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

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, \(j_m (\phi,\theta,\chi)=~e^{i m \phi} \delta (\theta - \theta_m) ~e^{i(j+1/2)\chi}\), which treats...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Rakitzis, T Peter, Koutrakis, Michail E, Katsoprinakis, George E
Format: Article
Language:English
Subjects:
Angular momentum
Angular velocity
Asymptotic properties
Clebsch-Gordan coefficients
Correlation analysis
Eigenvectors
Elementary particles
Euler angles
Gyromagnetic ratio
Operators (mathematics)
Quantum mechanics
Rotation
Wave functions
Wave packets
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Summary: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, \(j_m (\phi,\theta,\chi)=~e^{i m \phi} \delta (\theta - \theta_m) ~e^{i(j+1/2)\chi}\), which treats \(j\) in the \(|jm>\) state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; \(\phi\), \(\theta\), \(\chi\) are the Euler angles, and \(cos \theta_m=(m/|j|)\) is the Vector-Model polar angle. The \(j_m (\phi,\theta,\chi)\) 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 \(\Delta m \Delta \phi \), \(\Delta j \Delta \chi\), and \(\Delta\phi\Delta\theta\), 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 \(j_m (\phi,\theta,\chi)\) 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 \(j_m (\phi,\theta,\chi)\) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.
ISSN:2331-8422