Introduction to the Hydrogen Atom Model

The quantity ⏐Ψ⏐2 corresponds to the probability density function of the electron’s position in a hydrogen atom. In this example,

where M equals the mass of the nucleus and me represents the mass of an electron (9.1094·10−31 kg). The hydrogen nucleus consists of a single proton (more than 1800 times heavier than the electron), so the approximation of μ is valid with a reasonable accuracy. Thus you can treat the problem as a one-particle system.

The system’s potential energy is

where e equals the electron charge (1.602·10−19 C), ε0 represents the permittivity of vacuum (8.854·10−12 F/m), and r is the distance from the center of the atom.

In the model, the boundary condition for the perimeter of the computational domain is a zero probability for the electron to be outside the specified domain. This means that the probability of finding the electron inside the domain is 1. It is important to have this approximation in mind when solving for higher-energy eigenvalues because the solution of the physical problem might fall outside the domain, and no eigenvalues are found for the discretized problem. Ideally the domain is infinite, and higher-energy eigenvalues correspond to the electron being further away from the nucleus.