The Radial Equation for Hydrogen

Michel de Celles

Brian P. Anderson

Norman Henry March

Caio Lima Firme

This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).

Madraimova Zahro Dilmurod qizi, Worldly Knowledge Publishing Centre

This paper investigates the radial wave function of hydrogen-like atoms based on the Schrödinger equation. The radial equation is solved using Laguerre polynomials, and radial wave functions are calculated for the states n=1–5. For n=4 and n=5, the effect of the orbital quantum number on the function is analyzed graphically. The results show that the electron energy primarily depends on the principal quantum number.