Table of contents

A method of handling a number of generalized hypergeometric functions in terms of first-order partial differential matrix equations is introduced. This method has many advantages in formal manipulations and in numerical integration. In particular, it allows investigation of the energy dependence of matrix elements arising in scattering problems in quantum mechanics.

For pt.VIII see ibid., vol.4, p.665 (1971). The energy lower bound models HIP and SHIP may be substantially improved for small numbers of fermions by a simple modification to the limiting procedure in their derivation. It is shown that for intermediate values of N the modified SHIP model gives in some cases the best available lower bounds.

The leading divergences of single-loop scalar and vector effective Lagrangians in a six-dimensional Ricci-flat Riemannian space are evaluated.

It is shown that numerical data recently used to evaluate the residual entropy of Kagome ice can be adapted to form an expansion for the zero field free energy of a three-spin Ising model on a triangulated dice lattice. The expansion is used to test a recent conjecture of Wu (see J. Phys. C: Solid St. Phys. vol.7, p.181) that this model should exhibit two phase transitions.

Sykes and Gaunt (see ibid., vol.9, p.2131 (1976)) showed the mean cluster size in the two-dimensional Ising model to diverge as (T_c-T)^{- theta} with theta approximately=1.91 larger than the Ising susceptibility exponent gamma =1.75. This difference theta - gamma is explained here at least partly by Binder's cluster model (see Ann. Phys. NY. vol.98, p.390 (1976)) for critical points where theta >or= gamma + beta =1.875.

The radiation spectrum corresponding to transitions of a charge between quasienergetic states is obtained. The induced radiation is found.

Treating the Coulombic scattering of a charged particle by a random distribution of scatterers strictly as a two-body process, following an argument familiar in stellar dynamics, removes the divergence of the differential cross section towards small scattering angles.

A test particle is selected to characterize the average behaviour of an assembly of particles of the same kind in a plasma. The re-distribution of energy of this particle after binary interaction between it and the other members of the assembly is calculated. A comparison is made between the simple physical solution offered here and that of Kogan (1961). The present result can easily be applied to plasma problems, for example, the solar wind.

In a wide variety of solids the electronic structure is well described in terms of local atomic environment by the recursion method. A theory of perturbations, directed towards the calculation of transition matrix elements, is developed in terms of the local environment using the recursion method as the unperturbed solution. As well as embodying this physical concept, the theory permits calculations on systems with many more degrees of freedom than other methods.

Unbiased consistent estimators of the mean and variance of a correlated sequence (x(t_k)), k=1, 2,..., n, are derived. The sequence is obtained by Poisson sampling a random process x(t) with a known covariance function. The variances of the estimators are obtained. Finally, the application of the results to laser anemometry is discussed.

A large number of indefinite integrals of the form integral xⁿy₁y₂dx have been evaluated in terms of x, y₁, y₂ and their first derivatives; y₁ and y₂ are both solutions of the differential equation y"=xy. Some of these integrals can be applied to the quantum mechanical problem of a particle in a uniform field of force.

Extending the method of Havas (see J. Math. Phys. vol.16, p.1461 (1975)) for conservative systems, the separability of the Hamilton-Jacobi equation is investigated for mechanical systems described by a time-dependent Hamiltonian, including systems possessing a velocity-dependent potential energy. It is shown that for n degrees of freedom there exist n+1 different types of separable systems, of which the corresponding Hamiltonians are derived after constructing the separated differential equations. Herewith a more profound and systematic approach is given to the results of Iarov-Iarovoi (1963), which have been obtained on a more intuitive basis.

A generalized classical virial equation is derived, superior to the equations already existing in three respects: (i) applicability to assemblies of fluctuating numbers of particles; (ii) applicability to assemblies subjected to net external action; (iii) applicability to any arbitrarily selected part of a larger assembly. It is found that the momentum in- and out-flux densities, caused by particles crossing the limiting surfaces, contribute to the virial. The equation is applied to a homogeneous gas, and the ideal gas law is derived. Invariance criteria are studied, and translations and division into subsystems are discussed. Various internal and external contributions to the virial are discussed and compared for the new and the already existing virial equations.

A method, similar to the one previously used by the author in his derivation of a generalized classical virial theorem (see ibid., vol.10, no.4, p.507 (1977)), is used to derive a generalized quantum virial equation. This equation is applicable to any part of a larger system of particles. Furthermore, by the introduction of a flux density operator, it is possible to express the quantum surface flux virial in two alternative forms, as easily interpreted as the classical form. As an intermediate result, an equation of continuity for a general one-particle observable is obtained. In an appendix, an equation of motion of the reduced density matrix of the first order is derived.

The authors analyse certain realizations of the rotation group in which half-integer representations appear, even though he employs scalar fields only. This angular momentum is interpreted as being purely orbital, and is shown to apply only to massless particles. He discusses the double-valued nature of these representations, including transformation properties of the spinors, which are formed only from space coordinates x and delta / delta x. He describes also position and momentum operators which can be defined.

The operational meaning of fuzzy measurement of two spin components S_x and S_y is examined. Spectral densities assigning probabilities in each spin state to fuzzy simultaneous values of S_x and S_y are introduced, and their informational completeness is examined.

Dirac's equation has been considered in a Schwarzschild (Reissner-Nordstrom) background. Resonances in a continuum of states are found, similar to the Klein-Gordon case. Also the solutions of the Dirac and Klein-Gordon equations are investigated for an extended gravitating source. Special attention has been given to the case in which the radius of the source tends towards its Schwarzschild radius. A limiting charge-to-mass ratio for black holes is obtained.

Several authors have proposed criteria for the onset of Bose-Einstein condensation in finite systems. Using mathematical techniques developed by Pathria and collaborators (1972) these various proposals are systematically examined by applying them to an ideal Bose gas confined to a cuboidal enclosure under periodic boundary conditions. The 'transition points' thus obtained have been subjected to a comparative study, which indicates that the most useful general criterion is the macroscopic one based on the maximum of the specific heat C_v.

A non-translationally invariant spherical model, in which only a finite number of spins interact, is solved exactly. The model exhibits a phase transition in a non-zero uniform field, without spontaneous magnetization. The anomalous transition is attributed to the finite number of interacting spins taking on abnormally large values of order N^1/2 without contributing to the magnetization. The free energy of the model can be obtained from a spherical limit (n to infinity ) of a corresponding n-vector model. In zero field the free energy is of the Curie-Weiss (or mean-field) spherical form. The Curie-Weiss form can only be maintained in a field by admitting a non-uniform field of order N^1/2. This modified spherical model is also accessible from an n to infinity limit of a corresponding n-vector model.

The existence of a general class of similar solutions of the diffusion equation is demonstrated, when the boundary conditions vary as a simple power of time, and the transport coefficient varies non-linearly as a power of the concentration. Although earlier workers have described specific exact solutions, which are members of this class, the class as a whole has not previously been investigated. A simple method is given for the numerical integration of the characteristic differential equation of the profile for the case when an exact solution is not available. The use of these solutions in studies of laser interaction with solid targets, and as test problems for thermal conduction routines, is briefly discussed.

The combined Cerenkov output of a particle with an electric charge and a magnetic dipole moment moving uniformly in a generalized uniaxial medium is investigated in the rest frame of the particle. It is shown that the cross terms in the output add up to zero, so that the radiated powers of the electric and magnetic poles are simply additive. The integrals for the radiated energy are explicitly evaluated for an electric dipole in a singly anisotropic medium. From the results of this integration, the Cerenkov output of a magnetic dipole in a doubly anisotropic medium is obtained by making two sets of substitutions in succession.

The interaction between a quantized model of a semi-infinite plasma and a charged non-relativistic particle outside it is considered allowing fully for relativistic retardation in the electromagnetic field. The choice of gauge, which in the past has occasioned difficulty and imprecision, is elucidated. The Hamiltonian is written down in the strict Coulomb gauge having div A=0 everywhere. It is then transformed canonically to a more popular gauge where div A has a delta -function singularity on the interface, and where the scalar potential is zero in absence of the particle. The effective interaction ('dynamic image potential') is evaluated to second order in nu / omega _pZ but exactly in omega _pZ/c; (v is the particle velocity, Z is distance to interface, omega _p is the plasma frequency) and the results discussed.

The problem of saturated absorption by a gas of two-level molecules, including recoil effects, is formulated in a new quantum representation for the molecular motion. The relationship to other representations is clarified in a simple chart. The new formulation requires the solution of simultaneous linear differential equations with mixed boundary conditions, instead of linear difference equations in two variables followed by an integration over velocity. Analytic solutions are presented for one running wave with Doppler line broadening, and for a weak probe beam in the Doppler limit. The latter lineshape is analysed to show when power broadening obscures the recoil splitting.

Poisson's equation in cylindrical symmetry is solved numerically with the assumptions of zero field and zero electron energy at the cathode surface. The computed current is expressed in terms of an analytical solution obtained for the extreme relativistic case. Internal and external anode configurations are treated, with ratio of electrode radii extending to 10000 and potential difference up to 15 MV. Application of this analysis to the high-power field-emission diode is considered, with particular reference to the radially pumped ultraviolet laser.

The behaviour of a polymer dissolved in a poor solvent is studied using the self-interacting self-avoiding walk model. The properties of short polymers are generated by exact enumeration methods and the results extrapolated to large systems. Evidence supporting the existence of a phase transition is obtained. The exponents governing the length dependence of the partition function and moments of the distribution functions appear to be temperature dependent.