Quantum Theory Of Angular Momentum Research Articles

The $$q$$-Analog of the Quantum Theory of Angular Momentum: a Review from Special Functions

The $$q$$-Analog of the Quantum Theory of Angular Momentum: a Review from Special Functions

Classical problems in the theory of elasticity and the quantum theory of angular momentum

We show that applying the methods of the quantum theory of angular momentum enables us to obtain frequency equations for vibrational modes of a uniform isotropic elastic sphere, cylindrical rod, and infinite plate and results in a natural classification of these modes. We discuss how these models can be applied to describe vibrations of metal nanoparticles and semiconductor nanocrystals.

Классические задачи теории упругости и квантовая теория углового момента

Классические задачи теории упругости и квантовая теория углового момента

Recoupling Coefficients and Quantum Entropies

We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group is characterized by a quantum marginal problem -- namely, by the existence of quantum states of three particles with given eigenvalues for their reduced density operators. This generalizes Wigner's observation that the semiclassical behavior of the 6j-symbols for SU(2) -- fundamental to the quantum theory of angular momentum -- is governed by the existence of Euclidean tetrahedra. As a corollary, we deduce solely from symmetry considerations the strong subadditivity property of the von Neumann entropy. Lastly, we show that the problem of characterizing the eigenvalues of partial sums of Hermitian matrices arises as a special case of the quantum marginal problem. We establish a corresponding relation between the recoupling coefficients of the unitary and symmetric groups, generalizing a classical result of Littlewood and Murnaghan.

Deformation of a Quantum Many-Particle System by a Rotating Impurity

During the last 70 years, the quantum theory of angular momentum has been successfully applied to describing the properties of nuclei, atoms, and molecules, their interactions with each other as well as with external fields. Due to the properties of quantum rotations, the angular momentum algebra can be of tremendous complexity even for a few interacting particles, such as valence electrons of an atom, not to mention larger many-particle systems. In this work, we study an example of the latter: a rotating quantum impurity coupled to a many-body bosonic bath. In the regime of strong impurity-bath couplings the problem involves addition of an infinite number of angular momenta which renders it intractable using currently available techniques. Here, we introduce a novel canonical transformation which allows to eliminate the complex angular momentum algebra from such a class of many-body problems. In addition, the transformation exposes the problem's constants of motion, and renders it solvable exactly in the limit of a slowly-rotating impurity. We exemplify the technique by showing that there exists a critical rotational speed at which the impurity suddenly acquires one quantum of angular momentum from the many-particle bath. Such an instability is accompanied by the deformation of the phonon density in the frame rotating along with the impurity.

Fast and Accurate Evaluation of Wigner 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic

We present an efficient implementation for the evaluation of Wigner 3j, 6j, and 9j symbols. These represent numerical transformation coefficients that are used in the quantum theory of angular momentum. They can be expressed as sums and square roots of ratios of integers. The integers can be very large due to factorials. We avoid numerical precision loss due to cancellation through the use of multi-word integer arithmetic for exact accumulation of all sums. A fixed relative accuracy is maintained as the limited number of floating-point operations in the final step only incur rounding errors in the least significant bits. Time spent to evaluate large multi-word integers is in turn reduced by using explicit prime factorisation of the ingoing factorials, thereby improving execution speed. Comparison with existing routines shows the efficiency of our approach and we therefore provide a computer code based on this work.

Semiclassical analysis of the Wigner12jsymbol with one small angular momentum

We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner $9j$ symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave-functions to derive asymptotic formulas for the $9j$ symbol with small and large angular momenta. When applying the same technique to the $12j$ symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram for a $9j$ symbol. This illustrates a general geometric connection between asymptotic limits of the various $3nj$ symbols. This work contributes the first known asymptotic formula for the $12j$ symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner $15j$ symbol with two small angular momenta.

Semiclassical analysis of the Wigner9jsymbol with small and large angular momenta

We derive an asymptotic formula for the Wigner $9j$ symbol, in the limit of one small and eight large angular momenta, using a gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant noncanonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher $3\mathit{nj}$ symbols. We display without proof some asymptotic formulas for the $12j$ symbol and the $15j$ symbol in the Appendices. This work contributes an asymptotic formula of the Wigner $9j$ symbol to the quantum theory of angular momentum and serves as an example of a general method for deriving asymptotic formulas for $3\mathit{nj}$ symbols.

A note on recursive calculations of particular 9jcoefficients

The calculation of angular-momentum coupling transformation matrices can be very time consuming and alternative methods, even if they apply only in special cases, are helpful. We present a recursion relation for the calculation of particular 9j symbols used in the quantum theory of angular momentum.

On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman [SIGMA 4 (2008), 089, 18 pages] as a limit of the 9-j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a "poker dice" type probability model. The proof in this paper derives and makes use of the necessary and sufficient conditions of orthogonality in establishing orthogonality as well as indicating their geometrical significance. We also derive a 5-term recurrence relation satisfied by these polynomials.

ON SUMMATION THEOREMS FOR THE3F2(1) SERIES

Abstract. The intimate relation between the 3- j coefficient in QuantumTheory of Angular Momentum (QTAM) and the 3 F 2 (1) hypergeometricseries is exploited to derive new summation theorems, from formulas forthe 3- j coefficient. 1. IntroductionIn quantum mechanics, the algebra associated with angular momentum, isubiquitous and is extensively used in atomic, molecular and nuclear physicsstudies. The powerful tools developed over decades, from the inception of quan-tum mechanics, by E. P. Wigner (1931) and G. Racah (1942-1943), have becomean integral part of text books such as Rose [8], Edmonds [4] and Biedenharn andLouck [2]. The first connection between a Racah coefficient (or, angular mo-mentum re-coupling coefficient) and the generalized hypergeometric functionof unit argument, 4 F 3 (1), is given in the Appendix of “Multipole Fields” byRose [7]. “Angular Momentum in Quantum Mechanics” by Edmonds [4], alongwith “Elementary Theory of Angular Momentum” by Rose [8], became the firsttwo text books in this area. One of the most comprehensive review articlesabout Quantum Theory of Angular Momentum (QTAM) is by Smorodinskiiand Shelepin [10], which reveals the many facets of QTAM.The origin of the connection between the generalized hypergeometric se-ries [9] of unit argument and finite groups can be traced to the article entitled,

ON THE LINEAR COMBINANTS OF A BINARY PENCIL

AbstractLet A, B denote binary forms of order d, and let 2r−1 = (A, B)2r−1 be the sequence of their linear combinants for $1 \le r \le \lfloor\frac{d+1}{2}\rfloor$. It is known that 1, 3 together determine the pencil {A + λ B}λ∈P1 and hence indirectly the higher combinants 2r−1. In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2r−1 from the knowledge of 1 and 3. The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧2Sd for the group SL2. We give an example for the group SL3 to show that such a result may hold for other categories of representations.

Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and mathematical physics. This paper starts with an account of the basics of such a theory, which represents the most exhaustive framework in dealing with interacting many-angular momenta quantum systems. We then outline the essential features of this algebra, that can be encoded, for each fixed number N = (n + 1) of angular momentum variables, into a combinatorial object, the spin network graph, where vertices are associated with finite-dimensional, binary coupled Hilbert spaces while edges correspond to either phase or Racah transforms (implemented by 6j symbols) acting on states in such a way that the quantum transition amplitude between any pair of vertices is provided by a suitable 3nj symbol. Applications of such a combinatorial setting—both in fully quantum and in semiclassical regimes—are briefly discussed providing evidence of a unifying background structure.

Exact computation and large angular momentum asymptotics of 3nj symbols: Semiclassical disentangling of spin networks

Spin networks, namely, the 3nj symbols of quantum angular momentum theory and their generalizations to groups other than SU(2) and to quantum groups, permeate many areas of pure and applied science. The issues of their computation and characterization for large values of their entries are a challenge for diverse fields, such as spectroscopy and quantum chemistry, molecular and condensed matter physics, quantum computing, and the geometry of space time. Here we record progress both in their efficient calculation and in the study of the large j asymptotics. For the 9j symbol, a prototypical entangled network, we present and extensively check numerically formulas that illustrate the passage to the semiclassical limit, manifesting both the occurrence of disentangling and the discrete-continuum transition.

Unitary transformation between Cartesian- and polar-pixellated screens

A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian "Laguerre-Kravchuk" modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch-Gordan coefficients.

Subrepresentation semirings and an analog of 6j-symbols

Let V be a complex representation of the compact group G. The subrepresentation semiring associated to V is the set of subrepresentations of the algebra of linear endomorphisms of V with operations induced by the matrix operations. The study of these semirings has been motivated by recent advances in materials science, in which the search for microstructure-independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group SO(3). In this case, the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j-symbols familiar from the quantum theory of angular momentum. In this paper, we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by Mackey [“Multiplicity free representations of finite groups,” Pac. J. Math. 8, 503 (1958)]. We introduce a new class of symbols called twisted 6j-symbols for these groups, and we explicitly calculate the structure constants for subrepresentation semirings in terms of these symbols. Moreover, we show that these symbols satisfy analog of the standard properties of classical 6j-symbols.

Analytical scheme calculations of angular momentum coupling and recoupling coefficients

We investigate the Scheme programming language opportunities to analytically calculate the Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients that are used in the quantum theory of angular momentum. The considered coefficients are calculated by a direct evaluation of the sum formulas. The calculation results for large values of quantum angular momenta were compared with analogous calculations with FORTRAN and Java programming languages.

The bipartite Brill-Gordan locus and angular momentum

Given integers n,d,e with $1 \leqslant e < \frac{d}{2},$ let $X \subseteq {\Bbb P}^{\binom{d+n}{d}-1}$ denote the locus of degree d hypersurfaces in ${\Bbb P}^n$ which are supported on two hyperplanes with multiplicities d-e and e. Thus X is the Brill-Gordan locus associated to the partition (d-e,e). The main result of this paper is an exact determination of the Castelnuovo regularity of the ideal of X. Moreover, we show that X is r-normal for $r \geqslant 3.$ In the case of binary forms (i.e., for n = 1) we give an invariant-theoretic description of the ideal generators and, furthermore, exhibit a set of two covariants which define this locus set-theoretically. In addition to the standard cohomological tools in algebraic geometry, the proof crucially relies on the nonvanishing of certain 3j-symbols from the quantum theory of angular momentum.

Positivity of theN×Ndensity matrix expressed in terms of polarization operators

We use polarization operators known from the quantum theory of angular momentum to expand the (N × N)-dimensional density operators. Thereby we construct generalized Bloch vectors representing density matrices. We study their properties and derive positivity conditions for any N. We also apply the procedure to study Bloch vector space for a qubit and a qutrit.

Differential operators in terms of Clebsch-Gordon (C-G) coefficients and the wave equation of massless tensor fields

The quantum theory of angular momentum affords a treatment of tensors and vectors in a spherical basis. By using this theory we define the tensor differential operators: divergence, curl and gradient which act on a tensor of any rank, in terms of C-G coefficients. With these definitions we obtain a matrix representation and useful properties for those operators. An interesting application of this formalism is to find the wave equation of a tensor of any rank in a linear theory. This provides a new common way to look at the wave equations associated with both Maxwell's equations and the Maxwell-like equations for the linearized Weyl curvature tensor in gravitoelectromagnetism describing gravitational radiation on a Minkowski spacetime background.