The quantum theory of hydrogen is the starting point for the whole subject of atomic physics. Bohr's derivation of the quantized energies was one of the triumphs of early quantum theory, and makes a useful introduction to the notion of quantized energies and angular momenta. We, therefore, give a brief review of the Bohr model before moving to the main subject of the chapter, namely: the solution of the Schrödinger equation for the electron-nucleus system.
The Bohr Model of Hydrogen
The Bohr model is part of the “old” quantum theory of the atom (i.e., pre-quantum mechanics). It includes the quantization of energy and angular momentum, but uses classical mechanics to describe the motion of the electron. With the advent of quantum mechanics, we realize that this is an inconsistent approach, and therefore should not be pushed too far. Nevertheless, the Bohr model does give the correct quantized energy levels of hydrogen, and also gives a useful parameter (the Bohr radius) for quantifying the size of atoms. Hence, it remains a useful starting point to understand the basic structure of atoms.
It is well known from classical physics that planetary orbits are characterized by their energy and angular momentum. We shall see that these are also key quantities in the quantum theory of the hydrogen atom. In 1911, Rutherford discovered the nucleus, which led to the idea of atoms consisting of electrons in classical orbits where the central forces are provided by the Coulomb attraction to the positive nucleus, as shown in Figure 2.1. The problem with this idea is that the electron in the orbit is constantly accelerating. Accelerating charges emit radiation called bremsstrahlung, and so the electrons should be radiating all the time, losing energy. This would cause the electron to spiral into the nucleus, like an old satellite crashing to Earth.