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Journal of Mathematical Physics

Half theory fractional angular momentum and the application of fractional derivatives to quantum mechanics

Steve Bildstein

Published Feb 1, 2018• 7 citations
Publisher

Abstract

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.”

DOI
10.1063/1.4990102

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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.”
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 goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
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 the present study, we consider quasi-probability densities W[θ, p] and W1/2[θ, p] for pure states depending on two classical parameters θ and p, where −π ≤ θ ≤ π and p can take any real number. For integer values of p, the corresponding marginal distributions W[p] and W1/2[p] are positive and in accordance with Born’s rule in quantum mechanics for an angular momentum observable L. For the state ψ(θ) in Skagerstam and Rekdal [J. Opt. 26, 095201 (2024); arXiv:2312.16535v1 [quant-ph]], which can have half-integer angular momentum expectation values, negative values of W[θ, p] and W1/2[θ, p] can reveal non-classical features of ψ(θ) but in an ambiguous manner. In line with Franke-Arnold et al. [New J. Phys. 6, 103 (2004)], it is shown that experimental data of the uncertainties Δθ and ΔL can be sufficient to reveal quantum-mechanical features of such states ψ(θ) without necessarily making use of quasi-probability densities.