Fourier Transforms

Chem 3240 · Lecture 3.9

Davit Potoyan

Why Fourier?

Any signal can be built from waves.

  • Fourier series: periodic functions as sums of sines and cosines
  • Fourier transform: extends the idea to non-periodic functions
  • Connects position and momentum, time and frequency

Fourier Series

A periodic \(f(x)\) is a weighted sum of \(\sin\) and \(\cos\):

\[f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}a_n \cos(nx) + \sum_{n=1}^{\infty}b_n \sin(nx)\]

\(\sin\) and \(\cos\) form a complete orthogonal basis.

The Coefficients

Orthogonality picks out each coefficient:

\[a_n = \frac{1}{\pi} \int_{-{\pi}}^{\pi} f(x)\cos(nx)\,dx\]

\[b_n = \frac{1}{\pi} \int_{-{\pi}}^{\pi} f(x)\sin(nx)\,dx\]

The \(\frac{a_0}{2}\) term is the mean of \(f(x)\) over the period.

Complex Form

Euler’s formula folds \(\sin\) and \(\cos\) into exponentials:

\[f(x)=\sum_{n=0}^{\infty} c_{n} \exp\left(i \frac{2 \pi n x}{L}\right)\]

Introduce the wave number

\[k_{n}=\frac{2 \pi n}{L} = \frac{2 \pi}{\lambda}\]

From Series to Transform

Let the period \(L \to \infty\): the discrete sum becomes an integral.

\[\Delta k_{n}=\frac{2 \pi n}{L} \rightarrow d k\]

Discrete modes \(\to\) a continuum of wave numbers.

The Transform Pair

\[F(k)=\int_{-\infty}^{\infty} f(x)\, \mathrm{e}^{-i k x}\, d x\]

Inverse transform rebuilds the original:

\[f(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(k)\, \mathrm{e}^{i k x}\, d k\]

\(f(x)\) and \(F(k)\) are two views of the same object.

Narrow in x, Broad in k

Square pulse of width \(w\):

\[\psi(x) = \begin{cases} 1, & |x| \leq \frac{w}{2} \\ 0, & |x| > \frac{w}{2} \end{cases}\]

Its transform is a sinc:

\[\hat{\psi}(k) = w \, \text{sinc}(kw/2)\]

Shrink \(w\) in space \(\Rightarrow\) the sinc spreads in \(k\).

The Uncertainty Relation

\[\Delta x \, \Delta k \geq \frac{1}{2}\]

A function and its transform cannot both be sharp.

With de Broglie \(k = p/\hbar\), this becomes position vs momentum uncertainty.

Gaussian to Gaussian

A Gaussian transforms into another Gaussian:

\[f(t) = \exp\left(- \frac{t^2}{2\sigma^2}\right)\]

\[\mathcal{F}[f(t)] = \sqrt{2\pi}\, \exp\left(- \frac{\omega^2\sigma^2}{2}\right)\]

Broad in \(t\) \(\Rightarrow\) narrow in \(\omega\). The minimum-uncertainty shape.

Plane Wave and Delta Duality

Expand in momentum eigenstates:

\[\psi(x) = \frac{1}{\sqrt{2\pi}} \int \phi(k)\, e^{ikx}\, dk\]

  • Definite momentum \(k_0\): \(\;\phi(k)=\delta(k-k_0) \Rightarrow \psi(x)=e^{ik_0 x}\)
  • Localized at \(x_0\): \(\;\psi(x)=\delta(x-x_0) \Rightarrow \phi(k)=\frac{1}{\sqrt{2\pi}}e^{-ikx_0}\)

A plane wave is one momentum spread over all space; a spike in space is all momenta at once.

Takeaway

A function and its Fourier transform are conjugate views: sharpening one always blurs the other, and that trade-off is the uncertainty principle.