Hydrogen and One-Electron Atoms

Be T and so on, are one-electron atoms. Each consists of only two particles, a nucleus and an electron. The quantum mechanical description of one-electron atoms is the starting point for understanding the electronic structure of atoms and of molecules. It is a problem that can be solved analytically, and it is useful to work through the details. In the case of the hydrogen atom, the nuclear mass is about 2000 times that of an electron. Thus, the proton would be expected to make small excursions about the mass center relative to any excursions of the very light electron. That is, we expect the electron to be moving quickly about, and in effect, orbiting the nucleus. 

The Schrodinger equation for this two-body problem starts out the same as the general two-body Schrodinger equation (Equation 9.18) however, the potential function, V(r), is different from that of the vibrating-rotating diatomic molecule. It is an electrostatic attraction of two point charges, and its form is 

Z is the integer nuclear charge or atomic number e is the value of the fundamental charge of the electron 

This interaction between the electron and the nucleus is the product of their respective charges, in this case +Ze for the nucleus and -e for the electron, divided by the distance between them, r. 

The potential function for the hydrogen atom is dependent only on the spherical polar coordinate r. Thus, the separation of variables carried out in going from Equations 9.18 through 9.22 remains valid for this problem. That means we immediately know that the wavefunctions for the hydrogen atom consist of some type of radial fimction, R(r), 

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