The Features of the Quantum World

Quantum mechanics is different from most other branches of science in that it uses complex numbers in a fundamental way. Complex numbers were originally created as a mathematical curiosity: i = √−1 was the asserted “imaginary” solution to the polynomial equation x2 = −1. As time went on, an entire mathematical edifice was constructed with these “imaginary” numbers. Complex numbers have kept lonely mathematicians busy for centuries, while physicists successfully ignored these abstract creations. However, things changed with the systematic study of wave mechanics. After the introduction of Fourier analysis, researchers learned that a compact way to represent a wave was by using functions of complex numbers. As it turns out, this was an important step on the road to using complex numbers in quantum theory. Early quantum mechanics was largely based on wave mechanics. At first glance, we do not seem to experience complex numbers in the “real world.” The length of a rod is a real number, not a complex number. The temperature outside today is 73◦, not (32 − 14i)◦. The amount of time a chemical process takes is 32.543 seconds, not −14.65i seconds. One might wonder what possible role complex numbers can have in any discussion of the physical world. It will soon become apparent that they play an important, indeed an essential, role in quantum mechanics. We shall explore complex numbers in Chapters 1 and 2 of the text.