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arXiv:hep-ph/0209064v1 5 Sep 2002Calculation ofthe energy levels and sizes ofbaryonswith a noncentral harmonic potentialM´ ario Everaldo de SouzaDepartamento de F´isica, Universidade Federal de Sergipee-mail: mdesouza@sergipe.ufs.brPACS numbers: 12.40.Yx Hadron mass models and calculations. 12.39.Jh Non-relativistic quark model. 12.39.Pn Potential ModelsAbstract. It is considered that the effective interaction between any two quarks ina baryon can be approximately described by a simple harmonic potential. Also, itis made use of the nonrelativistic approximation since the effective (constituent)masses of quarks are not small. The problem is firstly solved in Cartesian coor-dinates in order to find the energy levels irrespective of their angular momentaand then it is also solved in polar cilindrical coordinates for taking into accountthe angular momenta of the levels. By making a comparison between the twosolutions the energies and the corresponding angular momenta (and parity) of al-most all baryon levels are described. The agreement with the experimental datais quite impressive. The solution in Cartesian coordinates also produces somevery important figures for the sizes of baryons and for the harmonic oscillatorconstant which is clearly related to confinement.1IntroductionAs is well known there are several important works that deal with the calcu-lation of the energy levels of baryons. One of the most important ones is thepioneering work of Gasiorowicz and Rosner [1] which has calculation of baryonlevels and magnetic moments of baryons using approximate wavefuncions. An-other important work is that of Isgur and Karl [2] which strongly suggests thatnon-relativistic quantum mechanics can be used in the calculation of baryonspectra. Other very important attempts towards the understanding of baryonspectra are the works of Capstick and Isgur [3], Bhaduri et al.[4], Murthy etal.[5], Murthy et al.[6], and Stassat et al.[7]. An important work attempting todescribe baryon spectra is the recent work of Hosaka, Toki and Takayama [8]published in 1998. This last work arrives at an important equation which hadalready been deduced by De Souza a long time ago, in 1992 [9]. Other worksby De Souza published before 1998 include it [10,11].The effective potential between any two quarks is not known and becauseof this several different potentials are found in the literature. In particular theharmonic approximation using a harmonic central potential has been widelyused. Since the three quarks of a baryon are always in a plane it is assumedthat the effective potential between any two quarks ofa baryon can be given by alinear harmonic oscillator. The motion of the plane of quarks is not consideredin this paper.This is a calculation quite different from those found in theliterature.1