Discrete variable method for non-integrable quantum systems

The states of an atom in external electric fields become quasi-bound since the electron can ionize by tunneling through the potential barrier into the continuum. Due to the external electric field the ionization threshold of the atom is lowered from the field-free value. This process becomes important for states close to the classical ionization energy or above. These resonance states can be studied using the complex coordinate method. In this method the Hamiltonian of the system is continued into the complex plane by a complex dilatation, therefore the Hamiltonian is no longer Hermitian and can support complex eigenenergies associated with decaying states. Resonances are uncovered by the rotated continuum spectra with complex eigenvalue and square-integrable (complex rotated) eigenfunctions. The basic idea is to combine this complex coordinate rotation method with the finite element method, and the discrete variable technique. These two methods have been successfully used to compute atomic data for the hydrogen atom in external magnetic and electric fields. We obtain a complex symmetric Hamiltonian matrix, which we solve using the implicitly restarted Arnoldi method (ARPACK). These methods have been extended to alkali atoms in external strong magnetic and electric fields by including model potentials and have also been successfully used in studying various effective one-particle problems.  1999 Elsevier Science B.V. All rights reserved. Despite its long history, the problem of a hydrogen atom in external magnetic and electric fields is still of significant interest. Since the late sixties evidence has been emerging that huge magnetic and electric fields exist in astrophysical “laboratories” such as neutron stars (B ≈ 107−109 T) and white dwarf stars (B ≈ 102−105 T). Those strong to very strong fields cause a drastic change in the atomic structure and perturbation theory is no longer applicable, making more advanced numerical methods necessary. Studying the external fields for the hydrogen atom gives also some information about shallow donor states, as the Hamiltonians for many systems are equivalent. E.g., for the 1 E-mail: [email protected]. donor in GaAs a magnetic field of 6.56 T will have the same impact as 4.7× 105 T for the hydrogen atom. On the other hand, the rapid advancement of laser spectroscopy has made it possible to produce atoms in a highly exited state. In these empirical studies, a Rydberg wave packet is formed from a low-lying state by a short laser pulse and the time evolution of this wave packet is probed by a second laser pulse. The examples mentioned above are non-integrable systems. One of the fascinating aspects of non-integrable, low-dimensional systems is given by their relation to quantumchaology, but also to fundamental research in all areas of physics. Non-integrability occurs if the number of commuting observables is smaller than the number of degrees of freedom of the system and hence non-integrability is rather the rule than the exception. 0010-4655/99/$ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0010-4655(99) 00 38 60 W. Schweizer, R. González-Férez / Computer Physics Communications 121–122 (1999) 480–482 481 In this paper we will report an effective numerical algorithm [1] based on discrete variable techniques combined with finite elements for solving nonintegrable three-dimensional quantum systems like those mentioned above [2]. Moreover we will discuss a simple computational method [3] for computing laser induced wave packet prolongation for alkali Rydberg atoms. The Hamiltonian of a hydrogen atom with infinite nuclear mass in an external magnetic field B and electric field F reads in spherical coordinates and atomic units as