Analytical structure and properties of Coulomb wave functions for real and complex energies

The radical Coulomb wave functions are analysed in their dependence on the energy E considered as a complex parameter. Repulsive and attractive fields are both considered. First turning to the function Φ_l ∝ r^?l?1F_l introduced by Briet, slightly modifying its definition, and assuming that the angular momentum is also a complex parameter, for which the notation L is used, it is proved that Φ_L is an entire function of both E and L. From an expansion of the regular Whittaker function given by Buchholz, the Taylor expansion of Φ_L in powers of E and a simple recurrence relation for its coefficients are easily obtained. The expansion of the regular function F_l is readily obtained from that of Φ_L for L = l, but the irregular function G_l contains Φ_l and $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ for L = l and ?l?1. Having proved that the expansion obtained for Φ_L in powers of E can also be regarded as a uniformly convergent series of entire functions of L, the derivative $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ can be obtained by term-by-term derivation. This method for obtaining the expansion of G_l is straightforward and leads to a final result involving essentially: (i) the conventional function $\text{h(η) =}\text{1}\text{2}\text{ψ(1 + iη) +}\text{1}\text{2}\text{ψ(1 ? iη) ?}\text{ln}\text{η}$ which is singular at η = ∞, i.e., at k = 0; (ii) two entire functions of E, namely Φ_l and Ψ_l; the terms of the expansion of the latter in powers of E contain only Bessel functions multiplied by Bernoulli numbers and coefficients easily obtained from a simple recurrence relation. As an application of the above results, the last sections contain: (i) an alternate from of G_l expansion useful in numerical computations; (ii) the definition and expansion of two linearly independent solutions of the Coulomb equation which are entire in E; (iii) the expansion and threshold properties of the outgoing and incoming solutions, O_l and I_l, corresponding to those we have obtained for F_l and G_l.     

Remark on a paper of Mņczyński

In his paper, Boolean Properties of Observables in Axiomatic Quantum Mechanics,M$\text{n}\text{?}$czyński formulates the notion of a Boolean representation of the magnitudes of a physical theory, and proves the theorem (4.1) that a family P of magnitudes has a Boolean representation if and only if there is a homomorphism from $\text{L}$into a Boolean algebra, where $\text{L}$ is the orthomodular poset associated with P. Exploiting Finch's representation theorem for orthomodular posets, M$\text{n}\text{?}$czyński is able to show that this condition is satisfied if and only if the Boolean subalgebras of $\text{L}$ have a nondegenerate direct limit. The direct limit is then a maximal Boolean representation of P. Homomorphic relations were introduced by Kochen and Specker as a weakening of the concept of an imbedding between partial algebras. The purpose of this remark is to show that the direct limit of the Boolean subalgebras of $\text{L}$ has a natural characterization in terms of the notion of a homomorphic relation between $\text{L}$ and the direct product of its Boolean subalgebras.     

Matter fields and metric deformations in multidimensional unified theories

This paper describes a first study of the effects due to including matter fields in generalized Kaluza-Klein (KK) theories with nonabelian compact gauge group G and nontrivial fibres $\text{V}$^K. The approach is based on the first-order Einstein-Cartan (EC) general relativity in (4 + K) dimensions. In the EC theory there are two basic mechanisms which can lead to a spontaneously compactified KK background geometry R⁴ × $\text{V}$^K: (A) a particular kind of energy-momentum density matter condensate in the quantized ground state, or (B) a particular kind of spin-density matter condensate. If (A) or (B) are operating, the inconsistencies usually found between the KK ansatz and the matter-free EC theory are avoided. Mechanism (B) works only when $\text{V}$^K is parallelizable. It is shown that the expansion of matter fields in normal modes on $\text{V}$^K implies that one must include deformations of the Yang-Mills (YM) potentials contained in the usual KK metrics. We discuss and characterize one class of such deformations. As a case study, we consider fibres $\text{V}$^K ～ G′, where G′ is a semisimple compact Lie group. We allow for the “maximal” YM gauge group G_L′ × G_R′. We carry out the harmonic analysis for spinor fields and study the mass spectrum and YM quantum numbers of the normal modes. We rely on mechanism (B) to provide a curvature-free connection (“parallelization”) on $\text{V}$^KG′ by means of a suitable vertical constant torsion. Minimal YM couplings are of size $\text{l}\text{L}\text{≡ g}$, where l is the Planck length and L is the length of the fibre; nonminimal YM couplings are of size L. Nonzero masses are of size L^?1. The massless modes are found and discussed. There would be no massless modes if the parallelizing vertical torsion were absent. This torsion also implies the vanishing of the cosmological constant. When the theory is restricted to massless modes, the YM deformations disappear and the dimensional reduction to four dimensions yields an effective YM theory, which is renormalizable at energies far below L^?1: the effective theory is obtained by letting L → 0 with g ? 1 fixed and by neglecting all masses of order L^?1; g corresponds to the bare YM coupling constant. The surviving effective YM gauge group is G_L′ and the matter fields are in a particular representation of G_L′ × G_R′, corresponding to the zero mass eigenvalue. Explicit examples are discussed for G′ = SU(2) and G′ = SU(3).     

Interaction de l'oxygène aux basses pressions avec la solution solide niobium-oxygène à haute température: I. Èquilibre de segrégation — chimisorption

Interactions between oxygen under low pressure and a niobium-oxygen solid solution had been studied, in the regime where adsorption is the rate-determining step, from 1000 to 1700 K. It is shown that at saturation of solid solution, there exists a constant limiting value Θ_l of superficial coverage, comparable to a limiting bulk concentration c_l. The ratios θ = Θ/Θ_l and ? = c/c_l are called “relative ratio of occupation” (superficial and bulk). $\text{K}$_SV is the equilibrium constant of segregation between adsorbed and dissolved oxygen atoms: (O_diss^?v) + σ ? (O_chim?σ) + v (σ and v being respectively surface and bulk sites), $\text{K}$_SV = [(1 ? θ)/θ] [?/(1 ? ?)]. The experimentally determined expression: $\text{K}$_SV = 5.7 exp[?(22.1 ? 12.1 θ)/ RT] shows that lateral superficial interactions have a large influence on the enthalpy of transfer between the bulk and the surface of the sample. Adsorption is direct and non activated. At the solubility limit, only a fraction of the superficial sites is occupied. We estimate it to be one half. The sticking probability b of oxygen on a niobium oxygen solid solution is given by b = (1 ? θ/2)², its value at zero coverage being estimated as unity.     

On the continuity of generalized eigenfunctions of the Schrödinger operator

We prove that the generalized eigenfunctions of the Schrödinger operator are continuous for potentials obeing the following assumptions: V=V⁺?V^?,V^±≥0,$\text{V}{}^{\mathrm{+}}\text{∈L}{}^{\mathrm{p}}{}_{\mathrm{<mtext>loc</mtext>}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$, $\text{V}{}^{\mathrm{?}}\text{∈L}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{l}}\text{)}$,$\text{p >}\text{l}\text{2}$.     

Limit cycle in a bound exciton recombination model in non-equilibrium semiconductors

A non-equilibrium semiconductor model involving the processes of photogeneration of electron-hole pairs (e-h) (rate G), stimulated creation of excitons from e-h (rate constant C) and decay of excitons on recombination centres (rate constant k) is analyzed in this paper for steady states and limit cycle behaviour. Considering the exciton decay to be similar to enzymatic processes in chemical reactions obeying a Michaelis-Menten law, and choosing units such that k = 1 = N, where N is the concentration of recombination centres, the model represents a 2-parameter (C and G) 2-dimensional (exciton and electron-hole concentrations x, n) dynamical system with a unique steady state (x₀,n₀) which is unstable in the region (l ? G)³?4C, the equality sign corresponding to the bifurcation curve in parameter space. In the region (l ? G)³ > 4C the system displays a unique stable limit cycle which is obtained in analytical form by employing a two-time-scales method for parameters in the neighbourhood of the bifurcation curve. The limit cycles are tilted ellipses with angular frequency $\text{}\text{?}$gw of the order of 10⁶ s^?1. In a realistic semiconductor situation G$\text{\$}\text{?}$10^?3.     

Semiclassical quantum mechanics. III. The large order asymptotics and more general states

We solve the time dependent Schrödinger Equation $\text{i?(}\text{?ψ}\text{?t}\text{) = ?(?}{}^2\text{/2m) Δψ + Vψ}$ modulo errors which have L² norms on the order of $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ for arbitrarily large l. The initial conditions we consider are fairly general states whose position and momentum uncertainties are proportional to $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

Phonon singularities on volt-ampere curves of niobium point contacts

The volt-ampere curves and their second derivatives were studied for niobium point contacts at low temperatures in the voltage range corresponding to the characteristic phonon energies. It was found that while for the dirty contacts in the normal state no PC spectra of phonons could be detected, in the superconducting state there were singularities in the I–V curves corresponding to maxima either in the first or in the second derivatives. The singularities observed were due to the energy dependence of the excess current. We suppose that the origin of these singularities is due to the inelastic transitions of electrons between chemical potentials of Cooper pairs at both sides of the contact, which differ in energy by eV. These transitions are possible if ξ(0) ? d ≈ Λ_?, (ξ(0) being the coherence length; d, the contact diameter; $\text{Λ}{}_{\mathrm{?}}\text{= (l}{}_{\mathrm{i}}\text{·l}{}_{\mathrm{?}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, where l_i and l_? being the elastic and inelastic electron mean free paths, respectively).     

Proton states in the N = 88 nuclei 149Pm, 151Eu and 153Tb populated in the (τ, d) and (α, t) reactions

The (τ, d) and (α, t) reaction on targets of ¹⁴⁸Nd, ¹⁵⁰Sm and ¹⁵²Gd have been studied, using beams of 24 MeV ³He and 27 MeV ⁴He from the McMaster University FN tandem Van de Graaff accelerator. The reaction products were analyzed with a magnetic spectrograph and detected with photographic emulsions. The (α, t) spectra were measured at two angles for each target, and the (τ, d) reactions were studied at 8 or 9 angles. The l-values for a number of low-spin states were determined from the (τ, d) angular distributions, and ratios of the (α, t) and (τ, d) cross sections were used to obtain l-values for several other states. There are some striking similarities in the observed structures of the three final nuclei, ¹⁴⁹Pm, ¹⁵¹Eu and ¹⁵³Tb. In each case there are low-lying strongly populated $\text{11}\text{2}$^? states and a higher lying l = 5 level somewhat below 1 MeV of excitation energy. Several states (10 in ¹⁴⁹Pm, 17 in ¹⁵¹Eu and 8 in ¹⁵³Tb) appear to be populated via l = 2 transitions, and there are strongly excited $\text{1}\text{2}$⁺ levels at $\text{≧ 1}\text{MeV}$ of excitation energy in each case. Of particular interest is a $\text{7}\text{2}$^? state located ≦ 50 keV above the lowest $\text{11}\text{2}$^? state in each nuclide. The relatively strong populations of these $\text{7}\text{2}$^? levels in the present experiments are contrary to expectations based on the simple shell model as there are no f_{$\text{7}\text{2}$} states in the 50 < Z < 82 shell.     

Electron drag of dislocations in thin metal plates

The electron force dragging dislocations in a thin metal plate of thickness d has been calculated and the dependence of the drag force F on the parameter $\text{d}\text{l}$ (l is the electron free path length) determined for the cases $\text{d}\text{l ? 1}$ and $\text{d}\text{l ? 1}$. With d ? l, the drag force depends upon the nature of electron scattering on the plate surface. The drag force dependence on the sample size has also been considered for a d.c. magnetic field H parallel to the plate faces. In the case of diffuse scattering the second derivative of F(H) reveals a square root singularity at the value of the magnetic field where the electron orbit diameter is equal to the plate thickness.     

On the definition of the pressure of an infinite system

A microcanonical system $\text{L}$^o_Λ is considered together with a manometer $\text{M}$_Λ. The thermodynamic limit Λ → ∞ is taken for the system $\text{L}$_Λ composed of $\text{L}$^o_Λ and $\text{M}$_Λ. This yields a definition of the pressure P(v, ε) of $\text{{}\text{L}{}^{\mathrm{o}}{}_{\mathrm{\Lambda }}\text{; Λ → ∞}}$ for given values ε and v of the energy and particle densities. P(v, ε) is shown to be equal to the thermodynamic function p(z, β) derived from the grand canonical ensemble for almost all values of ε and v provided the appropriate equilibrium values of the temperature β^?1 and the chemical potential μ = β^?1 log z are inserted.     

Electromagnetic waves near perfect conductors. I. Multiple scattering expansions. Distribution of modes

An expansion is established for the Green functions describing electromagnetic waves in the presence of a perfectly conducting boundary. Each term represents a process for which the wave scatters several times on the boundary and propagates freely in between. This multiple scattering expansion reduces to ray optics in the high frequency limit, and thus provides a general framework to study diffraction corrections. It also allows one to evaluate quantities averaged over the spectrum. Some symmetry properties of the expansion are exhibited, in particular, the replacement of magnetic fields by electric fields results in a change of sign at each scattering. Convergence is proven for any smooth boundary in the domain Im k ? | Re k | of complex wavenumbers k. The continuation of the convergence domain around k = 0 is shown to depend upon the topology of the boundary. The multiple scattering expansion method is applied to determine the distribution of electromagnetic eigenmodes in a conducting cavity. The density of modes ?(k) is analyzed in terms of closed classical rays, bouncing off the walls with mirror reflections. Paths of zero length yield the smooth part of ?, expanded as $\text{π}{}^{\mathrm{?2}}\text{[Vk}{}^2\text{-}\text{2}\text{3}\text{∫ dα / R}$ + $\text{O}$(k^?2)] where V is the volume of the cavity, and ∫ dα/R is the integral over the boundary of the mean curvature. Paths of finite length L yield contributions to the density ?(k), of the form Im(a exp ikL) appearing as regular oscillations in the bunching of eigenmodes. For an analytic boundary, inclusion of complex classical rays renders exact the analysis of the eigenmodes in terms of closed paths. As a consequence, the high temperature expansion for the energy of a small blackbody is obtained.     

The influence of the mean free path on the current induced superconducting/normal conducting transition of tin whiskers with indium impurities

The current induced step-like structure in the V-I characteristics of tin whiskers with indium impurities up to 4 at.% shows a zero voltage intercept I₀ of approximatively 0.5I_c. This current can be explained by the existence of phase slip centers above the critical current which carry a time averaged supercurrent of ≈0.5I_c. From the differential resistance associated with the first step a “healing length” L_n proportional to l^{$\text{1}\text{2}$} (l = mean free path) was obtained which can be related to the quasi-particle diffusion length Λ introduced by Tinkham et al. in the explanation of temperature independent nonequilibrium processes between pairs and quasiparticles at a phase slip center.     

Quantum theory of longitudinal dielectric response properties of a two-dimensional plasma in a magnetic field

An analysis of dynamic and nonlocal longitudinal dielectric response properties of a two-dimensional Landau-quantized plasma is carried out, using a thermodynamic Green's function formulation of the RPA with a two-dimensional thermal Green's function for electron propagation in a magnetic field developed in closed form. The longitudinal-electrostatic plasmon dispersion relation is discussed in the low wavenumber regime with nonlocal corrections, and Bernstein mode structure is studied for arbitrary wavenumber. All regimes of magnetic field strength and statistics are investigated. The class of integrals treated here should have broad applicability in other two-dimensional and finite slab plasma studies.The two-dimensional static shielding law in a magnetic field is analyzed for low wavenumber, and for large distances we find $\text{V(}\text{r}\text{) ～}\text{Q}\text{k}{}_0{}^2\text{r}{}^3$. The inverse screening length $\text{k}{}_0\text{=}\text{2πe}{}^2\text{??}\text{?ξ}$ (? = density, ξ = chemical potential) is evaluated in all regimes of magnetic field strength and all statistical regimes. k₀ exhibits violent DHVA oscillatory behavior in the degenerate zero-temperature case at higher field strengths, and the shielding is complete when $\text{ξ = r′}\text{l}\text{z.shtsls;}\text{ω}$, but there is no shielding when $\text{ξ ≠ r′}\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}$. A careful analysis confirms that there is no shielding at large distances in the degenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> ξ}$. Since shielding does persist in the nondegenerate quantum strong field limit $\text{l}\text{z.shtsls;}\text{ω}{}_{\mathrm{c}}\text{> KT}$, there should be a pronounced change in physical properties that depend on shielding if the system is driven through a high field statistical transition. (It should be noted that the static shielding law of semiclassical and classical models has no dependence on magnetic field in two dimensions, as in three dimensions.) Finally, we find that the zero field two-dimensional Freidel-Kohn “wiggle” static shielding phenomenon is destroyed by the dispersal of the zero field continuum of electron states into the discrete set of Landau-quantized orbitals due to the imposition of the magnetic field.     

Critical spin-relaxation in the hard magnetic direction of CrBr3

From ESR-linewidths measured at low microwave frequencies the kinetic coefficient of the hard magnetization is deduced. In the isotropic critical region, $\text{T}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{?1≡??0.1}$, L^⊥ increases as ?^-1.50 while in the anisotropic region (?<0.01)L^⊥ saturates and depends on field.     

Exciton binding energy in GaAs quantum wells deduced from magneto-optical absorption measurement

Magneto-optical absorption spectra due to exciton states and Landau-levels were measured in GaAs/AlAs multi-quantum-wells. By extrapolating the photon energies of the absorption peaks to zero magnetic field, the lowest state (1S) heavy hole exciton binding energy, E^B_$\text{h}$(1S), was obtained as a function of well size L_z in the range $\text{58}\text{A}\text{?}\text{?}\text{L}{}_{\mathrm{z}}\text{?252}\text{A}\text{?}$. The L_z dependence of E^B_h showed the change of the exciton character from three-dimensional to two-dimensional with decreasing L_z. The diamagnetic shift observed for the heavy hole exciton peak was larger than that for the light hole exciton peak, showing the anisotropic nature of the Luttinger-Kohn Hamiltonian. The diamagnetic shift of the heavy hole exciton peak became smaller as L_z was decreased, suggesting the enhancement of the two-dimensional exciton character.     

A comparison of πππ, Kππ, πKKKK system produced in the reactions meson +p→3 mesons + p

A comparison is made of the low-mass three-meson systems (πππ), (Kππ), $\text{(π}\text{K}\text{K}\text{)}$ and $\text{(}\text{K}\text{K}\text{K}\text{)}$ diffractively produced in the reaction meson + proton → three mesons + proton. Several striking similarities and a few important differences are observed: (i) the reactions are consistent with the assumption that the three mesons decay entirely into a 0^? meson and a 0⁺, 1^? or 2⁺ resonance; (ii) the three-meson mass spectra have a peak ≈ 250 MeV above the effective threshold M_eff of the dominant decay mode and then fall off approximately as (mass)^?3;(iii) the average spin 〈J〉 = 0.55 + 1.1 Q_eff, where Q_eff = M - M_eff; (iv) the average orbital angular momentum 〈l〉 increases according to 〈l〉 = 0.75 Q_eff; (v) the three-meson states are produced dominantly in unnatural spin-parity states and no evidence for their being resonant is found; (vi) the only natural spin-parity states found are the well-established 2⁺ resonances A₂ and K¹ (1420); they have similar properties to the non-resonant unnatural parity states except for a dip at t = 0 in the dσ/dt distributions; (vii) both the unnatural and natural spin-parity states are produced mostly by an exchange of natural parity; (viii) there is evidence for two types of production mechanism with different polarization properties, one approximately conserving helicity in the t-channel and the other in the s-channel.     

Exponents for the excluded volume problem as derived by the dynamic renormalization group method

By regarding the contour length L of a polymer chain as “time”, we derive the mean-square end-to-end distance 〈x²〉 for the polymer chain with excluded volume by means of the dynamic renormalization group. The result to first order in ? = 4 ? d (d is the dimension of the space) is $\text{〈x}{}^2\text{〉 =}\text{const}\text{× L}{}^{\mathrm{<mtext>1+?</mtext><mtext>8</mtext>}}$.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.