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Much work has been devoted to the study of nonstrange baryons belonging tothe $[\textbf{70},1^-]$ multiplet in the framework of the $1/N_c$ expansion.Using group theoretical arguments here we examine the relation between theexact wave function and the approximate one, customarily used in applicationswhere the system is separated into a ground state core and an excited quark. Weshow that the exact and approximate wave functions globally give similarresults for all of mass operators presented in this work. However we find thatthe inclusion of operators acting separately on the core and on the excitedquark deteriorates the fit and leads to unsatisfactory values for thecoefficients which encode the quark dynamics. Much better results are obtainedwhen we include operators acting on the whole system, both for the exact andthe approximate wave function.

The wave function of $[\bf{70},1^-]$ baryons in the $1/N_c$ expansion