Angular Momentum Commutation Relations Research Articles

Quantum field theory for spin operator of the photon

All elementary particles in nature can be classified as fermions with half-integer spin and bosons with integer spin. Within quantum electrodynamics (QED), even though the spin of the Dirac particle is well defined, there exist open questions on the quantized description of spin of the gauge field particle -- the photon. Using quantum field theory, we discover the quantum operators for the spin angular momentum (SAM) $\boldsymbol{S}_{M}=(1/c)\int d^{3}x\boldsymbol{\pi}\times\boldsymbol{A}$ and orbital angular momentum (OAM) $\boldsymbol{L}_{M}=-(1/c)\int d^{3}x\pi^{\mu}\boldsymbol{x}\times\boldsymbol{\nabla}A_{\mu}$ of the photon, where $\pi^{\mu}$ is the conjugate canonical momentum of the gauge field $A^{\mu}$. We also reveal a perfect symmetry between the angular momentum commutation relations for Dirac fields and Maxwell fields. We derive the well-known OAM and SAM of classical electromagnetic fields from the above defined quantum operators. Our work shows that the spin and OAM operators commute which is important for simultaneously observing and separating the SAM and OAM. The correct commutation relations of orbital and spin angular momentum of the photon have applications in quantum optics, topological photonics as well as nanophotonics and can be extended in the future for the spin structure of nucleons.

Planar squeezing by quantum non-demolition measurement in cold atomic ensembles

Planar squeezed states, i.e. quantum states which are squeezed in two orthogonal spin components, have recently attracted attention due to their applications in atomic interferometry and quantum information (He et al 2012 New J. Phys. 14 093012). While canonical variables such as quadratures of the radiation field can be squeezed in at most one component, simultaneous squeezing in two orthogonal spin components can be achieved due to the angular momentum commutation relations. We present a novel scheme for planar squeezing via quantum non-demolition (QND) measurements in spin-1 systems. The QND measurement is achieved via near-resonant paramagnetic Faraday rotation probing, and the planar squeezing is obtained by sequential QND measurement of two orthogonal spin components. We compute the achievable squeezing for a variety of optical depths, initial conditions and probing strategies. The planar squeezed states generated in this way contain entanglement detectable by spin-squeezing inequalities and give an advantage relative to non-squeezed states for any precession phase angle, a benefit for single-shot and high-bandwidth magnetometry.

Gauge invariance and canonical quantization applied in the study of internal structure of gauge field systems

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However, we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relations. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relations, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.

Observables of angular momentum as observables on the Fedosov quantized sphere

In this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the two-sphere S2 by implementing the Fedosov ∗-formalism. The Fedosov ∗ is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov ∗-formalism; essentially an undeformation of Fedosov’s algorithm.

Pauli Criterion and the Vector Aharonov–Bohm Effect

After discussing the commutation relations of the kinetic angular momentum of the electron in the vector Aharonov–Bohm effect, the author shows that the Pauli criterion for admissibility of the wave function is inapplicable. The point is that the kinetic angular momentum does not satisfy the fundamental commutation relations of the angular momentum. The inapplicability of the Pauli criterion reflects the breakdown of the symmetry of the electron's motion around the solenoid.

Rotation and wobbling motion in triaxially deformed nuclei.

A quantum mechanical method of rotation and wobbling motion in triaxially deformed nuclei is represented within the framework of time-dependent Hartree-Fock theory. For such systems, the intrinsic frame is defined by imposing constraints of principal-axis frame. With aid of the canonical formulation of the constrained system, the Dirac quantization of the classical system is performed. It is shown that the commutation relations of angular momentum in the intrinsic frame then exactly satisfy the body-fixed frame. Furthermore, a method of describing large amplitude collective motion in the constrained system is proposed by extending the self-consistent collective-coordinate method.

Spin matrices for gravitons and the humblet decomposition of the angular momentum of gravitational radiation in the linearized theory

We develop spin matrices for a classical gravitational field in the linearized theory which satisfy angular-momentum commutation relations and are appropriate for a spin angular momentum of two. The same spin matrices come out of a decomposition of the angular momentum density of the linearized gravitational field into orbital and spin parts, similar to that carried out by Humblet for the electromagnetic field. To achieve this decomposition, we use the momentum density for the gravitational field obtained from the Landau-Lifshitz pseudo-tensor in the weak gravity limit. We note a formal connection between the spin angular momenta of gravitational and electromagnetic fields using the Kaluza-Klein idea.

Use of a coherent-state formalism in superconductivity and Josephson-junction calculations

An investigation is made of the extent to which harmonic-oscillator coherent states can be used to describe superconductivity and Josephson tunneling phenomena. Recent work has utilized angular-momentum commutation relations for operators that create and annihilate pairs in a Josephson device. This formalism utilizes certain approximations in obtaining angular-momentum commutation rules for the superconducting tunnel junction. We apply the coupled harmonic-oscillator approach due to Schwinger to regain angular-momentum commutation relations.

Vanishing of Dipole Matrix Elements at Level Crossings

Demonstration that the vanishing of certain coupling matrix elements at level crossings follow from angular momentum commutation relations. A magnetic dipole transition having delta M = plus or minus 1, induced near a crossing of the levels in a nonzero magnetic field, is found to have a dipole matrix element comparable to or smaller than the quotient of the level separation and the field. This result also applies in the analogous electric field electric dipole case.

Fokker-Planck equation for a superradiant laser

Recently (L2) a theoretical model for a supe, rradiant laser has been treated, which is fully described by the dynamics of ~m)mic operators only. We derivc in this paper a Fokker-Planck (FP) equation for a classical distribution function which is closely related to the atomic-density apcrator. Our equation is particularly suitable to discuss the stationary behaviour of the, superradiant laser, and generalizes the treatment given in ref. (a) because it includes also ptlase-destroying effecls wbicb arc neglected in the heatbath-atom interaction model assumed therein. Ore' slarting t)oint is the superradiance master equation (SME) obtained in ref. (1). That equation is too hard to be solved exactly because it contains simultaneously single-atom and collective operators. We, have been able to derive the associated FP equation by using a general procedure (h~cribed in ref. (% A significant check on the validity of our al)proximations is that from our FP equation we derive the same rate equations obtained from the SME in ref. (1). We deal with a, N-two-level-atom system which is formally ident, ical with a Nst)in-~-particle systcm (5). Therefore, the generic /~-th atom cart be described in terms of the spin-ttip operators lt',)~ and R~- and the inversion operator/~3,- The :V-atom system 2l N can be described by means of the collective (6) operators R::-. ~R~ and R s=Z R3/, which still have angular-momentum commutation relations, u~ ,,-~ The SME from which we start is (~)

Reduction of the Fourth-Order Asymmetric-Rotor Hamiltonian

By use of the angular-momentum commutation relations, the fourth-order asymmetric-rotor Hamiltonian for molecules of orthorhombic symmetry has been considerably simplified. The number of rotational coefficients has been reduced to 19: three coefficients of second power, six coefficients of the fourth power, and 10 coefficients of sixth power in the angular momentum.