IIMAS - FENOMEC
UNAM
Two charges on a plane in a constant magnetic field
Resumen:
The classical and quantum mechanics of two Coulomb charges on a plane
(e1, m1) and (e2, m2) subject to a constant magnetic field
perpendicular to the plane is considered. Special "superintegrable"
trajectories (circular and linear) for which the distance between charges
remains unchanged are indicated and their constants of motion are found.
The number of the independent constants of motion for the special
trajectory is larger than for generic ones.
Low-lying bound states for the quantum problem of two Coulomb charges 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, but 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 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 in general.
Informes: coloquiomym@gmail.com, o al 5622-3564.