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

• URL: http://arxiv.org/abs/2305.11456v3
• Date: Tue, 5 Mar 2024 16:02:56 GMT
• Title: The Vector-Model Wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta
• Authors: T. Peter Rakitzis, Michail E. Koutrakis, George E. Katsoprinakis
• Abstract summary: In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum $j$. We show that a spatial wavefunction, $j_m (phi,theta,chi)$ gives a useful description of quantum-mechanical angular momentum.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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, $j_m (\phi,\theta,\chi)=~e^{i m \phi} \delta (\theta - \theta_m) ~e^{i(j+1/2)\chi}$, 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, $<j_3 m_3|j_{1X} j_{2X}|j_3 m_3>$. 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.

Related papers

• Klein-Gordon oscillators and Bergman spaces [55.2480439325792]
  We consider classical and quantum dynamics of relativistic oscillator in Minkowski space $mathbbR3,1$. The general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic functions on the K"ahler-Einstein manifold $Z_6$.
  arXiv  Detail & Related papers  (2024-05-23T09:20:56Z)
• Large Angular Momentum [0.0]
  We study the angular momentum/spin in the limit, $j to infty$. The state, $(J cdot n) | j, nrangle = j |j, n rangle $, where $J$ is the angular momentum operator and $n$ stands for a generic unit vector in $R3$, is found to behave as a classical angular momentum, $ j n $.
  arXiv  Detail & Related papers  (2024-04-23T11:15:10Z)
• Vacuum Force and Confinement [65.268245109828]
  We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
  arXiv  Detail & Related papers  (2024-02-09T13:42:34Z)
• Perfect quantum protractors [0.8246494848934447]
  Perfect quantum protractors can only exist for systems with a well-defined total angular momentum $j$. Perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around.
  arXiv  Detail & Related papers  (2023-10-19T18:00:01Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Spatial Wavefunctions of Spin [0.0]
  We present an alternative formulation of quantum mechanical angular momentum. The wavefunctions are Wigner D-functions, $D_n ms (phi, theta, chi)$. Some implications of the quantum number $n$ for fundamental particles are discussed.
  arXiv  Detail & Related papers  (2023-07-25T15:48:56Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots [0.0]
  $mathttd$-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) This paves the way to an efficient computation in either space of Shannon, R'enyi and Tsallis entropies, Onicescu energies and Fisher informations.
  arXiv  Detail & Related papers  (2021-03-24T11:08:31Z)
• The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere [0.46664938579243576]
  We show that the momentum for the particle on the hypersphere is the geometric one including the gauge potential. We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential.
  arXiv  Detail & Related papers  (2021-03-05T00:44:42Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.