The auxiliary field method is a new technique to obtain closed formulae for the solutions of eigen equations in quantum mechanics. The idea is to replace a Hamiltonian H for which analytical solutions are not known by another one ÌH including one or more auxiliary fields, for which they are known. For instance, a potential V(r)not solvable is replaced by another one P(r) more familiar, or a semirelativistic kinetic part is replaced by an equivalent nonrelativistic one. If the auxiliary fields are eliminated by an extremization procedure, the Hamiltonian ÌH reduces to Hamiltonian H . The approximation comes from the replacement of these fields by pure real constants. The approximant solutions for H, eigenvalues and eigenfunctions, are then obtained by the solutions of ÌH in which the auxiliary parameters are eliminated by an extremization procedure for the eigenenergies, which takes the form of a transcendental equation to solve.