4. Fourier Stuff¶
This note explains Fourier series and the Fourier transform in both discrete and continuous forms, and also explains why this is important in quantum mechanics.
Review of Discrete and Continuous Fourier \(\Sigma\) and \(\int\)¶
Discrete: consider \(f(x)\) in interval \(x\in[-\pi, \pi]\) that is periodic $f(x +2\pi) = f(x). Then
\[
f(x) = \frac{1}{2}A_0 + \sum_{n =1}^{\infty}\left[A_n\cos(nx) + B_n\sin(nx)\right]
\]
The series will converge if \(f(x), f'(x)\) are piecewise continuous on \((-\pi, \pi)\). In addition, \(A_n, B_n\) are found by multiplying by basis \(\sin(mx)\)
\[
A_m = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(mx)dx
\]
\[
B_m = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(mx)dx
\]
It can also be written in exponential form:
\[
f(x) = \frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty} c_ne^{inx}
\]
where
\[
c_m = \frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi} f(x) e^{-imx}dx
\]
A useful quantity to know also is the Kronecker delta:
\[
\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(n-m)}dx = \delta_{mn} = \begin{cases}1\qquad m = n\\0\qquad m\neq n\end{cases}
\]
which just show the dot product of two basis vectors are orthonormal.
In the continuous Forier transform, we find
\[
f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} g(k)e^{ikx}dk
\]
\[
g(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-ikx}dx
\]
Which is saying that you can transform \(f(x)\) and \(g(k)\) into each other, provided you have one.