Analytical Solutions of the Schrödinger Equation for 2 and 3 Electrons in a Magnetic

We found that the two–dimensional Schrödinger equation for 3 electrons in an homogeneous magnetic field (perpendicular to the plane) and a parabolic scalar confinement potential (frequency ω 0) has exact analytical solutions in the limit, where the expectation value of the center 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˜ω = ω 2 o + (ωc 2) 2. Further, for finite external fields, the total angular momenta must be M L = 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 3 particles and the exact solutions for 2 particles [13] can be written in an unified form. The first set of solutions reads Φ = − 1 2 ˜ ω n,m l r 2 l where p n,m (x) are certain finite polynomials and˜ω n,m is the spectrum 1 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˜ω 1 there is a second set of the form Φ = A a i<k (r i − r k) m ik exp − 1 2 ˜ ω 1 l r 2 l where A a is the antisymmetrizer and the pair angular momenta m ik can all be different integers. In both cases the first factor is a short– hand with the convention r m = r |m| e imα. 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 fulfills the following demands: it is exact for two particles and for 3 particles in the limit of small R and for the solvable external fields, and it is an eigenfuncton of the total orbital angular momentum. The Laughlin functions are special cases of this ansatz for infinite solvable fields and equal pair– angular– momenta.