A simple method is presented here for solving the wave mechanical problem of the hydrogen atom. The normal method of converting the Cartesian coordinates into polar coordinates is tedious and also requires an understanding of the Legendre and Lagurre polynomials. In this paper we are using an alternative method, which requires only minimal familiarity with mathematical concepts and techniques.
A simple method is presented here for solving the wave mechanical problem of the hydrogen atom. The normal method of converting the Cartesian coordinates into polar coordinates is tedious and also requires an understanding of the Legendre and Lagurre polynomials. In this paper we are using an alternative method, which requires only minimal familiarity with mathematical concepts and techniques.
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).
A partial separation of the variables is practicable for the solution of Schroedinger's temporally independent equation in cartesian coordinates x,y,z, which yields moderately simple algebraic formulae for the amplitude functions involving quantum numbers k,l,m, the same as in spherical polar coordinates. The properties of angular momentum are thus achieved with no angular variable. Several plots of surfaces of constant psi(x,y,z) are presented to illustrate the resemblance of the shapes of these surfaces to the shapes of surfaces of psi(r,theta,phi) with the corresponding quantum numbers.
The quantum mechanical problem of the hydrogen atom is treated by use of a finite difference equation in place of Schrödinger's differential equation. The exact solution leads to a wave vector energy expression that is readily converted to the Bohr-Rydberg formula. (The calculations here reported are limited to spherically symmetric states.) The wave vectors reduce to the familiar solutions of Schrödinger's equation as c --> infinity. The internal consistency and limiting behavior provide support for the view that the equations employed could well constitute an approach to a relativistic formulation of wave mechanics.
This chapter considers Schrodinger's model of hydrogen atom and analyzes in details the quantum wave mechanics approach to the motion of the electron in the hydrogen atom. The stationary Schrodinger equation is not easy to solve as it stands, and the whole problem looks rather complicated when expressed in terms of Cartesian coordinates. An essential difference between Bohr's and Schrodinger's models is that there are three quantum numbers, n , l , m , rather than the one number n describing a state of the electron. Moreover, in Bohr's model the atom is, at each instant, in one and only one energy state, that is., at each instant, the state of the atom is definite, but according to Schrodinger's model, the atom can exist in a superposition of states. In other words, the atom can be simultaneously in more than one energy state at a given instant. This is clearly incompatible with Bohr's model.
Based on an exposition of the underlying physics and applied mathematics in arXiv:1603.00899, this paper in five separate parts presents a description of the properties of the amplitude functions of the hydrogen atom according to wave mechanics in the coordinate representation in all four systems of coordinates in which Schroedinger's equations are separable, explicitly I spherical polar coordinates, II paraboloidal coordinates, III ellipsoidal coordinates and IV spheroconical coordinates; part V discusses the implications of these multiple systems in which these atomic orbitals can be directly calculated. Most published plots of orbitals are inaccurate. In particular, parts I - IV contain many accurate plots of the surfaces of amplitude functions in the various systems with a common criterion for the amplitude at which the surface is constructed. Although the plots are generated from amplitude functions in the specified systems of coordinates, the plots are translated to cartesian coordinates that the eye can recognise. The surfaces of the amplitude functions in ellipsoidal and spheroconical functions are previously unreported, with the accurate surfaces in spherical polar and paraboloidal coordinates for a direct comparison.
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).