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The Hydrogen Atom

Vinod Kumar Khanna

Published Jun 30, 2020• 0 citations

Abstract

During the solution of the Schrodinger equation for the hydrogen atom by separation of variables, we come across three quantum numbers, namely the azimuthal quantum number l , the magnetic quantum number m l , and the principal quantum number n . The quantum number l arises as a separation constant during the breaking apart of radial and angular segments of the Schrodinger equation and the quantum number m l occurs while separating the azimuthal component from the polar component of the angular part. The quantum number n is necessary to render the series for the non-asymptotic part of the radial wave function convergent. The three quantum numbers are interdependent because after selecting n , the selection of l is restricted to allowed values and similarly after choosing l , limited values are permitted for m l . The fourth quantum number m s representing electron spin is not a direct consequence of solving the Schrodinger equation. The detailed solution procedure of the Schrodinger equation including the normalization of the wave function, restrictions on the values of quantum numbers, and allied analysis are explained.

DOI: 10.1201/9781351204675-7

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During the solution of the Schrodinger equation for the hydrogen atom by separation of variables, we come across three quantum numbers, namely the azimuthal quantum number l , the magnetic quantum number m l , and the principal quantum number n . The quantum number l arises as a separation constant during the breaking apart of radial and angular segments of the Schrodinger equation and the quantum number m l occurs while separating the azimuthal component from the polar component of the angular part. The quantum number n is necessary to render the series for the non-asymptotic part of the radial wave function convergent. The three quantum numbers are interdependent because after selecting n , the selection of l is restricted to allowed values and similarly after choosing l , limited values are permitted for m l . The fourth quantum number m s representing electron spin is not a direct consequence of solving the Schrodinger equation. The detailed solution procedure of the Schrodinger equation including the normalization of the wave function, restrictions on the values of quantum numbers, and allied analysis are explained.

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.

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.

The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .

Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.

Based on the previous works, the Schrdinger equation of the hydrogen-like atom is analytically solved further. Three sets of analytical solutions are obtained if the factor r~ -nis not neglected. The first solution is the same as the traditional radial wave function, the second diverges, and the last one is quite different from the traditional solution. On the consideration of the finite size of the nuclear, the third wave function does not diverge when r approaches to zero. Its radial wave function has the following 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 nuclear than that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally.