Selected applications of hyperspherical harmonics in quantum theory

Avery, John

doi:10.1021/j100112a048

Cited by 42 publications

(63 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As before, the subscript u is an index that distinguishes different orthogonal basis functions in the same manifold; each index u is associated with a different set of good hyperspherical quantum numbers that satisfy Eqs. (19). Similarly, the new subscript a will later be used to index different antisymmetric states in the same manifold, but in this case, the K k and m k no longer constitute good quantum numbers.…”

Section: B Relative Hamiltonianmentioning

confidence: 99%

Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

2015

View full text Add to dashboard Cite

By separating the Schrödinger equation for N noninteracting spin-polarized fermions in twodimensional hyperspherical coordinates, we demonstrate that fractional quantum Hall (FQH) states emerge naturally from degeneracy patterns of the antisymmetric free-particle eigenfunctions. In the presence of Coulomb interactions, the FQH states split off from a degenerate manifold and become observable as distinct quantized energy eigenstates with an energy gap. This alternative classification scheme is based on an approximate separability of the interacting N -fermion Schrödinger equation in the hyperradial coordinate, which sheds light on the emergence of Laughlin states as well as other FQH states. An approximate good collective quantum number, the grand angular momentum K from K-harmonic few-body theory, is shown to correlate with known FQH states at many filling factors observed experimentally.

show abstract

Section: B Relative Hamiltonianmentioning

confidence: 99%

Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

2015

View full text Add to dashboard Cite

show abstract

“…62,63 We confine our attention to ferromagnetic (i.e. spinpolarized) systems in which each orbital with = 0, 1, .…”

Section: B Glda Exchange Functionalsmentioning

confidence: 99%

Exchange functionals based on finite uniform electron gases

Loos

2017

The Journal of Chemical Physics

View full text Add to dashboard Cite

We show how one can construct a simple exchange functional by extending the wellknow local-density approximation (LDA) to finite uniform electron gases. This new generalized local-density approximation (GLDA) functional uses only two quantities: the electron density ρ and the curvature of the Fermi hole α. This alternative "rung 2" functional can be easily coupled with generalized-gradient approximation (GGA) functionals to form a new family of "rung 3" meta-GGA (MGGA) functionals that we INTRODUCTIONDue to its moderate computational cost and its reasonable accuracy, Kohn-Sham (KS) density-functional theory 1,2 (DFT) has become the workhorse of electronic structure calculations for atoms, molecules and solids. 3 To obtain accurate results within DFT, one only requires the exchange and correlation functionals, which can be classified in various families depending on their physical input quantities. 4,5 These various types of functionals are classified by the Jacob's ladder of DFT 6,7 (see Fig. 1). The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density ρ. The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density ∇ρ as an extra ingredient. The third rung is composed by the so-called meta-GGA (MGGA) functionals 8 which uses, in addition to ρ and ∇ρ, the kinetic energy density τ = occ i |∇ψ i | 2 (where ψ i is an occupied molecular orbital). The infinite uniform electron gas (IUEG) or jellium 9-13 is a much studied and wellunderstood model system, and hence a logical starting point for local exchange-correlation approximations. 14-22 Though analytical models are scarce, we have recently discovered an entire new family of analytical models that one can use to develop new exchange and correlation functionals within DFT. 23-29 Indeed, we have shown that, by constraining n electrons on a surface of a three-dimensional sphere (or 3-sphere), one can create finite uniform electron gases (FUEGs). 13,27,30 Here, we show how to use these FUEGs to create a new type of exchange functionals applicable to any type of systems. We have already successfully applied this strategy to one-dimensional systems, 28,31,32 for which we have created a correlation functional based on this idea. 25,26 Moreover, we show that these alternative second-rung functionals can be easily coupled to GGA functionals to form a new family of third-rung MGGA functionals. Unless otherwise stated, we use atomic units throughout. II. THEORYWithin DFT, one can write the total exchange energy as the sum of its spin-up (σ = ↑) and spin-down (σ = ↓) contributions:2 Hartree Rung 1:Rung 2:Rung 2 : whereand ρ σ is the electron density of the spin-σ electrons. Although, for sake of simplicity, we sometimes remove the subscript σ, we only use spin-polarized quantities from hereon.The first-rung LDA exchange functional (or D30 33 ) is based on the IUEG 13 and readswhere3 A GGA functional (second rung) is defined aswhere F GGA x is th...

show abstract

“…Bochner essentially has this in his book [17]; also see Claus Müller's lectures on spherical harmonics [43] and [44]; it is probably in some notes of Calderon published in Argentina, but they are not widely available. Also see [10], [11], and recent papers [15] and [16].…”

Section: Expansion Formula For a Plane Wavementioning

confidence: 99%

Solution of the Cauchy problem for a time-dependent Schrödinger equation

Meiler

Cordero-Soto

Suslov

2008

Journal of Mathematical Physics

View full text Add to dashboard Cite

Abstract. We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.

show abstract

Selected applications of hyperspherical harmonics in quantum theory

Cited by 42 publications

References 7 publications

Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

Hyperspherical theory of the quantum Hall effect: The role of exceptional degeneracy

Exchange functionals based on finite uniform electron gases

Solution of the Cauchy problem for a time-dependent Schrödinger equation

Contact Info

Product

Resources

About