Spin Angular Momentum Operators Research Articles

Spin and Orbital Angular Momentum in Gauge Theories: Nucleon Spin Structure and Multipole Radiation Revisited

We address and solve the long-standing gauge-invariance problem of the nucleon spin structure. Explicitly gauge-invariant spin and orbital angular momentum operators of quarks and gluons are obtained. This was previously thought to be an impossible task and opens a more promising avenue towards the understanding of the nucleon spin. Our research also justifies the traditional use of the canonical, gauge-dependent angular momentum operators of photons and electrons in the multipole-radiation analysis and labeling of atomic states and sheds much light on the related energy-momentum problem in gauge theories, especially in connection with the nucleon momentum.

Energy dispersion relations of spin-split subbands in a quantum wire and electrostatic modulation of carrier spin polarization

We numerically calculate the energy dispersion relations of the spin-split subbands in a quantum wire subjected to a transverse magnetic field in the presence of Rashba and Dresselhaus spin-orbit interactions. The spin splitting energy at zero wave vector is found to be neither equal to the bare Zeeman splitting nor linear in the magnetic field in any subband. This happens because the expectation value of the spin angular momentum operator varies along the width of the wire, causing a spatial modulation of the spin density. We also show that spin splitting energy is subband dependent and has a complex dependence on the external magnetic field. In some subbands, it can vanish entirely at nonzero values of the external magnetic field. The effective spin polarization of carriers in any subband can be changed in both magnitude and sign with an external electrostatic potential, applied, for example, via a gate terminal. This has practical applications in quantum computing and other areas.

Comment on “Legendre-transform functionals for spin-density-functional theory” [J. Chem. Phys. 124, 224108 (2006)

In this Comment the authors propose an alternative formulation of the starting point to the analysis performed in the commented paper [P. W. Ayers and W. Yang, J. Chem. Phys.124, 224108 (2006)]. An important role of the z component of the vector of the total spin angular momentum operator for the considered many-electron system is indicated. The authors believe their approach to be mathematically cleaner and conceptually easier than the original treatment.

Spin squeezing and precision probing with light and samples of atoms in the Gaussian description

We consider an ensemble of trapped atoms interacting with a continuous wave laser field. For sufficiently polarized atoms and for a polarized light field, we may approximate the non-classical components of the collective spin angular momentum operator for the atoms and the Stokes vectors of the field by effective position and momentum variables for which we assume a gaussian state. Within this approximation, we present a theory for the squeezing of the atomic spin by polarization rotation measurements on the probe light. We derive analytical expressions for the squeezing with and without inclusion of the noise effects introduced by atomic decay and by photon absorption. The theory is readily adapted to the case of inhomogeneous light-atom coupling [A. Kuzmich and T.A.B. Kennedy, Phys. Rev. Lett. Vol. 92, 030407 (2004)]. As a special case, we show how to formulate the theory for an optically thick sample by slicing the gas into pieces each having only small photon absorption probability. Our analysis of a realistic probing and measurement scheme shows that it is the maximally squeezed component of the atomic gas that determines the accuracy of the measurement.

Generalized one-electron spin functions and self-similarity measures

A simple method for construction of eigenfunctions of one-electron spin angular momentum operators from products of the primitive one-electron spin functions is presented. Properties of these functions and their applications to the evaluation of some integrals met in theory of quantum similarity are briefly discussed.

Analysis of Dipolar-Coupling-Mediated Coherence Transfer in a Homonuclear Two Spin-[formula omitted] Solid-State System

Homonuclear dipolar-mediated coherence transfer (DCT), a through-space transfer of magnetization between like spins, can yield otherwise difficult-to-obtain structural information for macromolecules by measuring the internuclear distances between two sites of interest. The behavior of a spin-12 system under DCT is analyzed in detail by computing the time development of the density matrix using the product operator formalism. The effect of coherence transfer (CT) via the homonuclear isotropic scalar coupling on DCT is examined. Analytical and computational results that yield useful information on the frequencies, first-maxima, and first-zero of CT for a uniaxially oriented or a single-crystal solid-state system are presented. The results predict that the evolution of the spin angular momentum operators under the homonuclear dipolar coupling Hamiltonian leads to “cylindrical mixing” unlike “isotropic mixing” due to the strong scalar coupling Hamiltonian. These results will find relevance in both the design of RF pulse sequences for the structural studies of uniaxially oriented biological solids and the interpretation of solution NMR results from proteins embedded in partially oriented bicelles.

Momentum and spin of a particle with spin unity

The first part of this paper deals with the problem of expressing a given boost as the product of an operator which leaves the momentum of a given particle, massive or massless, invariant and a helicity conserving operator. This is done by multiplying boosts and rotations only. In this way more information is obtained than by applying the operators to the relevant vectors. An operator with the property of conserving spin while perhaps changing momentum is developed. The second part of the paper defines a spin angular momentum operator leaving two four-vectors invariant. These vectors are interpreted as the momentum and spin vectors of a massive particle and the theory is extended to accommodate a massless particle. This theory emphasizes the close association of spin and momentum in a relativistic theory. The spin angular momentum operator of a massless particle has invariance properties which resemble some of those of the gauge invariant electromagnetic theory.

Implications of color gauge symmetry for nucleon spin structure

We study the chromodynamical gauge symmetry in relation to the internal spin structure of the nucleon. We show that 1) even in the helicity eigenstates the gauge-dependent spin and orbital angular momentum operators do not have gauge-independent matrix element; 2) the evolution equations for the gluon spin take very different forms in the Feynman and axial gauges, but yield the same leading behavior in the asymptotic limit; 3) the complete evolution of the gauge-dependent orbital angular momenta appears intractable in the light-cone gauge. We define a new gluon orbital angular momentum distribution $L_g(x)$ which {\it is} an experimental observable and has a simple scale evolution. However, its physical interpretation makes sense only in the light-cone gauge just like the gluon helicity distribution $\Delta g(x)$y.

Hydrodynamic description of collective modes in heavy-fermion superconductors.

We present a hydrodynamical theory for the collective modes of a p-wave superconductor that is described by an order parameter d which transforms as a vector under rotations generated by the spin angular-momentum operator J=L+S. Such superconducting order is possible in metals with cubic, hexagonal, or tetragonal crystal symmetry whose pairing electrons experience strong spin-orbit coupling. This is true of the known heavy-fermion superconductors. The variables J and d follow a Hamiltonian dynamics that results in an internal Josephson effect between the populations of spin-up and spin-down Cooper pairs. The effect is similar to the longitudinal spin-resonance behavior observed in the superfluid phases of $^{3}\mathrm{He}$. Viscous corrections to the spin angular-momentum conservation of the pairs and to the momentum conservation of the bulk are introduced, serving to couple this Josephson mode to sound propagation. The coupling, which exists for both longitudinal and transverse sound, results in a resonant absorption contribution to the sound attenuation constant with a Lorentzian shape centered at the longitudinal spin-orbit resonance frequency. The height of this Lorentzian varies as ${\ensuremath{\omega}}^{2}$ with frequency, characteristic of hydrodynamics, and its width depends upon frequency and the amount of quasiparticle scattering. It is suggested that this mechanism accounts for the sound absorption peak recently observed in the heavy-fermion superconductor ${\mathrm{UBe}}_{13}$.

H and H2 NMR properties in amorphous hydrogenated silicon (a-Si:H).

It is shown that the basic NMR properties of ortho-${\mathrm{H}}_{2}$ molecules with a rotational angular momentum J and a spin angular momentum I under the influence of a completely asymmetric crystalline field in an amorphous matrix can be described by an effective nuclear spin Hamiltonian which contains only the nuclear spin angular momentum operators (${I}_{i}$), but is independent of the molecular rotational angular momentum operators (${J}_{i}$). By directly applying the existing magnetic-resonance theories to this effective nuclear spin Hamiltonian, a simple description is presented for various static and dynamic NMR properties of the ortho-${\mathrm{H}}_{2}$ NMR centers in amorphous hydrogenated silicon (a-Si:H), thereby resolving many difficulties and uncertainties encountered in understanding and explaining the H and ${\mathrm{H}}_{2}$ NMR observations in a-Si:H.

Microwave properties of (Fe0.82B0.18)0.09La0.05R0.05 amorphous metal alloys

Static magnetic field measurements have been reported on amorphous metal alloys of (Fe0.82B0.18)0.90La0.05R0.05, where R=La, Pr, Nd, Sm, Gd, Tb, Dy, Ho, and Er. Microwave properties of these alloys are now reported. The microwave measurements consisted of fixed (9 GHz) and swept (0.01–18 GHz) frequency magnetic resonance experiments. The ferromagnetic resonance (FMR) linewidth, ΔH and g factor were measured. ΔH has a minimum for R=Gd and is equal to 100 Oe. ΔH appears to scale with 〈L⋅S〉 where L and S are the spin and orbital angular momentum operators of the R ion. The measured g factor obeys the Wangsness relation, as R is varied.

Unitary transformations with spin angular momentum operators

Abstract Useful formulae for replacing an exponentiated spin scalar product by a finite sum of powers of this product, are presented in the context of Coulomb interactions between magnetic ions and conduction electrons in metals.

Stability analysis for singular non-Abelian magnetic monopoles

The Dirac magnetic monopole potential g A(r) = (g/4π)( r ̂ ×^n )(r− r· n ̂ ) −1 is a stable solution to the Abelian Maxwell equations. The simple generalization A = MA is a solution to the classical non-Abelian generalized Yang-Mills equations, where M is a matrix in the Lie algebra G of the gauge group. In this paper, the stability problem for these non-Abelian monopoles is posed and solved. Although A is essentially Abelian in that A× A = 0 , the stability analysis is non-trivial because it involves the full non-Abelian structure of the theory. It is first shown that the potential A leads to a rotationally invariant classical theory only if the quantization condition g β = 1 2 n β (g β = eigenvalue of M/4π; n β = integer = 0, ±1, ±2,…; 0⩽β⩽ dim G; gauge field coupling constant = e = 1) is satisfied. (In contrast, the Abelian Dirac quantization condition g/4π = 1 2 n is necessary only in quantum mechanics.) The stability analysis is performed by solving the linearized equations for the perturned potentials A(r) + a(r , t) . Thus the existence of a solution a(r ,t) which grows exponentially in time t is equivalent to the instability of A . Using a convenient choice for the basis of G , and the background gauge condition, the equation for the Fourier transform ψ( r,ω) of a is seen to be equivalent to the Schrödinger equation for a particle of unit spin, unit charge and unit anamolous magnetic moment moving in the potential 4πg β A . This equation is separated using the Wu-Yang monopole harmonics, generalized to include unit spin. The radial equation is then solved in terms of the eigenvalues of an operator related to the spin and orbital angular momentum operators. The result is that A is stable if and only if each integer n β is either 0 or ±1. The monopoles with |g β| 1 2 are thus unstable and therefore have no quantum-mechanical significance. This conclusion is used to speculate about the empirical absence of monopoles, the stability of the non-singular ('t Hooft-Polyakov) monopoles, and the existence of magnetic confinement.

The Heisenberg ferromagnet in spin coherent state representation

A new approach to Heisenberg ferromagnet using the spin coherent state representation is developed. The differential operator representation of spin angular momentum operators is used to derive thec-number analogs of the basic quantum mechanical equations, viz., the Schrodinger, Bloch and Liouville equations for the Heisenberg ferromagnet. As an important illustration of our formulation, which has noad hoc assumptions and does not use any boson representation, the excitation spectrum for one, two and three spin waves is obtained. In these cases it is also shown that eigenvalue spectrum can be obtained by completely ignoring the kinematical interactions.

Degeneracy of Relativistic Cyclotron Motion

The quantum-mechanical problem of the relativistic cyclotron motion of a charged particle in a uniform magnetic field is solved by consideration of the symmetry which the system obeys. It is shown that its symmetry is isomorphic to the Lie group called G(0,1) or G(1,0), and doubly degenerate infinite series of wave functions with a constant energy eigenvalue are labeled by the eigenvalues of the operators ${\mathcal{H}}^{2}$, ${L}_{z}+{S}_{z}$, and ${S}_{z}$. Here $\mathcal{H}$ is the relativistic Hamiltonian referred to in the present problem, and ${L}_{z}$ and ${S}_{z}$ are the usual orbital and spin angular momentum operators, respectively.

Intrinsic Invariance Groups, Mass Operators, and Linear Wave Equations

Relativity theory permits motions of a free-spinning particle in which the instantaneous velocity and linear momentum are not collinear. Therefore, it is necessary to specify two invariance groups in order to completely describe the space-time symmetry properties of a particle with intrinsic spin. If the quantal description of such a particle is given by a covariant linear field equation, then the external space-time symmetry is specified by the 10-element Poincar\'e group generated by the linear four-momentum and total angular momentum operators. However, invariance of the field equation under external inhomogeneous Lorentz transformations does not complete the algebra of the 10-element group generated by the instantaneous four-velocity and spin angular-momentum operators. Several formulations of linear field equations admitting a mass-spin spectrum are based on different choices for this latter, intrinsic, space-time symmetry group. We begin with the simplest choice, namely, intrinsic Poincar\'e invariance, and establish the formal connection between these several formulations by constructing a unitary transformation which generates an infinite sequence of linear wave equations describing ascendingly more complex intrinsic space-time symmetry, but linked by a common mass operator. Of special interest is the infinitesimal transformation which generates the familiar intrinsic DeSitter group. Finally, some kinematical properties of this transformation are discussed for various proposed mass operators.

Proper time formulation of the Dirac electron theory

In terms of the proper time of the particle, a covariant Hamiltonian formulation of Dirac's electron theory is given, and the formalism is applied first to discuss the motion of the particle and second, to construct a new conserved spin angular momentum operator.

Angular-Momentum Conditions for a Correlated Wavefunction

We obtain the conditions under which a correlation wavefunction containing pair correlation functions for the 2pn electrons of C(3P, 1D, 1S), N(4S), O(3P), and Ne will be an eigenfunction of the orbital and spin angular-momentum operators. These pair functions contain quite general powers of electronic and interelectronic coordinates along with their variational parameters and should avoid the convergence difficulties of configuration interaction. In all these cases, except carbon which has just one pair of 2p electrons, one cannot obtain such a pair function, ûij(1), just by minimizing its pair energy, ε̃ij(1), alone; i.e., minimizing the variational parameters of a ûij(1). Minimization of one pair at a time corresponds to working with just a part of the correlation wavefunction which by itself cannot converge to the right symmetry state. One must minimize sums of pair energies by taking a correlation wavefunction containing enough pair functions so as to have the right symmetry. The sets of variational parameters in these different ûij's are not independent but are coupled. These symmetry requirements now make the determination of pair correlation functions quite a bit more difficult. In carbon (3P), these conditions do not apply, but the ûij(1) of the 2p2 electrons of the 1D multiplet must be made orthogonal to the unoccupied orbitals of the open shell if it is to contain general powers of the interelectronic coordinate. This is in addition to the required orthogonality to occupied orbitals. We also obtain a form of the correlation wavefunction for 2p2(1S), a nonsingle-determinant Hartree—Fock (HF) state. This has not been previously known. We do this by applying the angular momentum stepdown operator to the correlation wavefunctions of multiplet states with single Slater determinant HF states and studying the resulting forms of the correlation wavefunction which, in turn, belong to some nonsingle-determinant states of these multiplets.

Relativistic two and three-particle states

Two-particle states transforming according to standard representations of the inhomogeneous Lorentz group are constructed which, in the centre-of-mass system, are eigenstates of orbital and spin angular momentum operators. The connection between these and helicity states is derived and a recoupling coefficient for three-particle states in the former scheme found.

Onσ-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules

1. The distortion due to molecular rotation causes the width of a spin multiplet to depend on $j$, and tends to uncouple the spin axis from quantization relative to the axis of figure, thus bringing about a gradual passage from Hund's case (a) to (b). Another rotational effect is the "sigma-type doubling" of spectral lines due to removal of the degeneracy associated with the equality in energy of left- and right-handed axial rotations in stationary molecules. The present paper treats these two effects, especially their interrelation.2. As a needed mathematical preliminary we calculate the perturbing matrix elements due to the components of angular momentum perpendicular to the figure axis, which are neglected in the usual treatment of the rotating molecule as a symmetrical top. This calculation would be similar to Kronig's and Wigner and Witmer's were it not for inclusion of the spin. This is handled by Pauli's scheme of two wave functions per electron, especially his method of transforming them from one Cartesian system to another by the Cayley-Klein parameters. The results also hold with Dirac's "quantum theory of the electron," as Dirac's quartet of wave functions transform under a rotation like two independent Pauli pairs. Although the orbital and spin angular momentum operators look superficially different, it is shown that their gyroscopic effects enter additively as commonly supposed, and that in the first approximation the effect of the spin is to make the rotational energy (except for an additive constant) that of the symmetrical top with ${\ensuremath{\sigma}}_{l}+{\ensuremath{\sigma}}_{s}$ in place of ${\ensuremath{\sigma}}_{l}$.3. Neglect of the relatively small sigma-doubling yields identically the formulas for the rotational distortion of spin multiplets which Hill and Van Vleck obtained by a different, alternative method that used Hund's case (b) rather than (a) as the unperturbed system.4.Singlet $P$ states should exhibit a sigma doubling proportional to $j(j+1)$ and $D$ states ordinarily a negligible doubling. Our technique of calculation differs slightly from that of Kronig, who obtained a similar result for singlet spectra (as did Hill and V. V. with a simple model), in that the degeneracy is removed in the final rather than in the initial approximation.5. In $^{2}P$ states the spin profoundly modifies the sigma doubling. In case (b) both spin components should exhibit equal doublings proportional to ${j}_{k}({j}_{k}+1)$, but in case (a) the ${P}_{\frac{3}{2}}$ sigma doubling should be negligible, but the ${P}_{\frac{1}{2}}$ fairly large and proportional to $j+\frac{1}{2}$. Formulas are developed for the sigma-doublet width which apply throughout the range between (a) and (b). The pronounced doubling of the ${P}_{\frac{1}{2}}$ component in case (a) is due to a rather complicated superposition of the rotational distortion on the magnetic coupling between the components of spin and orbital angular momentum which are perpendicular to the axis of figure.6.$^{2}S$ states. A similar superposition explains the so-called "rho-type doubling" in $^{2}S$ states whereby levels of like ${j}_{k}$ but unlike $j$ differ by small amounts proportional to ${j}_{k}+\frac{1}{2}$. This explanation differs from the usual interpretation of this doubling as due to magnetic moment developed by orbital motions, and seems to give a slightly larger effect.7.$^{3}P$ states. In case (b) all three components should have equal doublings proportional to ${j}_{k}({j}_{k}+1)$ but in (a) the ${P}_{0}$ and ${P}_{1}$ doublings should be respectively independent of $j$ and proportional to $j(j+1)$, while the ${P}_{2}$ is negligible. The large splitting of the ${P}_{0}$ component, whose exceptional behavior is due to its having $\ensuremath{\sigma}=0$, is due largely to magnetic action effective even in a stationary molecule, and so has a different origin than the usual sigma doubling due to rotational distortion.8.A summary and comparison with experimental data taken from Mulliken's following paper are finally given. There is a striking agreement between theory and experiment on the type of variation with $j$ in the various cases, especially the asymmetrical behavior of the various multiplet components in case (a). The theoretical orders of absolute magnitude are also confirmed, as evidenced by the reasonable values of the frequencies which must be assumed to give the observed separations.