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# Lecture 19: The Fourier Transform ## Motivation ### Why Fourier? * **Eigenfunction.** $e^{i k x}$ is an eigenfunction of translation. * $\psi(x) \rightarrow \psi(x-a)$. * Translations are important! (Space is homogeneous.) * **Diagonalize.** Fourier transform diagonalizes *any*...

# Lecture 19: The Fourier Transform ## Motivation ### Why Fourier? * **Eigenfunction.** $e^{i k x}$ is an eigenfunction of translation. * $\psi(x) \rightarrow \psi(x-a)$. * Translations are important! (Space is homogeneous.) * **Diagonalize.** Fourier transform diagonalizes *any* translation-invariant operator. * Like going to the eigenbasis of a matrix. * Makes life simple! * **Waves.** Quantum mechanics is all about waves ## Definition ### Fourier Transform $$ \begin{aligned} \tilde{f}(k) &=\int_{-\infty}^{\infty} d x e^{-i k x} f(x) \\ f(x) &=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k x} \tilde{f}(k) \end{aligned} $$ * **Also unitary.** (Like rotation to eigenbasis.) * **k often momentum.** (Since $p = \hbar k$) ### Some Conventions $$ \begin{aligned} \tilde{f}(k) &=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} d x e^{-i k x} f(x) \\ f(x) &=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} d k e^{i k x} \tilde{f}(k) \end{aligned} $$ * **Symmetric.** Looks nicer (and is more correct). * **Warning:** Factors of $2\pi$ are always tricky! ### Fourier Trick $$ \int_{-\infty}^{\infty} d x e^{i k x}=2 \pi \delta(k) $$ * **Very useful.** * **Example.** $$ \begin{aligned} \int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k x} e^{-i k x^{\prime}} &=\delta\left(x-x^{\prime}\right) \\ \int_{-\infty}^{\infty} d x e^{-i k x} e^{i k^{\prime} x} &=2 \pi \delta\left(k-k^{\prime}\right) \end{aligned} $$ ## Properties ### Derivatives $$ \begin{aligned} \tilde{f}(k) &=\int_{-\infty}^{\infty} d x e^{-i k x} f(x) \\ \frac{d}{d k} \tilde{f}(k) &=\int_{-\infty}^{\infty} d x(-i x) e^{-i k x} f(x) \end{aligned} $$ * **Therefore** $$ \left(\frac{d}{d k}\right) \widetilde{f(x)}=-i \widetilde{x f(x)} $$ * **Also** $$ \begin{aligned} f(x) &=\int_{-\infty}^{\infty} \frac{d k}{2 \pi} e^{i k x} \tilde{f}(k) \\ \frac{d}{d x} f(x) &=\int_{-\infty}^{\infty} \frac{d k}{2 \pi}(i k) e^{i k x} \tilde{f}(k) \end{aligned} $$ * **Therefore** $$ \widehat{\left(\frac{d}{d x} f(x)\right)}=i k \tilde{f}(k) $$ ### Example: Gaussian $$ f(x)=e^{-a x^{2}} $$ * **Do the integral.** $$ \begin{aligned} \tilde{f}(k) &=\int_{-\infty}^{\infty} d x e^{-i k x} e^{-a x^{2}} \\ &=\int_{-\infty}^{\infty} d x e^{-a\left(x^{2}+i \frac{k}{a} x\right)} \\ &=\int_{-\infty}^{\infty} d x e^{-a\left(x+i \frac{k}{2 a}\right)^{2}} e^{-\frac{k^{2}}{4 a}} \\ &=e^{-\frac{k^{2}}{4 a}} \int_{-\infty}^{\infty} d x e^{-a x^{2}} \\ &=\sqrt{\frac{\pi}{a}} e^{-\frac{k^{2}}{4 a}} \end{aligned} $$ ### Fourier of Gaussian (2) $$ \begin{aligned} f(x) &=e^{-a x^{2}} \\ \tilde{f}(k) &=\sqrt{\frac{\pi}{a}} e^{-\frac{k^{2}}{4 a}} \end{aligned} $$ * **Big in real space = small in k-space.** * **Small in real space = big in k-space.** * **Uncertainty principle.** $$ \begin{aligned} \Delta x \Delta k &\geq 1 \\ \Delta x \Delta p &\geq \hbar \end{aligned} $$ ### Convolution $$ (f * g)(x) \equiv \int_{-\infty}^{\infty} d y f(x-y) g(y) $$ * **"Smoothing".** * **Take Fourier transform.** $$ \begin{aligned} \widehat{(f * g)}(k) &=\int_{-\infty}^{\infty} d x e^{-i k x} \int_{-\infty}^{\infty} d y f(x-y) g(y) \\ &=\int_{-\infty}^{\infty} d y \int_{-\infty}^{\infty} d x e^{-i k x} f(x-y) g(y) \\ &=\int_{-\infty}^{\infty} d y \int_{-\infty}^{\infty} d z e^{-i k(z+y)} f(z) g(y) \\ &=\int_{-\infty}^{\infty} d z e^{-i k z} f(z) \int_{-\infty}^{\infty} d y e^{-i k y} g(y) \\ &=\tilde{f}(k) \tilde{g}(k) \end{aligned} $$ * **Convolution becomes multiplication!** * **Multiplication becomes convolution!** ### Parseval's Theorem $$ \int_{-\infty}^{\infty} d x|f(x)|^{2}=\int_{-\infty}^{\infty} \frac{d k}{2 \pi}|\tilde{f}(k)|^{2} $$ * **Proof.** $$ \begin{aligned} \int_{-\infty}^{\infty} d x|f(x)|^{2} &=\int d x f^{*}(x) f(x) \\ &=\int d x\left(\int \frac{d k}{2 \pi} e^{-i k x} \tilde{f}^{*}(k)\right)\left(\int \frac{d k^{\prime}}{2 \pi} e^{i k^{\prime} x} \tilde{f}\left(k^{\prime}\right)\right) \\ &=\int \frac{d k}{2 \pi} \int \frac{d k^{\prime}}{2 \pi} \tilde{f}^{*}(k) \tilde{f}\left(k^{\prime}\right) \underbrace{\int d x e^{i\left(k^{\prime}-k\right) x}}_{2 \pi \delta\left(k^{\prime}-k\right)} \\ &=\int \frac{d k}{2 \pi}|\tilde{f}(k)|^{2} \end{aligned} $$ * **Also called "Plancherel's Theorem".** * **Energy is conserved.** * **Unitary transformation.** ## Examples ### Free Particle $$ H=\frac{p^{2}}{2 m}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} $$ * **Time-independent Schrödinger equation.** $$ -\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi(x)=E \psi(x) $$ * **Fourier transform.** $$ \frac{\hbar^{2} k^{2}}{2 m} \tilde{\psi}(k)=E \tilde{\psi}(k) $$ * **Therefore** $$ \tilde{\psi}(k) \propto \delta\left(\frac{\hbar^{2} k^{2}}{2 m}-E\right) $$ * **Integrate to find:** $$ \tilde{\psi}(k) \propto \delta\left(k \pm \frac{\sqrt{2 m E}}{\hbar}\right) $$