The fundamental solutions of the Schroedinger equation for the hydrogen atom

Abstract

Using the power series method, we obtained an exact analytic expression of a second linearly independent solution of the Schrödinger equation for the hydrogen atom. The solution consists of a sum of two parts, one of which increases indefinitely over long distances, while the other is limited and contains a logarithmic term. This feature is peculiar to all values of the orbital angular momentums. In known for us textbooks on quantum mechanics, the first regular solution is considered only. To exclude the second linearly independent solution from the general solution, different textbooks give various arguments.