THE HYDROGEN-ATOM IN INTENSE MAGNETIC-FIELDS - EXCITONS IN 2-DIMENSIONS AND 3-DIMENSIONS

As a model of excitonic behaviour the paper concerns itself with the o ne-electron states generated by a bare Coulomb potential -Ze(2)/tau, p lus an applied magnetic field H which is assumed intense. A quantity w hich reflects the entire one-electron level spectrum epsilon(i) and th e corresponding wave functions psi(i)is the canonical density matrix C (tau, tau, beta) = Sigma(i) psi(i)(tau)psi(i)(tau)exp(-beta epsilon(i) ) whose trace is the partition function Z(beta). The inverse Laplace t ransform of C/beta yields the Dirac density matrix rho(tau,tau,E), its diagonal element being the integrated local density of states. For fr ee electrons in an intense field rho(o)(tau,E) = H(E - H)(1/2) in thre e dimensions and in the presence of the Coulomb potential the Thomas-F ermi approximation replaces E by (E + Ze(2)/tau). One can approximate, albeit somewhat crudely, the lowest energy state by finding the energ y, E(l) say, such that integral(-infinity)(El) rho TF (E) = 2. Such a procedure leads to a large error in E(l) for H = O but is expected to be a better approximation in a large field. Applying the same aproxima tion in two dimensions, with the magnetic field perpendicular to the p lane of confinement assumed for the electrons, leads again to an estim ate for the lowest one-electron state in an intense field. This motiva tes then a further study of this two-dimensional case, in which the pu re cylindrical symmetry of the two-dimensional zero field case is pres erved. Finally, in an Appendix, the bare Coulomb limit of a recent, se lf-consistent field, treatment by Lieb et al. of atoms in a 'hyperstro ng field' such that, in suitable units, H >> Z(3) is set out.