On the singular solution of Schrödinger equation for the hydrogen atom


Purpose. The authors of known for us textbooks on quantum mechanics pay attention only to the first regular solution of Schrödinger equation for the hydrogen atom. To exclude the second linearly independent solution from the general solution, different textbooks give various arguments such as invalid boundary condition in the coordinate origin, the appearance of Dirac delta function or divergence of the kinetic energy in the origin.
Methods. Using the power series method, we obtained an exact analytic expression for the second independent solution of Schrödinger equation for the hydrogen atom.
Results. 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.
Conclusions. On the example of the hydrogen atom, we demonstrated the mathematically correct algorithm of construction of the independent solutions for the power series method. In particular, this algorithm is important in the case of quantum systems with coupled channels which are described by two or more coupled Schrödinger equations

Keywords: hydrogen atom, regular solution, singular solution, ordinary differential equation, indicial equation