The present study is aimed at solving the problem of the Schrodinger equation model for the hydrogen atom. An important clarification is the absence of the topic of direct postulation of the discreteness of eigenvalues, and the transition through boundary and initial conditions. The relevant conclusions of the study were the found forms of the psi function of the probability density, reflecting various orbitals and properties of the electron cloud. Special attention is paid to the mathematical distribution of the theory with an overview of many forms of solutions, from the idealized classical model to the Dirac equation. The research is also aimed at further continuation and extrapolation of the method relative to more complex multiparticle systems, implying an extension of the original task.