Undergraduate students come across the concepts of atomic physics at various stages during their degree programs. For example, the Bohr model is a central part of introductory courses on quantum physics, while the hydrogen atom is a key element in a first course on quantum mechanics. After that, the more advanced topics could either be a component of a second, broad quantum physics module, or a stand-alone unit. This book is designed for the latter approach, without necessarily excluding its usefulness for the former, where it might be used, for example, in conjunction with a text on nuclear physics.

The book evolved from a detailed set of lecture notes prepared for a thirdyear module at the University of Sheffield. The notes were prepared to respond to the lack of a short text at the right level. The subject material was either scattered across various chapters of large quantum physics texts, or was included in introductory sections of more advanced texts. Neither case was particularly suited to the needs of the students.

The range of topics included within the book aims to cover the core curriculum on atomic physics set out by the Institute of Physics, and might be useful either to second- or third-year students within the United Kingdom, depending on how a particular university subdivides the syllabus. For readers outside the United Kingdom, the text is pitched at intermediate-level students. It assumes basic familiarity with the techniques of quantum mechanics, but does not have the depth required for masters-level courses.

The course notes have been freely available on the Internet for several years, and I was approached by several publishers who thought they could form the basis for a textbook. Having already written two textbooks, I was well aware of the extra effort required to turn a set of lecture notes into a book and resisted the approaches I received. However, I then discovered the Cambridge Student's Guide series, and realized that it is the right place for the material. Its inclusion within the series makes it clear that the book does not claim to be an authoritative reference work, but rather an intermediate-level text aimed at explaining the basic concepts to undergraduate students.

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.