Table of contents

A serious error in the work of Misra et al. (see abstr. A35611 of 1973) on static solutions of the Einstein-Maxwell equations is pointed out and corrected.

A modification of the recently published derivation of Clippe and Evrard (see ibid., vol.7 L89, 1974) of an expression for the free energy of a hard-sphere gas permits its evaluation in closed form.

Lowering operators for O⁺(9) to O⁺(3) and O⁺(7) to O⁺(3) are explicitly constructed. This solves the problem of obtaining an O⁺(3) basis (non-orthogonal) for irreducible representations of U(9) and U(7).

A convenient series expansion of the solutions of the confluent hypergeometric equation is developed for angular momentum quantum numbers l=2 and 3. Following a procedure proposed by Kuhn (1951) the expansion coefficients are deduced for the two independent solutions which may then be combined to give a series representation of the eigenfunctions. The results are couched in terms suitable for application to physical situations involving hydrogenic wavefunctions. The Coulomb (Whittaker) functions are written in terms of the Kuhn asymptotic series expansion with respect to a parameter related to the principal quantum number. The solutions are applied to find electron energy eigenvalues in the neighbourhood of a positive ion of finite size.

An N-dimensional linear Lagrange equation is given which generalizes some classical problems. Particular interest is paid to the equation generalizing a rotation. The general solution is obtained with the use of a matrix exponential method. When applied to the three-dimensional motion of a heavy particle near the surface of the earth, the results are in agreement with known results found by other methods.

The first four energy levels of the anharmonic oscillator H=p²+q²-kp^2M are numerically evaluated for the M values 2, 3, 4, up to the tenth order of perturbation theory, and for M=1, for all orders of perturbation theory. A recursion formula derived in a recent paper of the author on applications of hypervirial and Hellmann-Feynman theorems is used. The results indicate that for M>1, the perturbation expansion in k is divergent. The ratios of absolute values of successive terms of the perturbation series form a monotonically increasing sequence, in agreement with an earlier result of Bender and Wu. The rate of divergence of this sequence of ratios is small. Furthermore, for k=10^-9, the individual values of the ratios are small-less than 10^-4. Thus for sufficiently small k values, and for the purpose of practical calculations, the energy expansion in k effectively behaves like a convergent series up to very high orders of perturbation theory.

The body-fixed symmetry of the ideal asymmetric top is that of the double four (or quaternion) group. For integral spin this is equivalent to the ordinary four-group and the secular determinant breaks down diagonally into four distinct blocks, as in the standard theory. For half-odd-integral spin, however, one obtains two equivalent blocks, the levels being doubly degenerate. A generalization is then presented which entails making the top wavefunction a spinor-valued quantity. This is expanded in spinor hyperspherical harmonics which are generalizations of the ordinary hyperspherical harmonics, the D_MN^L. A systematic procedure is given for finding the elements of the secular determinant. The motivation for the generalization is that of spinor fields in a frozen Mixmaster universe.

The notions of systematic and accidental degeneracy are discussed for a quantum-mechanical particle in a two-dimensional box. Separability of coordinates is shown to admit extra systematic degeneracy over that expected from the point-group symmetry. The 'accidental' degeneracy for square and triangular boxes is discussed in detail and related to geometric symmetries-described by rings of operators, however, and not by a symmetry group.

The manner in which the extension of a small wave packet moving in a circular orbit in a central potential V(r) changes with time is examined. It is shown that in general, the extension in the orbital plane increases indefinitely at large times unless rV"-V'=0 at the radius of the orbit. The question whether there exist any special 'coherent' wave packets which retain a finite size for arbitrary times in a Coulomb potential is investigated, and the answer is shown to be in the negative.

A prescription is given by which the distorted waves and the on-energy-shell T matrix for particles interacting via Coulomb- like potentials can be determined from the corresponding screened distorted waves and on-energy-shell screened T matrix.

A method developed by Glasser in 1973 is used to evaluate exactly lattice sums in 2, 4, 6 and 8 dimensions. Amongst the many results given are the Madelung sums. A certain three-dimensional sum first evaluated by Glasser but inaccurately presented, is given correctly.

For pt. II see abstr. A40429 of 1972. The star cluster expansion for the specific heat of a three-dimensional Ising ferromagnet is described. The classification of stars by ascending number of edges is studied and a canonical description developed. The construction of a general star-graph list for high temperature expansions is described.

Some results relevant to the derivation of the high temperature series expansion for the zero-field free energy of the Ising model are collected from the literature. The general theory of the star cluster expansion is summarized and some methods for the derivation of the appropriate weights are given in outline.

Generalized codes are defined for the simple quadratic lattice and it is shown how they may be used to derive high magnetic field or low temperature expansions for two Ising-type models with four-spin interactions. One of these models, namely Baxter's eight-vertex model, has critical exponents which are known to depend on x, the ratio of the strengths of the four-spin to two-spin interactions. The series so derived are analysed to yield estimates of the critical exponent delta for the magnetization as a function of magnetic field at T_c. The results for both models are consistent with a constant value delta =15 independent of x as predicted by the scaling laws.

For pt. I see abstr. A11864 of 1974. A study is made of the ground state orderings of the one-dimensional Ising system, under the assumption that the interaction is of finite range r and has inversion symmetry. It is shown that the ground state energy is effected by the regular chain either of an irreducible block without or with one symmetry point of r or (r+1) sites, or of a symmetric irreducible block with two symmetry points. For the case of the one-dimensional Ising magnet under a uniform external field, it is effected by two, four, seven and twenty regular orderings, when the range r of the interaction is one, two, three and four, respectively.

High-field expansions are obtained for the three-state Potts model. These expansions are used to investigate the critical isotherm of the three-state system on a square lattice, and to determine the general properties of the transition in face-centred cubic systems.

An alternative method has been found to display the information contained in the Fourier transform of a helical particle. This allows a strong selection rule to be defined for the transform on the layer lines and consequently the discrimination between signal and noise contributions to the data can be improved.

A particular model for N coupled waves in non-thermal equilibrium is studied in detail. Exact closed forms for the one- and two-wave intensity distribution functions and an exact recurrence relation for the photon counting distribution function are obtained. The 'thermodynamic limit', N to infinity , is discussed with special reference to the multi-mode laser, and a mean field type phase transition is shown to occur at laser threshold.

A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.

It is shown that certain features of the Weisskopf-Wigner theory of natural line width can be generalized for the construction of a systematic, mathematically and logically autonomous approximation scheme for treating the interaction of a bound electron with photons. This scheme is equipped with its own hierarchy of orders which are not ruled by powers of the coupling constant. Under very weak conditions on the bound electron all Weisskopf-Wigner theories of finite order exist as ordinary quantum theories on ordinary Hilbert spaces H_I and have strictly unitary time evolution operators U_I(t), t< infinity . This implies in particular that in such theories the interaction of any finite number of states of the Dirac hydrogen atom with any bounded number of photons can never lead to divergencies. If an infrared catastrophe is banned by a small photon mass mu >0, Weisskopf-Wigner type theories of the interaction of an 'm-level atom', m< infinity , with any unbounded number of photons also exist and have strictly unitary time evolution operators. Some examples of novel applications of Wiesskopf-Wigner type theories are given.

In quasi-stellar objects the redshifts of spectral lines are generally different in absorption and emission. The possibility that the redshift difference in most cases is gravitational is not ruled out. The effect can be seen in a Fowler-Hoyle type model. There exists the possibility that most (but not all) of lines with z_ab<z_em might be associated with quasistellar objects.