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arXiv:2301.12433

Spherical harmonics for fractional quantum numbers l=1/n

Lv, Qingzhang

Published Jan 29, 2023• 0 citations
arXivHTMLPDF

Abstract

The angular momentum quantum number L of spherical harmonic Y_l_,_m based on an associated Legendre polynomial is nonnegative integer 0 1 2 ... and must never be a fraction. But the study in this paper found that the quantum number L corresponding to other series of solutions of the associated Legendre equation should be fractions. This paper not only proposed the spherical harmonics Y for L=1/2, but also spherical harmonics Y for L = 1/n = 1/3 1/4 1/5 ... In addition to the spin s=1/2 of electron-like particles, the fractional spin such as 1/3, 1/4, 1/5,...were boldly speculated to be verified in this paper. Setting the spin of a particle with only two spin components of up and down to s = 1/2 is not necessarily correct. Based on the symmetry of the plots of Ys, three different spin classes of particles are predicted. The first class of particles s = 1/2 1/6 ... resembles electrons, particles with the parallel spins tend to move away from each other, and the second class of particles s = 1/3 1/5 ... does not repel each other regardless of whether their spins are parallel or not, and the third class of particles s = 1/4 1/8 ... always repels each other and tend to move away regardless of whether their spins are parallel or not. The use of fractional spin can be well illustrated for electrons and protons as reported in the literature. This view may be important for the study of quantum mechanics and elementary practices.

DOI
10.48550/arxiv.2301.12433

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The angular momentum quantum number L of spherical harmonic Y_l_,_m based on an associated Legendre polynomial is nonnegative integer 0 1 2 ... and must never be a fraction. But the study in this paper found that the quantum number L corresponding to other series of solutions of the associated Legendre equation should be fractions. This paper not only proposed the spherical harmonics Y for L=1/2, but also spherical harmonics Y for L = 1/n = 1/3 1/4 1/5 ... In addition to the spin s=1/2 of electron-like particles, the fractional spin such as 1/3, 1/4, 1/5,...were boldly speculated to be verified in this paper. Setting the spin of a particle with only two spin components of up and down to s = 1/2 is not necessarily correct. Based on the symmetry of the plots of Ys, three different spin classes of particles are predicted. The first class of particles s = 1/2 1/6 ... resembles electrons, particles with the parallel spins tend to move away from each other, and the second class of particles s = 1/3 1/5 ... does not repel each other regardless of whether their spins are parallel or not, and the third class of particles s = 1/4 1/8 ... always repels each other and tend to move away regardless of whether their spins are parallel or not. The use of fractional spin can be well illustrated for electrons and protons as reported in the literature. This view may be important for the study of quantum mechanics and elementary practices.
The Fermion Spherical harmonics [Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for half-odd-integer ℓ\ellℓ and mmm - presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for integer ℓ\ellℓ and mmm]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators M+M_+M+​ (M−M_-M−​) that ascend (descend) the sequence of harmonics over the values of mmm for a given value of ℓ\ellℓ, do not produce the expected result {\em in just one case}: when the value of mmm changes from ±1/2\pm{1/2}±1/2 to ∓1/2\mp{1/2}∓1/2; i.e. when mmm changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take mmm outside the range: −ℓ≤m≤+ℓ-\ell\le m\le +\ell−ℓ≤m≤+ℓ. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles, θθθ and ϕϕϕ.
In this paper, we review the fractional derivative and apply it to various problems in quantum mechanics. Among other things, we find fractional angular momentum, with “fractional spherical harmonics” as solutions to the squared quantum mechanical angular momentum operator, but with non-integer eigenvalues. These fractional functions might be interpreted as intermediary states visited by an orbiting electron as it jumps between the more stable, integer angular momentum values in an atom. Alternatively, these fractional states may be of interest in the mechanisms of chemical bonding. The justification for these new states, which are normalizable solutions to Hermitian operators, depends of course on experiment. We remember Feynmann’s statement about quantum mechanics, “Everything that is not expressly forbidden, is mandatory.”
The recently established existence of spherical harmonic functions, Yℓm(θ,ϕ)Y_\ell^{m}(θ,ϕ)Yℓm​(θ,ϕ) for half-odd-integer values of ℓ\ellℓ and mmm, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles θs,ϕsθ_s, ϕ_sθs​,ϕs​, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, S2,SzS^2, S_zS2,Sz​, are represented by the usual differential operators in spherical polar coordinates (commonly used for L2,LzL^2, L_zL2,Lz​), and their electron-spin eigenfunctions are sin⁡θsexp⁡(±ϕs/2)\sqrt{\sinθ_s} \exp(\pmϕ_s/2)sinθs​​exp(±ϕs​/2). This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, α,β,α, β,α,β, that ``integration over spin coordinates'' is a true integration (over the angles θs,ϕsθ_s, ϕ_sθs​,ϕs​). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.
Two vector operators aimed at shifting angular momentum quantum number l in spherical harmonics |lm>, primarily proposed by Prof. X. L. Ka in 2001, are further studied. For a given magnetic quantum number m, specific states |lm> in spherical harmonics with the lowest angular momentum quantum numbers l are obtained and the state with minimum angular momentum quantum number in whole set of the spherical harmonics is |0,0>. How to use these states to generate whole set of spherical harmonics is illustrated.
Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.
The spin quantum number of particles with fractional spin is neither an integer nor a half-integer. Quasiparticles knownas anyons are the most familiar examples of such particles. However, despite the considerable interest in anyons dueto their potential applications in quantum computation, a systematic analysis within the framework of quantummechanics addressing the fundamental question of whether elementary particles with fractional spin can exist is stilllacking. Two major obstacles are the requirement that spin operators satisfy an su(2) algebra and the spin-statisticstheorem. Focusing on spin-1/4 particles, we show that the first obstacle can be overcome by assuming that their spinoperators obey a deformed su(2) algebra with unequal structure constants. This symmetry breaking leads to a uniquerotational symmetry axis and imposes that the two particles forming a bound state have aligned symmetry axes andidentical spin projections. The second obstacle can be overcome by assuming that the binding interaction acts betweenparticles of different species, aligning their spins and confining their relative motion to the plane perpendicular to thesymmetry axis. Under these conditions, the braid operators describing the interchange of two spin-1/4 particles can berepresented in terms of the spin-component operator associated with the symmetry axis, closely paralleling theconventional spin-statistics connection. These results establish a framework in which spin-1/4 particles can beconsistently incorporated into quantum mechanics. Finally, we discuss the potential relevance of the proposedframework to theories beyond the Standard Model in which quarks are regarded as composite particles.