The Schrödinger Equation and the Confluent Hypergeometric Functions

Chen Gang

In this paper, the three-dimensional radial position-dependent mass Schrödinger equation is exactly solved through mapping this wave equation into the constant mass Schrödinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.

Franz Schwabl

Guruh Sukarno Putra, Hamdi Akhsan

The energy for each level of the hydrogen atom was triggered by Neils Bohr, where the amount of energy decreases at every level of orbit. The value of energy depends on the principal quantum number which only applies to atoms with the s subshell of an atom with the same characteristics as the hydrogen atom. With the advancement of science, specifically in quantum physics, many potential equations aimed to solve the time-independent Schrodinger equation with the solution of the estimated value of energy. One of the applicable potential equations is the Hulthen potential which is explained about the potential at a small distance and only applies to atoms and micro-scale objects in physics and only has a solution at the azimuth quantum number value l = 0 or groups of atoms subshell s. Using the confluent hypergeometric function, the combination of the Hulthén potential and time-independent Schrödinger equation generates the estimated energy value of the hydrogen atom level, which can be used as a comparison with the estimated energy value of Bohr’s hydrogen atom. This study found that the comparison between Hulthén and Bohr energy level lies in the same ratio 1:1

Franz Schwabl

V. V. Kudryashov, A. V. Baran

The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schrödinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.