https://haifei-home.pages.dev/en/tags/wave-packets/Recent content in Wave Packets on Haifei's HomeHugoenhfwang132@gmail.com (hfwang132)hfwang132@gmail.com (hfwang132)This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Wed, 04 May 2022 15:39:54 +0800https://haifei-home.pages.dev/en/post_20220504_%E6%B3%A2%E5%8C%85-%E7%BE%A4%E9%80%9F%E5%BA%A6%E4%B8%8E%E8%89%B2%E6%95%A3%E5%85%B3%E7%B3%BB/Wed, 04 May 2022 15:39:54 +0800hfwang132@gmail.com (hfwang132)https://haifei-home.pages.dev/en/post_20220504_%E6%B3%A2%E5%8C%85-%E7%BE%A4%E9%80%9F%E5%BA%A6%E4%B8%8E%E8%89%B2%E6%95%A3%E5%85%B3%E7%B3%BB/<h2 id="introduction">Introduction</h2> <p>When I first began studying physics, I was somewhat confused about the concepts of wave packets and group velocity. Looking back now, they are actually very simple; it is just that some textbooks do not explain them clearly. This article reviews the concept of group velocity and several examples of dispersion relations.</p> <h2 id="momentum-eigenstates-and-fourier-transforms">Momentum Eigenstates and Fourier Transforms</h2> <p>The wave function of a plane wave</p> \[|\bm p\rangle=\frac{1}{(2\pi\hbar)^{3/2}}\exp(\mathrm{i}\frac{\bm{p}}{\hbar}\cdot\bm{x})\]<p>cannot be normalized.</p> <p>According to the postulates of quantum mechanics, all physically existing wave functions are normalizable. Therefore, momentum eigenstates \(|p\rangle\) do not represent physically existing states. In other words, momentum cannot be measured with complete precision.</p>