Special analytical solutions of the Schrödinger equation for two and three electrons in a magnetic field and ad hoc generalizations to N particles

Abstract

We found that the two-dimensional Schrödinger equation for three electrons in a homogeneous magnetic field (perpendicular to the plane) and a parabolic scalar confinement potential (frequency ω0) has analytical solutions in the limit where the expectation value of the centre-of-mass vector R is small compared with the average distance between the electrons. These analytical solutions exist only for certain discrete values of the effective frequency . Furthermore, for finite external fields, the total angular momenta must be ML = 3m with m = integer, and spins have to be parallel. The analytically solvable states are always cusp states, and take the components of higher Landau levels into account. These special analytical solutions for three particles and the exact solutions for two particles (Taut M 1994 J. Phys. A: Math. Gen. 27 1045 and Taut M 1994 J. Phys. A: Math. Gen. 27 4723 (erratum))can be written in a unified form. The first set of solutions reads where thepn,m(x)are certain finite polynomials and n,m is the spectrum of the fields. The pair angular momentum m has to be an odd integer and the integer n defines the number of terms in the polynomials. For infinite solvable fields 1, there is a second set of the form where Aa is the antisymmetrizer and the pair angular momentummik can all be different integers. In both cases the first factor is a short-hand form with the convention rm = r|m| eim. These formulae, when ad hoc generalized to N coordinates, can be discussed as an ansatz for the wave function of the N-particle system. This ansatz fulfils the following requirements: it is exact for two particles and for three particles in the limit of small R and for the solvable external fields, and it is an eigenfunction of the total orbital angular momentum. The Laughlin functions are special cases of this ansatz for infinite solvable fields and equal pair angular momenta.