Fourier Analysis

Abstract

Fourier analysis is of utmost importance in many areas of mathematics, physics, and engineering. In quantum mechanics, the Fourier transform is an essential tool for the solution and the interpretation of the Schrödinger equation. It will help you to understand how a wave function can describe simultaneously the localization properties and the momentum distribution of a particle.

In this chapter we collect many results from Fourier analysis which will be used frequently in later chapters. In passing, you will be introduced to the most important mathematical concepts of quantum mechanics, such as Hilbert spaces and linear operators. Moreover, you will learn that the famous uncertainty relation is just a property of the Fourier transformation. If you need some more motivation, you may read Chapter 3, Sections 3.1 and 3.2 first.

This chapter starts by describing the Fourier series of a complex-valued periodic function. The Fourier series describes the given function as an infinite linear combination of stationary plane waves, each characterized by an amplitude and a wave number. In order to understand in which sense the Fourier sum converges, we need to introduce the concept of a Hilbert space.

As the period of the complex-valued function goes to infinity, the Fourier series becomes a Fourier integral which represents the function as a “continuous superposition” of stationary plane waves. The spectrum of wave numbers is described by a function on “/k-space.” This is the space of all possible wave numbers, which in the context of quantum mechanics is called the momentum space. It is a very important observation that the original function and the function describing the continuous spectrum of wave numbers depend on each other in a very symmetrical way. This relationship—the Fourier transformation—can be described as a linear operator acting in the Hilbert space of square integrable functions.

The properties of the Fourier transformation make it a very useful tool in quantum mechanics. For example, the derivative of a function corresponds via the Fourier transformation to a simple multiplication by k in momentum space. This fact will be exploited in Chapter 3 to solve the free Schrödinger equation with arbitrary initial conditions.

While this chapter contains some material that is indispensable for a thorough description of quantum mechanics, there are some mathematically more elaborate sections that may be skipped at first reading. These sections are labeled “special topics.”