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Arabian journal of science.

RADIAL WAVE FUNCTION AND SOLUTIONS OF HYDROGEN-LIKE ATOMS

Madraimova Zahro Dilmurod qizi, Worldly Knowledge Publishing Centre

Published Aug 12, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21897371

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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.
The simplest atom consisting of a proton and an electron is a hydrogen atom. The energy level and the probability of finding an electron in a hydrogen atom can be determined based on the function of radial wave. The function of radial wave of a hydrogen atom is influenced by the principal quantum number ( n ) and the orbital quantum number ( l ). Completion of the function of radial wave of a hydrogen atom in the principal quantum numbers( n ) 4 and 5 uses the time-independent Schrodinger equation approach in spherical coordinates, variable separation method and uses the associated Laguerre function. The normalized of the function of radial wave of a hydrogen atom R nl ( r ) at n = 4 produces four wave functions:R 40 ( r ); R 41 ( r ); R 42 ( r ); R 43 ( r ), whereas at n = 5 produces five wave functions:R 50 ( r ); R 51 ( r ); R 52 ( r ); R 53 ( r ); R 54 ( r ). The results of this research in the form of the functions of radial wave of hydrogen atom can be used to determine the behavior of electron in a hydrogen atom.
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.
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).
In this chapter we solve the radial TISE for the case of a radial potential, specifically the radial potential of an electron attracted to the proton which is the nucleus of a hydrogen atom. Again, we rely on solutions to a related equation solved by a French mathematician, this time named Laguerre, whose polynomials are, in this chapter, multiplied by a simple exponential function to produce the radial wave functions. These functions contain the principle quantum number, which corresponds to the row number in the periodic table. Multiplying these radial solutions by the spherical harmonics from Chapter 14 produces the full three-dimensional solutions for the hydrogen atom, also known as the atomic orbitals. The relationship between the quantum numbers n , l , and m arise naturally from the possible combinations of radial solutions with spherical harmonics.
In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.