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Jurnal Penelitian Pendidikan IPA

Legendre Polynomials are Associated in the Schrodinger Polar Equation of Helium Ions

Bambang Supriadi, Eryna Dwicahyaning Krisnawanto, Ratih Wahyudianti, Mohammad Fathorrosi, Muhammad Najih Hamdi, Eka Citra Rahayu, Haniyah Alivia

Published May 25, 2024• 0 citations
Publisher

Abstract

The helium ion is one of the ions classified as a hydrogenic atom because it has only one electron in its outer orbital so that the helium ion problem can be solved using the Schrodinger equation approach. The Schrodinger equation of the polar part of the helium ion can be solved by the associated legendary polynomial function, . The polar function expresses the angle formed by the vector of the electron's position relative to the z-axis of the helium ion's nuclear position. And it is produced that the polar function of the hydrogenic atom has the same polar function both in position space and in momentum space .

DOI
10.29303/jppipa.v10i5.5913

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The helium ion is one of the ions classified as a hydrogenic atom because it has only one electron in its outer orbital so that the helium ion problem can be solved using the Schrodinger equation approach. The Schrodinger equation of the polar part of the helium ion can be solved by the associated legendary polynomial function, . The polar function expresses the angle formed by the vector of the electron's position relative to the z-axis of the helium ion's nuclear position. And it is produced that the polar function of the hydrogenic atom has the same polar function both in position space and in momentum space .
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).
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).
Because of considerable mathematical difficulties the Schrödinger equation can be solved exactly for only a few simple quantum systems. In particular for atoms with at least two electrons no such exact solutions exist. The most important system where an exact solution is indeed possible consists of a single electron with charge q 1 = - e in motion under the influence of a Coulomb potential about a nucleus with a positive charge q 2 = Ζe , for Ζ ≥ 1 an integer. Such a system is called a hydrogenic or hydrogen-like atom , with atomic number Z . One may view the nucleus as consisting of Ζ protons. For Ζ = 1 we obtain the hydrogen atom. For Ζ = 2, for example, the system is singly ionized helium He + ; this is helium with its two protons as usual, but stripped of one of its two electrons. Since it has only one electron but two protons it is positively charged, as indicated by the plus sign +. Similarly the lithium atom Li consists of three electrons and three protons (and four neutrons). If we strip off two electrons we obtain the ion Li ++ —doubly ionized lithium, which is the case Ζ = 3. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.