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diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html deleted file mode 100644 index b256e52..0000000 --- a/static/know/concept/probability-current/index.html +++ /dev/null @@ -1,108 +0,0 @@ -<!DOCTYPE html> -<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> -<head> - <meta charset="utf-8" /> - <meta name="generator" content="pandoc" /> - <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> - <title>Prefetch | Probability current</title> - <link rel="icon" href="data:,"> - <style> - body { - background:#ddd; - color:#222; - max-width:80ch; - text-align:justify; - margin:auto; - padding:1em 0; - font-family:sans-serif; - line-height:1.3; - } - a {text-decoration:none;color:#00f;} - h1,h2,h3 {text-align:center} - h1 {font-size:200%;} - h2 {font-size:160%;} - h3 {font-size:120%;} - .nav {height:3rem;font-size:250%;} - .nav a:link,a:visited {color:#222;} - .nav a:hover,a:focus,a:active {color:#00f;} - .navl {width:30%;float:left;text-align:left;} - .navr {width:70%;float:left;text-align:right;} - pre {filter:invert(100%);} - @media (prefers-color-scheme: dark) { - body {background:#222;filter:invert(100%);} - } </style> - <script> - MathJax = { - loader: {load: ["[tex]/physics"]}, - tex: {packages: {"[+]": ["physics"]}} - }; - </script> - <script src="/mathjax/tex-svg.js" type="text/javascript"></script> - </head> -<body> -<div class="nav"> -<div class="navl"><a href="/">PREFETCH</a></div> -<div class="navr"> -<a href="/blog/">blog</a>  -<a href="/code/">code</a>  -<a href="/know/">know</a> -</div> -</div> -<hr> -<h1 id="probability-current">Probability current</h1> -<p>In quantum mechanics, the <em>probability current</em> describes the movement of the probability of finding a particle at given point in space. In other words, it treats the particle as a heterogeneous fluid with density <span class="math inline">\(|\psi|^2\)</span>. Now, the probability of finding the particle within a volume <span class="math inline">\(V\)</span> is:</p> -<p><span class="math display">\[\begin{aligned} - P = \int_{V} | \psi |^2 \dd[3]{\vec{r}} -\end{aligned}\]</span></p> -<p>As the system evolves in time, this probability may change, so we take its derivative with respect to time <span class="math inline">\(t\)</span>, and when necessary substitute in the other side of the Schrödinger equation to get:</p> -<p><span class="math display">\[\begin{aligned} - \pdv{P}{t} - &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \dd[3]{\vec{r}} - = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} - \\ - &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) - - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}} - \\ - &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}} - = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} -\end{aligned}\]</span></p> -<p>Where we have defined the probability current <span class="math inline">\(\vec{J}\)</span> as follows in the <span class="math inline">\(\vec{r}\)</span>-basis:</p> -<p><span class="math display">\[\begin{aligned} - \vec{J} - = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) - = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} -\end{aligned}\]</span></p> -<p>Let us rewrite this using the momentum operator <span class="math inline">\(\hat{p} = -i \hbar \nabla\)</span> as follows, noting that <span class="math inline">\(\hat{p} / m\)</span> is simply the velocity operator <span class="math inline">\(\hat{v}\)</span>:</p> -<p><span class="math display">\[\begin{aligned} - \boxed{ - \vec{J} - = \frac{1}{2 m} ( \psi^* \hat{p} \psi - \psi \hat{p} \psi^*) - = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p}}{m} \psi \Big\} - = \mathrm{Re} \{ \psi^* \hat{v} \psi \} - } -\end{aligned}\]</span></p> -<p>Returning to the derivation of <span class="math inline">\(\vec{J}\)</span>, we now have the following equation:</p> -<p><span class="math display">\[\begin{aligned} - \pdv{P}{t} - = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}} - = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} -\end{aligned}\]</span></p> -<p>By removing the integrals, we thus arrive at the <em>continuity equation</em> for <span class="math inline">\(\vec{J}\)</span>:</p> -<p><span class="math display">\[\begin{aligned} - \boxed{ - \nabla \cdot \vec{J} - = - \pdv{|\psi|^2}{t} - } -\end{aligned}\]</span></p> -<p>This states that the total probability is conserved, and is reminiscent of charge conservation in electromagnetism. In other words, the probability at a point can only change by letting it “flow” towards or away from it. Thus <span class="math inline">\(\vec{J}\)</span> represents the flow of probability, which is analogous to the motion of a particle.</p> -<p>As a bonus, this still holds for a particle in an electromagnetic vector potential <span class="math inline">\(\vec{A}\)</span>, thanks to the gauge invariance of the Schrödinger equation. We can thus extend the definition to a particle with charge <span class="math inline">\(q\)</span> in an SI-unit field, neglecting spin:</p> -<p><span class="math display">\[\begin{aligned} - \boxed{ - \vec{J} - = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p} - q \vec{A}}{m} \psi \Big\} - } -\end{aligned}\]</span></p> -<hr> -© "Prefetch". Licensed under <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA 4.0</a>. -</body> -</html> |