Special Analytical Solutions of the Schr\"odinger Equation for 2 and 3 Electrons in a Magnetic Field and ad hoc Generalizations to N Particles

Abstract

We found that the two-dimensional Schr\"odinger equation for 3 electrons in an homogeneous magnetic field (perpendicular to the plane) and a parabolic scalar confinement potential (frequency $\omega_0$) has exact analytical solutions in the limit, where the expectation value of the center of mass vector $\bf R$ is small compared with the average distance between the electrons. These analytical solutions exist only for certain discrete values of the effective frequency $\tilde \omega=\sqrt{\omega_o^2 + ({\omega_c \over 2} )^2}$. Further, for finite external fields, the total angular momenta must be M_L=3 m 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 previously published exact analytical solutions for 2 particles can be written in an unified form. These formulae, when {\em 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 $\bf R$ 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.