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Correspondence identities.

Bohr's correspondence principle relates quantum phenomena to classical mechanics in the limit ~h / S→0, as the dynamical action variables S become large in comparison with Planck's constant. Relations between quantum and classical mechanics which hold even for low quantum numbers and relatively small values of the classical action are called correspondence identities. For the Coulomb potential the following three correspondence identities are known: 1) The Rutherford Scattering identity The quantum-mechanical and classical angular differential cross sections for the scattering of a charged particle by a fixed charge are the same. (2) The Bohr-Sommerfeld identity The old quantum theory, which postulated that only those orbits occur for which the action around the classical path of a periodic system is a multiple of 2π~h gives the correct energy levels of the hydrogen atom and hydrogenic ions. (3) The Fock identity The classical and quantal microcanonical distributions in momentum of the electron in the hydrogen atom are equal for all values of the classical energy equal to the levels En. These correspondence identities concern the system of electron and proton and in this thesis it is shown how each of the identities follows from a complete correspondence identity whereby the non-relativistic quantum dynamics of the system is obtained from the solution of the corresponding classical problem. A complete correspondence identity is provided by expressing the kernel of the spectral operator IE = δ(E — H) in momentum representation for all real non-zero energies E, as a sum over paths of terms containing the classical action. For the bound states the paths are the classical paths. For positive energies they are the generalised classical paths which arise from the analytic continuation in energy of the bound state paths. The generalised classical paths are built up from the paths of scattering of both electrons and positrons and are needed to obtain the quantal barrier penetration in momentum space. Because of the similarities between the techniques used in this thesis to provide a complete correspondence identity and those of the phase-integral approximation the results are compared wherever possible with those of Gutzwiller (1967). Finally, a general derivation of a scattering cross section from the spectral operator is presented which does not require an explicit treatment of the long-range distortion in the case of the Coulomb potential

Industrial Change in Old Port Areas, the Case of the Port of Toronto

The classical association of ports with industry needs revising. Traditionally, industry in the port area was concerned with the port operation itself, with ships, and with the cargoes transported by the ships. In recent years, as a result of technological changes, land use in the central water-front has tended to switch into residential, recreational, commercial and institutional uses, while new industries have been attracted to the adjacent old port-areas. These new industries have an entirely different role, being much more closely tied to the function of the city itself. They include: industry oriented to serving the local urban market; noxious industry; and labour-oriented industry. Many port-related industries are now located downriver or at deep-water greenfield locations where extensive sites are available. The illustration of these changes is presented with a case study of Toronto and the example of some major Canadian port cities.L'association classique entre les ports et les industries doit être réexaminée. Traditionnellement, les industries situées dans les zones portuaires étaient dépendantes du fonctionnement du port lui-même, des navires, et des marchandises transportées par ces navires. Récemment, par suite de changements technologiques, la zone située à proximité des quais centraux a vu apparaître des usages résidentiel, récréationnel, commercial et institutionnel, alors que de nouvelles industries ont été attirées par les vieilles zones portuaires qui sont contiguës. Ces nouvelles industries ont un rôle tout différent, car elles sont associées aux fonctions de la ville elle-même. Elles comprennent les industries associées au marché urbain régional, les industries nuisibles pour l'environnement et les industries étroitement dépendantes de la main-d'oeuvre. Beaucoup d'industries dépendant directement du port ont émigré dans les zones situées en aval ou dans des sites d'eau profonde à proximité d'espaces vacants. Ces changements sont illustrés à l'aide du cas du port de Toronto et de l'exemple de quelques grandes villes portuaires canadiennes