J. Phys. B: At. Mol. Phys. zyxwvut 15 (1982) 1195-1204. Printed in Great Britain zyxw Semiclassical quantisation of the hydrogen atom in crossed electric and magnetic fields T P Grozdanovt and E A Solov’evS tInstitute of Physics, PO Box zyxwvu 57, Belgrade, Yugoslavia $Faculty of Physics, Leningrad State University, USSR Received 12 October 1981, in final form 19 January 1982 Abstract. This method, based on the adiabatic invariance of semiclassical quantisation conditions (Solov’ev 1978), is used to calculate the energy levels of a three-dimensional, non-integrable system: a hydrogen atom in crossed electric and magnetic fields. The results are presented for a ground state and two excited states, for various ratios of field strengths and angles between the fields. The limitations of the method are discussed. 1. Introduction Semiclassical quantisation of non-integrable systems is important both for fundamental and practical reasons. It contributes to the clarification of the not completely under- stood relations between the classical and quantum mechanics, and also appears to be competitive with the corresponding quantal calculations for large quantum numbers. In the last case the quantal treatments become very complicated and expensive, due to the large number of basis states involved in calculations. In classical mechanics, non-integrable systems are described by non-linear equations (Hamilton-Jacobi equation or equations of motion), which are qualitatively more complicated than the corresponding linear equations of quantum mechanics, and so far, the problem of quantisation has not been rigorously formulated for them. Most practical semiclassical calculations of energy spectra of non-integrable systems performed up to now, used the Einstein-Brillouin-Keller (EBK) quantisation procedure (see, for example, Percival (1977) and references therein). The main points of this procedure are briefly summarised below. An integrable classical system of s degrees of freedom can be characterised by a set of fundamental frequencies zyxwv {wj, zyxwvut j zyxwvu = 1,2, . . . , s}, and a set of canonically conjugate, angle (linear in time) and action variables: ej = wjt + zyxwvu sj where zyxwvut Si are initial phases, (Pk, qk) are canonically conjugate momenta and coordinates and Cj are closed paths in phase space, defined by: 0 s ej < 2 ~ , ei = constant for i # j. Action variables Ii are single-valued integrals of motion. By fixing them one defines an s-dimensional hypersurface in phase space, called the invariant toroid. Any path 0022-3700/82/081195 + 10$02.00 @ 1982 The Institute of Physics 1195