Two charges on plane in a magnetic field I. “Quasi-equal” charges and neutral quantum system at rest cases Original Research Article

Low-lying bound states for the problem of two Coulomb charges of finite masses on a plane subject to a constant magnetic field image perpendicular to the plane are considered. Major emphasis is given to two systems: two charges with the equal charge-to-mass ratio (quasi-equal charges) and neutral systems with concrete results for the hydrogen atom and two electrons (quantum dot). It is shown that for these two cases, when a neutral system is at rest (the center-of-mass momentum is zero), some outstanding properties occur: in double polar coordinates in CMS image and relative image coordinate systems (i) the eigenfunctions are factorizable, all factors except for image-dependent are found analytically, they have definite relative angular momentum, (ii) dynamics in image-direction is the same for both systems being described by a funnel-type potential; (iii) at some discrete values of dimensionless magnetic fields image the system becomes quasi-exactly-solvable and a finite number of eigenfunctions in image are polynomials. The variational method is employed. Trial functions are based on combining for the phase of a wavefunction (a) the WKB expansion at large distances, (b) the perturbation theory at small distances (c) with a form of the known analytically (quasi-exactly-solvable) eigenfunctions. Such a form of trial function appears as a compact uniform approximation for lowest eigenfunctions. For the lowest states with relative magnetic quantum numbers image this approximation gives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy image for magnetic fields image (hydrogen atom) and image (two electrons). The evolution of nodes of excited states with the magnetic field change is indicated. In the framework of convergent perturbation theory the corrections to proposed approximations are evaluated.