Table of contents

Path integration for a one dimensional three body problem is discussed. This involves essentially the path integration of a Lagrangian in plane polar coordinates with a potential energy term depending on both the radial and angular coordinates.

It is pointed out that the exponent gamma (for which the mean-square end-to-end length of an n step walk approximately= constant n^gamma ) is associated also with other purely topological properties of the lattice. It is therefore suggested that a better numerical estimate of gamma may be possible, since the problem of the 'distance' between two sites of a network has been bypassed.

Treats the propagation of longitudinal electric waves in a cold plasma, which can occur if the photon has a nonzero rest mass.

The hyperfine interaction of excited nuclear levels with deeply ionized atoms is discussed. A measurement is reported for ¹⁸O(2⁺) at 1.98 MeV, interacting with 7⁺ ions in the 1s state. Limits for the g factor of this level are reported as: 0.2< mod g mod <0.36. The excitation pattern of highly stripped ions is considered and it is concluded that such ions are formed in two distinct patterns, one characterized by predominant ground state occupation and the other by prolific excitation. The feature which determines the pattern is the degree of isolation of the ground state of the ions.

The connection between single particle and collective motion is discussed on the basis of the generator coordinate method. Earlier and recent work on the subject is reviewed.

The basic framework for discussing quantum mechanics on multiply connected spaces is presented using the covering space concept. The theorem of Laidlaw and DeWitt is rederived and extended to the case of field theory. It is pointed out that chiral dynamics is similar to Skyrme's nonlinear theory and forms another example of Finkelstein's kink idea. The possible existence of ' pi geons' is raised, and the fact that the pion manifold may be any one of the Clifford-Klein constant curvature space-forms, rather than just the whole three-sphere, is suggested. The related formalism for quantum mechanics on homogeneous spaces is given in general terms.

The relativistic Kepler problem in Dirac form is solved by means of a quaternionic method for particles having both electric and magnetic charges. The variables in spherical coordinates can be separated in a simple way and the results obtained are similar to those which are valid for the usual hydrogen atom.

The two-particle self-energy operator is derived. The method is an extension of the Kadanoff and Baym method for the one-particle self-energy operator. The self-energy is then expanded to illustrate the similarities and differences with the one-particle case. An example is used to study the structure of the expansion. In conclusion a brief discussion of the one-particle method to derive the two-particle Green function is given.

The different kinds of motion to be expected are classified in terms of the relative magnitudes, expressed in time scales, of inertia, viscosity, barrier energies, and temperature. In particular, and as a comparison for other cases, the idealized case of a strictly inextensible, but otherwise freely flexible chain, is solved in detail. The nonlinear nature of the constraint is shown to lead to the generation of a Fokker-Planck equation. This Fokker-Planck equation shows that there are wave-like solutions to the motion for long wave fluctuations in the conformation of the chain (second sound) in the absence of viscosity. In the high viscosity limit the equation has Rouse-like solutions which correspond to short wave changes in conformation. In general, the equilibrium functions are modified so that if the chain is considered as a series of connected mass points, the energy is kT per point, not ³/₂kT, a result which though intuitively obvious, is difficult to establish. This idealized model can be related to several realistic cases but a reinterpretation of the constants is involved.

The problem of finding and calculating numerically a conformal mapping which simultaneously uniformizes several branch points in energy of a partial wave scattering amplitude is studied. Arguing from a potential scattering model a general prescription is given for the construction of such conformal mappings in terms of automorphic forms. The automorphic forms are defined with respect to a group each of whose generators arises from one of the cuts in the energy plane of the scattering amplitude. The automorphic forms are calculated numerically from their Fourier series. The method is illustrated in detail for a relativistic single particle exchange model. Finally it is shown how the conformal mapping leads to simple representations of the analytic properties of the amplitude in terms of the uniformizing variable.

It is proposed that thermodynamic relations to be used for lambda transitions in mixtures will normally follow the lines of Pippard's relations for lambda transitions in pure substances. The relations given by Stout, treating C_p as finite but discontinuous, will usually be irrelevant.

For pt. I see ibid., vol.5, 1004 (1972). On the basis of the behaviour of magnetic and electric susceptibilities, and of magnetostriction and electrostriction, the possibility is investigated of obtaining additional Pippard relations for the effect of change of composition on (magnetic or electric) Curie points and Neel points of mixtures. The question is aired of whether further such relations might not sometimes be obtainable on the basis of the behaviour of properties other than equilibrium thermodynamic properties.

For pt. II see ibid., vol.5, 1015 (1972). On the basis of the discussion in part I of the possibility of obtaining Pippard's relations involving the dependence of a lambda temperature on the composition of mixtures, together with standard relations for liquid- vapour equilibria generally, comparisons are made between the behaviour of the specific heat of solutions of ³He in liquid ⁴He and (a) the change in the composition of the saturated vapour as the lambda temperature is approached, and (b) the change in the slope of the vapour pressure curve as the lambda temperature is approached.

For pt. I see abstr. A81745 of 1971. The classical radiation field is examined, clarifying some concepts and definitions, and the results of I are applied to derive the usual expressions for the real photon processes of conventional quantum electrodynamics.

The quantization of a massive spin one particle field, satisfying the first order system of equations due to Proca, is summarized. The interaction of such a field with an arbitrary field is then considered. Equivalent theories in terms of vector and antisymmetric second rank tensor spin one fields are then constructed in a simple manner, providing a generalization of an equivalence theorem due to the author. In addition, this approach, involving the use of first order systems of equations, allows a simple extension to the case of equivalence theorems for arbitrary integral spin. The nature of this extension is indicated.

Projection operators appropriate to the general multispinor representations of SU(n) are constructed in a systematic way from the components of a single generic SU(n) vector transforming as the adjoint representation. The techniques have been devised with the problems of writing explicit forms for finite SU(n) rotations and nonlinear chiral Lagrangians kept specifically in mind. In particular, the general second rank tensors constructed from a single vector are found, counted, and exhibited in a very tractable form.

A method for the numerical integration of the cascade equations of the longitudinal development of electromagnetic cascades is described. Numerical results on the longitudinal development of electron induced cascades in lead are presented in graphical form. Comparisons with data from Monte Carlo simulations and with experimental data are presented. The accuracy of approximations A and B of conventional cascade theory is investigated.

The levels of ³⁴S from 5.4 MeV to 7.3 MeV excitation energy have been studied by particle- gamma ray angular correlation experiments using the reaction ³⁴P( alpha ,p)³⁴S. Spin assignments of 2, 3, 4, 4, 1, 2 and 2 have been made to the levels at 5.680, 6.173, 6.250, 6.415, 6.480, 7.112 and 7.220 MeV respectively. Spin restrictions have been placed on other levels and some previous spin assignments have been confirmed. Branching ratios have been determined for many of the observed decay modes and multipole mixing ratios obtained for several transitions.

The total neutron cross section of ³¹P has been measured from a neutron energy of 0.75 MeV to 9.00 MeV. Good agreement is obtained with the calculated values using the generalized optical model potential of Wilmore and Hodgson.

Results are presented for a low frequency self-excited oscillation in an inhomogeneous r.f. collisional plasma. By using a method of dynamic stabilization to reduce the instability to a low level frequency measurements have been made in the linear regime where most linearized theories apply. Also, by analysing the subsequent return of the oscillations to its saturation level, the linear growth rate has been obtained. Comparison of these growth rate and frequency measurements in the linear regime with a previous theory of the drift-dissipative instability shows reasonably good agreement.

Vlasov's equation and the full set of Maxwell's equations are solved as an initial value problem in a semi-infinite plasma. On specifying boundary conditions, a dispersion relation is obtained for surface waves in two situations, one without a wave incident on the boundary, the other with such a wave. The former case includes all previous results as special cases. In the latter case, it is found that surface waves cannot be excited by a wave incident on the boundary.

An experiment is described in which plane Langmuir probes with different orientations with respect to ion flow were used to measure ion velocity in the side arm of a low pressure mercury discharge. The ions appear to be supersonic, a result which confirms recent theoretical predictions.