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Fourier Transforms

Expansion of a function in terms of Fourier Series proved to be an effective way to represent functions that were periodic in an interval $x \in (-\lambda/2 ,-\lambda/2)$ . Useful insights into ``what makes up a function'' are obtained by considering the amplitudes of the harmonics (i.e., each of the sub-periodic trigonometric or complex oscillatory functions) that compose the Fourier series. That is, the component harmonics can be quantified by inspecting their amplitudes. For instance, one could quantitatively compare the same note generated from a Stradivarius to an ordinary violin by comparing the amplitudes of the Fourier components of the notes component frequencies.

However there are many physical examples of phenomena that involve nearly, but not completely, periodic phenomena--and of course, quantum mechanics provides many examples of isolated events that are composed of wave-like functions.

It proves to be very useful to extend the Fourier analysis to functions that are not periodic. Not only are the same interpretations of contributions of the elementary functions that compose a more complicated object available, but there are many others to be obtained.

For example:

momentum/position

The wavenumber $k_n = 2 \pi n/\lambda$ turns out to be proportional to the momentum in quantum mechanics. The position of a function, $f(x)$ , can be expanded in terms of a series of wave-like functions with amplitudes that depend on each component momentum--this is the essence of the Heisenberg uncertainty principle.

diffraction

Bragg's law, which formulates the conditions of constructive and destructive interference of photons diffracting off of a set of atoms, is much easier to derive using a Fourier representation of the atom positions and photons.

To extend Fourier series to non-periodic functions, the domain of periodicity will extended to infinity, that is the limit of $\lambda \rightarrow \infty$ will be considered. This extension will be worked out in a heuristic manner in this lecture--the formulas will be correct, but the rigorous details are left for the math textbooks.

Recall that the complex form of the Fourier series was written as:

$$\begin{split}f(x) = & \sum_{n=-\infty}^{\infty} \mathcal{A}_{... ..._{-\lambda/2}^{\lambda/2} f(x) e^{-\imath k_n x} dx \end{split}$$ (18-1)

where $\mathcal{A}_{k_n}$ is the complex amplitude associated with the $k_n = 2 \pi n/\lambda$ reciprocal wavelength or wavenumber.

This can be written in a more symmetric form by scaling the amplitudes with $\lambda$ --let $\mathcal{A}_{k_n} = \sqrt{2 \pi} \mathcal{C}_{k_n}/\lambda$ , then

$$\begin{split}f(x) = & \sum_{n=-\infty}^{\infty} \frac{\sqrt{2... ..._{-\lambda/2}^{\lambda/2} f(x) e^{-\imath k_n x} dx \end{split}$$ (18-2)

Considering the first sum, note that the difference in wave-numbers can be written as:

$$\Delta k = k_{n+1} - k_n = \frac{2 \pi}{\lambda}$$ (18-3)

which will become infinitesimal in the limit as $\lambda \rightarrow \infty$ . Substituting $\Delta k/(2 \pi)$ for $1/\lambda$ in the sum, the more ``symmetric result'' appears,

$$\begin{split}f(x) = & \frac{1}{\sqrt{2 \pi}} \sum_{n=-\infty}... ..._{-\lambda/2}^{\lambda/2} f(x) e^{-\imath k_n x} dx \end{split}$$ (18-4)

Now, the limit $\lambda \rightarrow \infty$ can be obtained an the summation becomes an integral over a continuous spectrum of wave-numbers; the amplitudes become a continuous function of wave-numbers, $\mathcal{C}_{k_n} \rightarrow g(k)$ :

$$\begin{split}f(x) = & \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{... ...i}} \int_{-\infty}^{\infty} f(x) e^{-\imath k x} dx \end{split}$$ (18-5)

The function $g(k=2 \pi/\lambda)$ represents the density of the amplitudes of the periodic functions that make up $f(x)$ . The function $g(k)$ is called the Fourier Transform of $f(x)$ . The function $f(x)$ is called the Inverse Fourier Transform of $g(k)$ , and $f(x)$ and $g(k)$ are a the Fourier Transform Pair.

Higher Dimensional Fourier Transforms

Of course, many interesting periodic phenomena occur in two dimensions (e.g., two spatial dimensions, or one spatial plus one temporal), three dimensions (e.g., three spatial dimensions or two spatial plus one temporal), or more.

The Fourier transform that integrates $\frac{dx}{\sqrt{2 \pi}}$ over all $x$ can be extended straightforwardly to a two dimensional integral of a function $f(\vec{r}) = f(x,y)$ by $\frac{dx dy}{2 \pi}$ over all $x$ and $y$ --or to a three-dimensional integral of $f(\vec{r})\frac{dx dy dz}{\sqrt{(2 \pi})^3}$ over an infinite three-dimensional volume.

A wavenumber appears for each new spatial direction and they represent the periodicities in the $x$ -, $y$ -, and $z$ -directions. It is natural to turn the wave-numbers into a wave-vector

$$\vec{k} = (k_x , k_y , k_z) = ( \frac{2 \pi}{\lambda_x} , \frac{2 \pi}{\lambda_y} , \frac{2 \pi}{\lambda_y} )$$ (18-6)

where $\lambda_i$ is the wavelength of the wave-function in the $i^{th}$ direction.

The three dimensional Fourier transform pair takes the form:

$$\begin{split}f(\vec{x}) = & \frac{1}{\sqrt{(2 \pi)^3}} \iiint... ...f(\vec{x}) e^{-\imath \vec{k}\cdot\vec{x}} dx dy dz \end{split}$$ (18-7)