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

• Published in: Annals of physics Vol. 340; no. 1; pp. 37 - 59
• Main Authors: Escobar-Ruiz, M.A., Turbiner, A.V.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.01.2014; Elsevier BV
• Subjects: ANGULAR MOMENTUM; APPROXIMATIONS; BOUND STATE; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Coulomb friction; EIGENFUNCTIONS; ELECTRONS; EXACT SOLUTIONS; EXCITED STATES; HYDROGEN; Hydrogen atoms; Magnetic field; MAGNETIC FIELDS; MASS; Mathematical analysis; PERTURBATION THEORY; Planes; POLYNOMIALS; QUANTUM DOTS; QUANTUM NUMBERS; Quantum theory; Rest; Two-body planar Coulomb system; VARIATIONAL METHODS; WAVE FUNCTIONS; Two-body planar Coulomb system; Magnetic field
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2013.10.010

Low-lying bound states for the problem of two Coulomb charges of finite masses on a plane subject to a constant magnetic field B 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 (R,ϕ) and relative (ρ,φ) coordinate systems (i) the eigenfunctions are factorizable, all factors except for ρ-dependent are found analytically, they have definite relative angular momentum, (ii) dynamics in ρ-direction is the same for both systems being described by a funnel-type potential; (iii) at some discrete values of dimensionless magnetic fields b≤1 the system becomes quasi-exactly-solvable and a finite number of eigenfunctions in ρ 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 s=0,1,2 this approximation gives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy E(B) for magnetic fields 0.049a.u.<B<2000a.u. (hydrogen atom) and 0.025a.u.⋜B⋜1000a.u. (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. •Approximate solution for the low-lying eigenstates for two Coulomb charges given.•Factorization of eigenfunctions in double polar center-of-mass coordinates uncovered.•A magnetic field range varies from weak to ultra-strong, 1000–2000 a.u.