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arXiv:2504.20218 · APL Quantum

Fractional angular momentum and quasi-probability densities for angular degrees of freedom

Bo-Sture Skagerstam, Per Kristian Rekdal

Published Aug 21, 2026• 0 citations

Abstract

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.

DOI: 10.1063/5.0338403

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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.

In the present paper we construct a properly defined quantum state expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position

θ

and the corresponding angular momentum operator

L = -id/dθ

. The quantum uncertainties

Δθ

and

ΔL

for the state are well-defined and are, e.g., shown to give a lower value of the uncertainty product

ΔθΔL

than the minimal uncertainty states of Ref.\cite{Padgett_2004}. The mean value

< L >

of the state is not required to be an integer. In the case of any half-integer mean value

< L >

the state constructed exhibits a remarkable critical behavior with upper and lower bounds

Δθ< \sqrt{π^2/3 -2}

and

ΔL > 1/2

.

Abstract In the present paper we construct a properly defined quantum state ψ ( θ ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L = − i d / d θ in units of ℏ = 1 . The quantum uncertainties Δ θ and Δ L for the state are well-defined and are shown to give a lower value of the uncertainty product Δ θ Δ L in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨ L ⟩ of the state ψ ( θ ) is not required to be an integer. In the case of any half-integer mean value ⟨ L ⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δ θ = π 2 / 3 − 2 and Δ L = 1 / 2 .

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.”

Coherent angular momentum states are defined for the two-dimensional isotropic harmonic oscillator. They share many attractive properties with the familiar (Cartesian) coherent states, but are in general distinct from those states. The probabilities of obtaining particular values for the radial and angular momentum quantum numbers follow independent Poisson distributions in the new states, but not in the old. In a quasiclassical description of the oscillator, corresponding to a given classical trajectory, the uncertainty in the angular momentum of the system is smaller if the new states are used rather than the old. The new states are the natural analogs of the coherent angular momentum states introduced for the three-dimensional oscillator by Bracken and Leemon [A. J. Bracken and H. I. Leemon, J. Math. Phys. 22, 719 (1981)].

This Letter presents a derivation of the relationship between the quantum and classical descriptions of angular momentum polarization. The results involve an "uncertainty broadening" term that directly expresses the restrictions imposed by the uncertainty principle. It is argued that neglect of this term can lead to error in the interpretation of theoretical or experimental angular momentum polarization data. Functions that take the uncertainty broadening into account, appropriate for use in quantum or quasiclassical descriptions of spatial distributions of angular momenta, are defined.