 143-159
Author(s):  
ASHISH BANSAL ◽  
AJAY KUMAR ◽  
JYOTI SHARMA
Keyword(s):  
Irreducible Representation ◽  
Lie Group ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Gabor Transform ◽  
Nilpotent Lie Group ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Hardy’S Theorem ◽  
Compact Abelian Groups

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

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