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local-search.xml
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<?xml version="1.0" encoding="utf-8"?>
<search>
<entry>
<title>备忘</title>
<link href="/2024/09/23/memo-math/"/>
<url>/2024/09/23/memo-math/</url>
<content type="html"><![CDATA[<h1 id="delta-函数"><span class="math inline">\(\delta\)</span>函数</h1><p><span class="github-emoji"><span>🍎</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f34e.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 如果函数 <span class="math inline">\(f(x)\)</span> 的实根<span class="math inline">\(x_i\)</span> 全是单根，那么有关系 <span class="math display">\[\delta(f(x)) = \sum_i \frac{\delta(x-x_i)}{|f'(x_i)|}\]</span></p><p><span class="github-emoji"><span>1⃣</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/0031-20e3.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 首先证明 <span class="math inline">\(\delta(ax) =\frac{\delta(x)}{|a|}\)</span> <span class="math display">\[\begin{align} \int^\infty_{-\infty} \mathrm{d}x \delta (ax) &=\int^{\infty\cdot a}_{-\infty\cdot a} \mathrm{d}y \frac{\delta (y)}{a} \\ &= \int^{\infty}_{-\infty} \mathrm{d}y \frac{\delta (y)}{|a|}\end{align}\]</span> <span class="github-emoji"><span>2⃣</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/0032-20e3.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 假设函数 <span class="math inline">\(f(x)\)</span>具有一系列零点 <span class="math inline">\(x_i\)</span> 。附近展开 <span class="math inline">\(f(x) = f(x_i) + f'(x_i)(x-x_i) +\frac{1}{2}f''(x_i)(x-x_i)^2\)</span></p><p><span class="math display">\[\begin{align} \int^\infty_{-\infty} \delta(f(x)) \mathrm{d}x &= \sum_i\lim_{\epsilon\rightarrow 0} \int^{x_i+\epsilon}_{x_i-\epsilon}\delta(f(x)) \mathrm{d}x \tag{1.1} \label{eq1} \\ &= \sum_i \lim_{\epsilon\rightarrow 0}\int^{x_i+\epsilon}_{x_i-\epsilon} \delta\biggl( f'(x_i)(x-x_i) +\frac{1}{2}f''(x_i)(x-x_i)^2 \biggr) \mathrm{d}x \tag{1.2}\label{eq2} \\ &= \sum_i \lim_{\epsilon\rightarrow 0}\int^{x_i+\epsilon}_{x_i-\epsilon} \delta\biggl(\underbrace{f'(x_i)(x-x_i)}_{\epsilon} +\underbrace{\frac{1}{2}f''(x_i)(x-x_i)^2}_{\epsilon^2} \biggr)\mathrm{d}x \tag{1.3} \label{eq3} \\ &= \sum_i \int^{\infty}_{-\infty} \delta\biggl(f'(x_i)(x-x_i) \biggr) \tag{1.4} \label{eq4} \\ &= \sum_i \frac{\delta (x-x_i)}{|f'(x_i)|} \tag{1.5}\label{eq5}\end{align}\]</span></p><p>其中</p><ul><li><span class="math inline">\(\eqref{eq1}\)</span>是将整个积分区间分成了最多只包含一个零点的一系列小区间，因为 <span class="math inline">\(\delta\)</span> 函数的存在， 只有包含 <span class="math inline">\(f(x)\)</span>零点的那些小的积分区间才会有贡献。并且由于 <span class="math inline">\(\delta\)</span>函数的性质，严格说来只有零点附近那“一点点”才有贡献，我们用一个无穷小量<span class="math inline">\(\epsilon\)</span> 来衡量。</li><li><span class="math inline">\(\eqref{eq2}\)</span>是将那一个小区间的函数在零点附近做展开。因为 <span class="math inline">\(x_i\)</span> 是 <span class="math inline">\(f(x)\)</span>的单根，所以最低可以是一阶导数。</li><li><span class="math inline">\(\eqref{eq3}\)</span>是把求极限操作直接拿到 <span class="math inline">\(\delta\)</span>函数里面考虑（这么做可能有点不太好嘿嘿<span class="github-emoji"><span>😋</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f60b.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span>）这样的话二阶导对应的是高阶小量</li><li><span class="math inline">\(\eqref{eq4}\)</span>补上了剩下的积分区间，因为其他地方在 <span class="math inline">\(\delta\)</span> 函数的保护下为零</li><li><span class="math inline">\(\eqref{eq4}\)</span> 中的 <span class="math inline">\(f'(x_i)\)</span> 是一个常数，利用前面的第<span class="github-emoji"><span>1⃣</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/0031-20e3.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 点就可以得到 <span class="math inline">\(\eqref{eq5}\)</span></li></ul><p><span class="math inline">\(\delta (x^2-a^2) = \frac{1}{2|a|}[\delta(x-a)+\delta (x+a)]\)</span></p>]]></content>
</entry>
<entry>
<title>非拓扑孤子</title>
<link href="/2024/08/05/nontopo-soliton/"/>
<url>/2024/08/05/nontopo-soliton/</url>
<content type="html"><![CDATA[<p>今天下午还在看老李的场论讲义，因为他的第七章是为数不多愿意在场论书里比较物理地介绍孤子的吧。Ryder的 QFT，还有 QFT for the Gifted Amateur也有一章节提到过，但感觉讲得没有老李这章简明但完整，从经典的到量子的，从拓扑到非拓扑。当然最核心的是非拓扑的孤子，Q-ball的稳定性分析。这毕竟也是老李一个很重要的工作，他与提出了最早的一种实现非拓扑孤子的方式，讨论了三维时候稳定性的分析<a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.13.2739">1</a>。讲义里面也对这一部分作了很物理的介绍，没有无意义的计算，从老早前就开始看，断断续续地，到现在才明白一些吧。最后用集体坐标量子化的方式我现在还没有太明白。</p><p><img src="/images/lee-qft.png"></p><p>遗憾的是中文版有不少错误，114 页第二段里面漏掉一个式子；116页下方文中的“无穷大”应该是“无穷小”；122 页的 (7.54) 式中的 <span class="math inline">\(x\)</span>应该在指数上。对比英文版才发现这些奇怪的地方是中文版的小错误。</p>]]></content>
</entry>
<entry>
<title>碎碎念</title>
<link href="/2024/01/24/%E6%96%B0%E5%BB%BA%20Markdown/"/>
<url>/2024/01/24/%E6%96%B0%E5%BB%BA%20Markdown/</url>
<content type="html"><![CDATA[]]></content>
<tags>
<tag>physics</tag>
</tags>
</entry>
<entry>
<title></title>
<link href="/2024/01/17/confused/"/>
<url>/2024/01/17/confused/</url>
<content type="html"><![CDATA[]]></content>
</entry>
<entry>
<title>新年好</title>
<link href="/2024/01/01/newyear-2024/"/>
<url>/2024/01/01/newyear-2024/</url>
<content type="html"><![CDATA[<p>2024 新年好啊！</p>]]></content>
</entry>
<entry>
<title>first</title>
<link href="/2023/09/10/first/"/>
<url>/2023/09/10/first/</url>
<content type="html"><![CDATA[<p>这是一篇测试文章</p><p>今天是10月 <span class="github-emoji"><span>😄</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f604.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></p><h1 id="h1-标题">h1 标题</h1><img src="/2023/09/10/first/ocean.jpg" class="" title="图片引用方法一"><figure><img src="/images/ocean.jpg" alt="图片引用方法三"><figcaption aria-hidden="true">图片引用方法三</figcaption></figure><h2 id="h2-标题">h2 标题</h2><p>引擎可以选择 <code>mathjax</code> 或 <code>katex</code>，下面使用的<a href="https://docs.mathjax.org/en/latest/input/tex/extensions/centernot.html?highlight=slash#centernot">mathjax</a></p><p><span class="math display">\[ \int f(x)^2 =\begin{pmatrix} \hbar & \not\gamma \\ 0 & 0\end{pmatrix}\\\Set{ x\in\mathbf{R}^2 | 0<{|x|}<5 }\]</span></p><p>这是行内 <span class="math inline">\(\Braket{ \phi |\frac{\partial^2}{\partial t^2} | \psi }\)</span>。</p><p><span class="math display">\[E=mc^2\tag{1}\]</span></p>]]></content>
<tags>
<tag>test</tag>
</tags>
</entry>
<entry>
<title>Hello World</title>
<link href="/2023/09/10/hello-world/"/>
<url>/2023/09/10/hello-world/</url>
<content type="html"><![CDATA[<p>Welcome to <a href="https://hexo.io/">Hexo</a>! This is your veryfirst post. Check <a href="https://hexo.io/docs/">documentation</a> formore info. If you get any problems when using Hexo, you can find theanswer in <a href="https://hexo.io/docs/troubleshooting.html">troubleshooting</a> oryou can ask me on <a href="https://github.com/hexojs/hexo/issues">GitHub</a>.</p><h2 id="quick-start">Quick Start</h2><h3 id="create-a-new-post">Create a new post</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo new <span class="hljs-string">"My New Post"</span><br></code></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/writing.html">Writing</a></p><h3 id="run-server">Run server</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo server<br></code></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/server.html">Server</a></p><h3 id="generate-static-files">Generate static files</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo generate<br></code></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/generating.html">Generating</a></p><h3 id="deploy-to-remote-sites">Deploy to remote sites</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ hexo deploy<br></code></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/one-command-deployment.html">Deployment</a></p>]]></content>
</entry>
</search>