A quantum-mechanical generalisation of Carathéodory’s theorem in classical dynamics is established. Several related properties of classical canonical transformations are also generalised to the quantum case.

The representation of the group SU(2) afforded by the symplecton calculus of Biedenharn and Louck is mathematically related to the older treatments of the representations of this group. The method used is similar to the phase space description of quantum mechanics, and considerably simplifies important calculations.

Simplified formulae for the effective electromagnetic transition matrix elements and the core polarization contribution to the effective two-nucleon interaction are derived. From these general expressions, the polarization effects in any other physical quantity of interest can easily be written down. It is also proved that the usual RPA eigenvalue problem corresponding to a 2n×2n matrix$$\left( \begin{gathered} AB \hfill \\ - B - A \hfill \\ \end{gathered} \right)$$ is equivalent to the diagonalization of an×n matrix (A+B) (A−B).

A medium with periodic variation of the dielectric tensor is considered. It is assumed to have helical symmetry. The response to external fields is described by the most general linear law—that is, by the methods of spatial dispersion theory. The propagation of a wave is described by a mutually consistent electric field and polarisation. It is shown that the presence of the medium produces changes in the polarisation and wave vector of the electric field, and the selection rules governing these changes are derived from symmetry. The results generalise previous work on the Oseen model for a cholesteric liquid crystal to the case when the molecules are not perpendicular to the helical axis. This can arise in an external magnetic field applied along this axis.

Two ion sources of the charged particle oscillator type are described. The discharge chamber consists of an outer cylinder as cathode and an axial wire or a pair of wires along the axis as the anode. The energy spectra and ionic composition of the ion beam have been measured with a double focusing mass spectrometer of the Mattauch-Herzog design constructed in the laboratory. A high percentage of dissociated ions and multiply charged ions has been observed in the ion beam. A brief description of the mass spectrometer is also given.