Solution of the Schrödinger Equation for the Hydrogen Atom in Rectangular Coordinates

A.B. Datzeff

Bambang Supriadi, Zainur Rasyid Ridlo, F Fuadah +3 more

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 .

Augustine C. Chen

A coordinate transformation which exhibits the rotational invariance of the hydrogen atom in four-dimensional Hilbert space is introduced. The coordinates are shown to be directly related to the spherical polar and parabolic coordinates in position space. With the use of the transformation, the Schr\"odinger equation for the hydrogen atom left-multiplied by $4r$ is transformed into one for a four-dimensional harmonic oscillator. Solutions are obtained and related to the hydrogenic wave functions. Group-theoretical implications of the transformation and its application to the hydrogen Stark problem are briefly discussed.

Philip R. Wallace

James M. Feagin

J. Weiner, Frederico Dias Nunes

This chapter begins with a discussion of radiation by a microscopic classical oscillator with radiative damping. It then turns to the Schrödinger equation for the hydrogen atom, and an extended analysis of the solutions to the Schrödinger equation for the hydrogen atom in polar coordinates is then presented. Real orbitals are constructed from linear combinations of the complex Schrödinger equation solutions. The significance of real orbitals for molecular structure and the consequent introduction of rotating and vibrating dipoles are then introduced. A semiclassical model (quantal atom and classical E-M field) for the absorption and emission of radiation is developed and compared to the Einstein A and B coefficients. It is shown how the time-dependence of the radiatively coupled non-stationary states of the hydrogen atom are analogous to the classical harmonic oscillator. The essentials of atomic transitions, as well as molecular rotational, vibrational, and electronic spectroscopy complete the chapter.