The Essence of Quantum Mechanics Part 2: Complex Numbers

In the first article in this series we outlined some basic physical intuition and described some of the ways in which quantum physics diverges from the classical physics of everyday life. The most important difference between classical and quantum physics is that quantum physics tends to resist intuitive understanding and is therefore best understood in terms of an abstract mathematical formalism. Before we can move on to building this formalism, we need to go over some mathematical foundations.

The first piece of that foundation has to be a familiarity with complex numbers. Most of the mathematical formalism of quantum physics is expressed in terms of complex numbers, and to express that formalism solely in real numbers would be extremely cumbersome if not impossible.

Fields

The best way to explain the complex numbers is to introduce them as an extension of the field of real numbers. Our first step must therefore be to explain what a field is. A field (F, +, ×), or simply F, is a set of objects combined with two binary operations + and ×, called addition and multiplication (Note that fields are very general objects and these operations may have nothing to do at all with the usual arithmetic multiplication and addition) that satisfies the field axioms. We often skip the multiplication symbol and simply write “ab” instead of a×b. A binary operation is one that assigns a value to a pair of objects. Additions is an example of a binary operation, adding two…