Hydrogen Atom

Radial wavefunction plots for s (R10, R20, & R30) and p (R21, R31, & R41) states. Notice that as binding energy decreases, the spatial extent of the wavefunction increases, that is to say the electron is more delocalized.

Introduction

Hydrogen and hydrogenic atoms or ions are systems with a single electron bound to the nucleus. These systems are the only systems for which the Schrodinger’s equation can be solved exactly. The solution of the Schrodinger’s equation for hydrogen is important because atoms containing more than one electron could be modeled using the hydrogen atom. This means the entire periodic table can be constructed using hydrogenic wavefunctions. Thus the hydrogen atom is the fundamental model that is used for describing all systems made up of nuclei and electrons such as atoms, molecules, polymers, nanostructures, and bulk three dimensional materials.

This article discusses the solution of the Schrodinger’s equation for hydrogenic systems. We shall focus only on the radial solution which is needed for obtaining the quantization condition for the energy levels of the system. Because all atoms are spherical, the spherical solutions which are called spherical harmonics are the same for all atoms.

The Schrodinger Equation for Hydrogenic Atoms

Radial wavefunction plots for s (R10, R20, & R30) and p (R21, R31, & R41) states. Notice that as binding energy decreases, the spatial extent of the wavefunction increases, that is to say the electron is more delocalized.

In summary, we’ve discussed in a step-by-step manner the solution of the Schrodinger’s equation for hydrogen-like atoms. We also plotted the radial wavefunctions for s and p states.

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