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arXiv:2305.11456

The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta

Theodore Peter Rakitzis, Michail E. Koutrakis, George E. Katsoprinakis

Published May 19, 2023• 0 citations
arXivHTMLPDF

Abstract

In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. In contrast, here we show that a spatial wavefunction, $j_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}$, which treats $j$ in the $|jm>$ state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; $ϕ$, $θ$, $χ$ are the Euler angles, and $cos θ_m=(m/|j|)$ is the Vector-Model polar angle. The $j_m (ϕ,θ,χ)$ gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in $j$ and $m$) which predicts the effective wavepacket angular uncertainty relations for $Δm Δϕ$, $Δj Δχ$, and $ΔϕΔθ$, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive $j$-rotation measurements. We also use the $j_m (ϕ,θ,χ)$ to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, $g=2$, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to $j=1/2$, these expressions are either exact (the last two) or excellent approximations (the first two), showing that $j_m (ϕ,θ,χ)$ gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.

DOI
10.48550/arxiv.2305.11456

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Abstract We present an alternative formulation of quantum mechanical angular momentum, based on spatial wavefunctions that depend on the Euler angles φ , θ , χ , and have an additional internal projection n . The wavefunctions are Wigner D-functions, D nm s ( φ , θ , χ ) , for which the body-fixed projection quantum number n has the unusual value n = s = s s + 1 = s , or n = 0 . We show that the states D s m s ( φ , θ , χ ) of elementary particles with spin s give a gyromagnetic ratio of g = 2 for s > 0 , and we identify these as the spatial angular-momentum wavefunctions of known fundamental charged particles with spin. All known Standard-Model particles can be categorized with either value n = s or n = 0 , and all known particle reactions are consistent with the conservation of its projection in the internal frame, and with internal-frame Clebsch-Gordan coefficients of unity. Therefore, we make the case that the D nm s ( φ , θ , χ ) are useful as spatial wavefunctions for spin. Some implications and new predictions related to the quantum number n for fundamental particles are discussed, such as the proposed Dirac-fermion nature of the neutrino, the explanation of some Standard-Model structure, and some proposed dark-matter candidates.
Abstract Angular momentum in classical mechanics is given by a vector. The plane perpendicular to this vector, in accordance to central field theory, determines the space in which particle motion takes place. No such simple picture exists in quantum mechanics. The states of a particle in a central field are proportional to spherical harmonics which do not define any plane of motion. In the first part of this paper we discuss the angular distribution of particle position and compare it to the classical probabilistic approach. In the second part, the matter of addition of angular momenta is discussed. In classical mechanics this means addition of vectors, while in quantum mechanics Clebsch–Gordan coefficients have to be used. We have found classical approximations to quantum coefficients and the limit of their applicability. This analysis gives a basis for the so-called ‘vector addition model’ used in some elementary textbooks on atomic physics. It can help to better understand the addition of angular momenta in quantum mechanics.
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum ppp is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group E(2)E(2)E(2) of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the sincsincsinc function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.