Two charges on plane in a magnetic field I. “Quasi-equal” charges and neutral quantum system at rest cases
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.<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. -- Highlights: •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.
- OSTI ID:
- 22224278
- Journal Information:
- Annals of Physics (New York), Vol. 340, Issue 1; Other Information: Copyright (c) 2013 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
- Country of Publication:
- United States
- Language:
- English
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