Terms Of Angular Momentum Research Articles

Die Pauli-Gleichung mit Differentialoperatoren für den Spin/ The Pauli Equation with Differential Operators for the Spin

The spin operator s = (ħ/2) σ in the Pauli equation fulfills the commutation relation of the angular momentum and leads to half-integer eigenvalues of the eigenfunctions for s. If one tries to express s by canonically conjugated operators Φ and π = (ħ/i) ∂/∂Φ the formal angular momentum term s = Φ X π fails because it leads only to whole-integer eigenvalues. However, the modification of this term in the form s = 1/2 {π + Φ(Φ π) + Φ X π} leads to the required result. The eigenfunction system belonging to this differential operator s(Φ π) consists of (2s + 1) spin eigenfunctions ξm (Φ) which are given explicitly. They form a basis for the wave functions of a particle of spin s. Applying this formalism to particles with s = 1/2, agreement is reached with Pauli’s spin theory. The function s(Φ π) follows from the theory of rotating rigid bodies. The continuous spinvariable Φ = ((Φx , Φy, Φz) can be interpreted classically as a “turning vector” which defines the orientation in space of a rigid body. Φ is the positioning coordinate of the rigid body or the spin coordinate of the particle in analogy to the cartesian coordinate x. The spin s is a vector fixed to the body.

The multimode Jahn‐Teller coupling for an orbital triplet in cubic crystals

AbstractAn investigation is made of the properties of an iron group ion which has an orbital triplet ground state and which interacts with the lattice vibrations of its nearest neighbours. The interaction with e‐type vibrations is considered only. For the phonon spectrum of the host crystal the Debye and Einstein approximations are used. Explicit formulae are obtained for the Jahn‐Teller energy, the displacements of the neighbours nearest to an impurity, and for the first‐order reduction factor of orbital angular momentum in terms of the phonon spectrum parameters. All cubic coordinations are considered: fourfold, sixfold, and eightfold.

Generalized invariants for the time-dependent harmonic oscillator

A generalized class of invariants, I (t), for the three-dimensional, time-dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time-independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time-dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.

Volume dependence of the electronic structure of lanthanum and cerium

AbstractResults of a band structure calculation for the f.c.c. lattices of lanthanum, γ‐and α‐cerium are presented. The computation is based on Altmann's rigorous cellular method. A local potential is used generated according to the standard method. Charge densities of the free atoms are obtained from a SCFXα‐integration of the Dirac equation. Earlier band calculations using the full Slater exchange led to f‐levels far too low in energy. Instead the more adequate Gaspar‐Kohn‐Sham value of 2/3 is used for the exchange scaling factor, which yields reasonable f‐band energies. The Permi level proves to be well below the f‐bands for both metals. As follows from an analysis of the occupied band states in terms of angular momenta, there are f‐admixtures of 0.3 and 0.6 electrons/atom for La and Ce, respectively, which increase slightly under pressure. As the pressure rises the Fermi level moves across three van Hove singularities in the electronic density of states at –15, 10, and at 60 kbar. This is also reflected in topological changes of the Fermi surface which can give rise to isostructural phase transitions. Recent Bremsstrahlung‐ and XPS‐spectra of La and Ce are in fair agreement with the results on the density of states.

Angular momentum flow in the sun

Theories of the solar core rotation are first reviewed. The sun's surface rotation rate being relatively low as stars go has led many authors to believe the sun has lost most of its angular momentum and has this low rotation rate throughout. This low rotation rate is supported by heliometer observations of the roundness of the sun around the turn of the century as well as by recent experiments of Hill and colleagues. Dicke and Goldenberg, however, observe a more oblate appearance indicative of a rapidly rotating solar core. This rapid rotation appears to be explicable in terms of angular momentum released as the solar core contracts due to hydrogen burning. Three calculations of solar core rotation rates yield values of the pierod near one day. The author contends that the sun has a rapidly rotating core, but not fast enough to favour the Brans-Dicke theory of general relativity over the Einstein theory. The surface rotation of the sun is also reviewed. Four theories which appear as candidates for the explanation of the sun's differential rotation are anisotropic viscosities, Rossby waves, giant cell convection, and solar wind drag at the poles. All appear reasonable and there does not appear to be overwhelming support for any one.

Relations between a criterion for polar wander and some conditions for absolute plate motion

Jurdy and Van der Voo's (1974) definition of polar wander is shown to be related to equations which Lliboutry (1914a, b) and Solomon and Sleep (1974) use for calculating absolute plate motions. A restatement of their polar wander criterion in terms of the net angular momentum of infinitely thin lithospheric plates is compared by a suggestive but nonrigorous argument with Solomon and Sleep's (1974) condition that the net torque acting on the lithosphere be zero, The small amount of polar wander calculated by Jurdy and Van der Voo (1974, 1975) since Early Cretaceous suggests that the abovementioned conditions for absolute plate motion have some validity.

Orthonormal Expansions of Angular Momentum Functions

A method is given for expanding an arbitrary function of a component of angular momentum in terms of orthonormal operators. The orthonormal operators are given explicitly in terms of Chebyshev’s polynomials. A general formula for the expansion coefficients is derived and applied to the rotation operator.

Impact parameter representations from the point of view of the Poincar� group

The impact parameter representations for two-particle scattering amplitude are deduced by means of an expansion of twoparticle state are groups isomorphic to the group of motions of the E(2)-plane. Infinitesimal generiltors of these groups are expressed in terms of total angular momentum and relative momentum operators. As a result the impact parameter operator and the states of two particles with fixed impact parameter are well defined in terms of the Poincare group. (auth)

Excited state wavefunctions for helium

Using several angular momentum terms and a parametric expansion for the radial function, wavefunctions are obtained for certain excited S, P, D and F states of helium, the virial theorem being satisfied exactly.

Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

Energy and angular momentum flow into a black hole

The area of the event horizon round a rotating black hole will increase in the presence of a non-axisymmetric or time dependent perturbation. If the perturbation is a matter field, the area increase is related to the fluxes of energy and of angular momentum into the black hole in such a way as to maintain the formula for the area in the Kerr solution. For purely gravitational perturbations one cannot define angular momentum locally but one can use the area increase and the expression for area in terms of mass and angular momentum to calculate the slowing down of a black hole caused by a non-axisymmetric distribution of matter at a distance. It seems that the coupling between the rotation of a black hole and the orbit of a particle going round it can be significant if the angular momentum of the black hole is close to its maximum possible value and if the angular velocity of the particle is nearly equal to that of the black hole.

Phenomenological Description of Nuclear Rotational Spectra by the Use of Rational Fractions in Terms of Angular Momentum

The use of rational fractions in terms of a suitably defined angular momentum variable is further examined for describing the rotational spectra below the Yrast region (I ≲ 30). As a specific example, a four-parameter expression for the moment of inertia is applied and found to fit well the spectra of many good and several soft rotors with spins I ⩽ 22. The systematics of the parameters are also studied. The work is limited to ground-state rotational bands of even-even nuclei.

Boson theory of angular momentum: Heisenberg ferromagnetism at low temperatures

The representation of angular momentum in terms of the boson operators originally introduced by Holstein and Primakoff is studied, and the resulting formalism is used to evaluate the low-temperature properties of the Heisenberg ferromagnet. As in the Holstein-Primakoff method, the one-to-one correspondence between spin-deviation number in the spin space and occupation number, n r, in the proper boson subspace (0 ⩽ n r ⩽ 2 S) is exploited to obtain boson representations of spin operators, and the matrix elements of these operators between improper boson states ( n r > 2 S) are treated as somewhat arbitrary adjustable parameters. Each spin operator in the boson representation may be split into a sum of proper and improper parts, and all matrix elements of the improper parts vanish within the proper subspace. Since the proper parts of the spin operators yield the correct commutation rules and matrix elements within the proper subspace, they constitute a transformation equivalent to that of Holdtein and Primakoff in which the square roots have been replaced by finite series without any special assumptions of large spin and/or low density. In general, the boson Hamiltonian may also be split into a sum of proper and improper parts, and the matrix elements of the proper part are in one-to-one correspondence with the matrix elements of the spin Hamiltonian within the proper subspace. The matrix elements of the improper part vanish within the proper subspace, and are arbitrary parameters within the improper subspace. The improper matrix elements may be chosen so as to obtain an arbitrarily large energy gap between the ground state of the spin system and the lowest improper state; in particular, the energy gap may be infinite, since the actual spin system cannot be excited into an improper state, and in this way we make contact with the hard-core model of Morita and Van Kranendonk. Finally, it is shown that in the limit of large spin the boson Hamiltonian used here reduces in form to a hermitian equivalent (H D + H D + ) 2 of Dyson's Hamiltonian, H D , thus providing a hermitian Hamiltonian which may be used to describe the Heisenberg ferromagnet with arbitrary spin when the exclusion principle for spin deviations is ignored. In applications, there are two important features of this formulation: First, since the interactions among spin waves are treated purely mechanically there are no cumber-some restrictions on the boson partition function; second, since the lowest improper boson state is separated from the ground state of the spin system by an arbitrarily large energy gap, the contribution of improper states to the ensemble average of all physical quantities may be made arbitrarily small. Without ignoring any interactions, this formulation is used to study the low-temperature properties of the Heisenberg ferromagnet. The two-particle bound states of Wortis and Hanus and the ideal scattering states of Dyson are reproduced for arbitrary spin, and moreover these results are found to be completely independent of all improper matrix elements. In evaluating the dynamical properties, we obtain off-shell terms in the boson self-energy which do not appear in previous theories, and theses are associated with the exclusion principle for spin deviations, albeit they are not exponentially small. However, it is shown that, to first order in the density of spin waves, these off-shell terms do not modify the quasiparticle spectrum found when the exclusion principle is ignored. In fact, by showing that all purely off-shell terms mutually cancel in the expression for the transverse susceptibility, we prove explicitly that the exclusion principle plays no role in the low-temperature dynamics of spin waves, at least to first order in the density. This result is a generalization of Dyson's proof that the kinematical interaction may be ignored in evaluating the low-temperature thermodynamics. Finally, we confirm that the purely off-shell terms do not contribute to the free energy, F, and thereby obtain Dyson's original result for F, including exponentially small terms.

Calculation of Predissociation Rates of the B 3Π+u State of I2

A comparison of the purely radiative decay rates of vibrational levels of the B 3Π0+u state of I2 with total decay rates obtained from several independent lifetime measurements has indicated a significant amount of spontaneous predissociation of the B state. We present here a calculation of the observed relative predissociation rates by means of a model consisting of a repulsive 1u state which is coupled to the υ′J′ levels of the B state by rotational and electronic angular-momentum terms neglected in the zeroth-order Hamiltonian. The heterogeneous predissociation rate between a bound υ′J′ level and a continuum level at the same energy is calculated from Kronig's theory using exact wavefunctions for both levels. The dependence of this rate on the rotational level J′ is taken into account, together with the fractional population of the υ′J′ levels which were excited in the measurement of the total decay rates. Excellent agreement is found between the calculated and observed relative predissociation rates at regions about υ′ of 14, 25, and 50 of the B state for one position of the repulsive state lying very near the repulsive wall of the B state over the energy range 17 300–20 000 cm−1. The position of this state relative to the B 3Π0+u and A 3Π1u states agrees well with Mulliken's prediction of the location of the Π1u1 state of I2.

Orbital-Angular-Momentum Decomposition of the Three-Body Coupling Scheme: Muonic Molecules and the Hydrogen Molecular Ion

In obtaining the wave functions of three-body Coulomb systems in which two heavy equally charged, and one light oppositely charged, particles are involved, one can use a type of Born-Oppenheimer coupling scheme, to obtain nonadiabatic effects. In this paper we decompose the coupled differential equations of such a scheme in terms of the total orbital angular momentum and parity of the states, and reduce these equations to one-dimensional ones in a form suitable for calculation.

Multipole expansions for stationary fields in general relativity

Stationary solutions of the Einstein field equations with no spatial symmetries are considered in order to discover the number of arbitrary constants, corresponding to the moments of the source in the Newtonian field, which appear in the solution. It is shown that the stationary field contains twice the number of moments which appear in the static field, apart from the mass term which appears singly. These additional moments are essentially angular momentum terms, which in Newtonian theory make no contribution to the field.

Spin Dependence of High-Energy Scattering Amplitudes. I

A formalism, useful for discussing spin dependence of scattering amplitudes in terms of complex angular momentum, is given. The question of the spin dependence of elastic scattering amplitudes, its relationship to coupling to Regge poles, fixed poles in the angular-momentum plane, and superconvergence relations are discussed. It is concluded that general principles permit the coupling at a vertex of a Regge pole of integer spin ${J}_{0}$ and two particles whose spin projection is greater than ${J}_{0}$, provided the Regge pole has the wrong signature to be a particle of spin ${J}_{0}$. In particular, the Pomeranchuk trajectory at $t=0$ (${J}_{0}=1$) can flip helicities by two or more units and hence yield spin-dependent cross sections. Several models which support this conclusion are given. The possibility that a particular Regge pole's coupling does vanish is not ruled out, however.

Eigenvectors of a slightly asymmetric rotator

The normalized eigenvectors of a slightly asymmetric rotator are obtained through fourth order in the asymmetry parameter. The perturbation method is described in a previous paper. The commutator relations which are developed in connection with the asymmetric rotator should be useful for calculating the corrections to the energy eigenvalues which arise from the higher order angular momentum terms in the vibration-rotation Hamiltonian for a molecule.

Satellite Orbits

The various types of satellite and projectile orbits are described and related to the conditions, in terms of energy and angular momentum, under which the ideal orbits occur. The transient nature of the initial and final part of a satellite's path and the conditions necessary for placing a satellite in a given orbit are discussed.

Structure and Evolution of Globular Clusters.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Structure and Evolution of Globular Clusters. Michie, Richard W. Abstract A three-parameter distribution function in terms of energy and angular momentum and with a cutoff at the energy of escape is used to describe the structure of a spherical stellar system. The corresponding density distribution for a specific set of parameter values is obtained numerically and the result agrees very well with observations of 47 Tuc from one to 50 parsecs by Gascoigne and Burr. In obtaining the evolution of a spherical cluster, the basic assumption is that the continuous evolutionary process can be approximated by a series of successive equilibrium states. Therefore, the distribution function is assumed invariant in form and the problem is the calculation of the changes in the three parameters after a time interval. This is done by taking three moments of the Boltzmann equation with the assumption of dynamical equilibrium during the interval of time. These moments are mass, the sum over all stars of the kinetic energy, and the sum over all stars of the angular momentum squared. An evolutionary time step of three quarters of a billion years is obtained numerically with an IBM 650. The results show a flow of stars inward within one parsec from the center and an expansion beyond twenty parsecs. The core of the cluster (within 1.6 parsecs from the center) contracts. In the inner regions the mean-square velocity decreases, and beyond twenty parsecs the mean-square velocity increases. The isotropy of the distribution in velocity space increases in the inner regions and decreases for the outer sections. This result can be explained by a few stars suffering encounters and being thrown into orbits of high energy and low angular momentum. Publication: The Astronomical Journal Pub Date: 1961 DOI: 10.1086/108494 Bibcode: 1961AJ.....66Q..49M full text sources ADS |