scholarly journals Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids

2021 ◽  
Vol 19 (2) ◽  
Author(s):  
Arya Bangun ◽  
Arash Behboodi ◽  
Rudolf Mathar
Keyword(s):  
Quantum Mechanics ◽  
Angular Momentum ◽  
Lower Bound ◽  
Rotation Group ◽  
Inner Product ◽  
Mutual Coherence ◽  
Sensing Matrix ◽  
Regular Sampling ◽  
Theoretical Results ◽  
Sensing Matrices

AbstractMany practical sampling patterns for function approximation on the rotation group utilizes regular samples on the parameter axes. In this paper, we analyze the mutual coherence for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which appear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

scholarly journals WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

2007 ◽  
Vol 22 (35) ◽  
pp. 2675-2687 ◽  
Author(s):  
LUIS F. BARRAGÁN-GIL ◽  
ABEL CAMACHO
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Field ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Approximation Method ◽  
Bound States ◽  
The Other ◽  
Homogeneous Gravitational Field ◽  
Definition Of ◽  
Ground Energy

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

Relativity and the periodic table

2020 ◽  
Vol 378 (2180) ◽  
pp. 20190305
Author(s):  
N. C. Pyper
Keyword(s):  
Quantum Mechanics ◽  
Angular Momentum ◽  
Periodic Table ◽  
Relativistic Effect ◽  
Relativistic Quantum ◽  
Binding Energies ◽  
Heavy Elements ◽  
Oxidation States ◽  
Relativistic Quantum Mechanics ◽  
Chemical Behaviour

The periodic table provides a deep unifying principle for understanding chemical behaviour by relating the properties of different elements. For those belonging to the fifth and earlier rows, the observations concerning these properties and their interrelationships acquired a sound theoretical basis by the understanding of electronic behaviour provided by non-relativistic quantum mechanics. However, for elements of high nuclear charge, such as occur in the sixth and higher rows of the periodic table, the systematic behaviour explained by non-relativistic quantum mechanics begins to fail. These problems are resolved by realizing that relativistic quantum mechanics is required in heavy elements where electrons velocities can reach significant fractions of the velocity of light. An essentially non-mathematical description of relativistic quantum mechanics explains how relativity modifies valence electron behaviour in heavy elements. The direct relativistic effect, arising from the relativistic increase of the electron mass with velocity, contracts orbitals of low angular momentum, increasing their binding energies. The indirect relativistic effect causes valence orbitals of high angular momentum to be more effectively screened as a result of the relativistic contraction of the core orbitals. In the alkali and alkaline earths, the s orbital contractions reverse the chemical trends on descending these groups, with heavy elements becoming less reactive. For valence d and f electrons, the indirect relativistic effect enhances the reductions in their binding energies on descending the periodic table. The d electrons in the heavier coinage metals thus become more chemically active, which causes these elements to exhibit higher oxidation states. The indirect effect on d orbitals causes the chemistries of the sixth-row transition elements to differ significantly from the very similar behaviours of the fourth and fifth-row transition series. The relativistic destabilization of f orbitals causes lanthanides to be chemically similar, forming mainly ionic compounds in oxidation state three, while allowing the earlier actinides to show a richer range of chemical behaviour with several higher oxidation states. For the 7p series of elements, relativity divides the non-relativistic p shell of three degenerate orbitals into one of much lower energy with the energies of the remaining two being substantially increased. These orbitals have angular shapes and spin distributions so different from those of the non-relativistic ones that the ability of the 7p elements to form covalent bonds is greatly inhibited. This article is part of the theme issue ‘Mendeleev and the periodic table’.

Sign in / Sign up

Export Citation Format

Share Document