Schrödinger Equation in One-Electron System: Hydrogenic Atom

Abstract

In Chap. 6, Schrödinger equation for quantum particle was derived by the introduction of wave-particle duality. Kinetic and potential energies were expressed as the general form. In this chapter, we consider the specific case: hydrogenic atom, where one-electron interacts with atomic nucleus. The Hamiltonian is constituted by operators of electron kinetic energy, nuclear kinetic energy and Coulomb potential energy between electron and atomic nucleus. Schrödinger equation is analytically solved in hydrogenic atom. Introducing polar coordinates, quantum wave function is separated into two parts. The operation is called separation of variables in differential equation. As the result, Schrödinger equation is transformed into two equations. One is for radial quantum wave function. The other is for angular quantum wave function. These functions are classified using three quantum numbers such as principal quantum number, angular momentum quantum number and magnetic quantum number. Shell type (K, L, M etc.) and subshell type (s, p, d etc.) are designated by principal quantum number and angular momentum quantum number, respectively. Though 2p_z atomic orbital is directly expressed by the use of spherical harmonics and radial quantum wave functions, 2p_x and 2p_y atomic orbitals are represented by linear combination of spherical harmonics. In the same manner, four of five 3d atomic orbitals are represented by linear combination of spherical harmonics. The total energy for hydrogenic atom is also analytically obtained. It is demonstrated how to estimate the excitation energy from the formula. Finally, pictorial representation of atomic orbital is explained.