The Schrödinger Equation for a One-Electron System

L. P. Ginzburg

Galina A. Nemova

A.P. French, Edwin F. Taylor

Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.

Jochen Autschbach

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.

Jerry Goodisman

Z. Ficek

The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.