Improved variational method that solves the energy eigenvalue problem of the hydrogen atom

In most quantum mechanics textbooks for graduate studies, the hydrogen atom is studied in an approximate way by means of the variational method. The type of trial functions commonly used are the Gaussian and the Lorenzian. In this paper we consider a natural generalization of the Mei method to improve the trial wave functions [1] applied to hydrogen atom. We propose a sequence of functions as trial wave functions to calculate the eigen-energies of the hydrogen atom. These trial wavefunctions are given in terms of three variational parameters, one of them is fixed by means of the normalization condition and the other two are adjustable parameters. One of these parameters can be chosen as being an integer number, then this parameters will define the sequence of functions. We will show that when the integer parameter approaches to infinity, the ground state, first excited state and second excited state converge to the exact results.