The semiclassical resonance spectrum of hydrogen in a constant magnetic field

We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with Lz = 0. The classical system is unbound and completely chaotic for a scaled energy ∼ EB−2/3 larger than a critical value c > 0. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron–nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Lett. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein–Brillouin–Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds. PACS numbers: 0545, 0365, 3115, 3230 The hydrogen atom in a uniform magnetic field has become one of the most important examples for studying the correspondence between quantum mechanics and classical chaos. The Hamiltonian is known to a high accuracy and furthermore, the system is experimentally accessible in the laboratory. The angular momentum in the direction of the magnetic field is conserved, which reduces the classical system to a problem with two degrees of freedom. The classical flow in phase space covers a wide range of Hamiltonian dynamics reaching from bound, nearly integrable behaviour to completely chaotic and unbound motion by varying one parameter, the scaled energy, . In the 1980s, the system served as a catalyst for quantum chaos. Modulations in the absorption spectra of highly excited hydrogen atoms in a magnetic field, the so-called quasiLandau levels [1, 2], could be assigned to classical trajectories [3, 4] and could be understood ‡ email: [email protected] ‖ email: [email protected] + email: [email protected] ∗ Present address: NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark. Also at: Physics Department, University of Oslo, Box 1048, Blindern, N-0316 Oslo, Norway. 0951-7715/96/061641+30$19.50 c © 1996 IOP Publishing Ltd and LMS Publishing Ltd 1641