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diff --git a/content/know/category/quantum-mechanics.md b/content/know/category/quantum-mechanics.md index dcb6eb6..d302dfe 100644 --- a/content/know/category/quantum-mechanics.md +++ b/content/know/category/quantum-mechanics.md @@ -11,3 +11,6 @@ Alphabetical list of concepts in this category. ## D * [Dirac notation](/know/concept/dirac-notation/) + +## P +* [Probability current](/know/concept/probability-current/) diff --git a/content/know/concept/index.md b/content/know/concept/index.md index 6604aba..19f2027 100644 --- a/content/know/concept/index.md +++ b/content/know/concept/index.md @@ -11,3 +11,6 @@ Alphabetical list of concepts in this knowledge base. ## D * [Dirac notation](/know/concept/dirac-notation/) + +## P +* [Probability current](/know/concept/probability-current/) diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md new file mode 100644 index 0000000..a6780f7 --- /dev/null +++ b/latex/know/concept/probability-current/source.md @@ -0,0 +1,85 @@ +# Probability current + +In quantum mechanics, the *probability current* expresses the movement +of the probability of finding a particle. Or in other words, it treats +the particle as a heterogeneous fluid with density $|\psi|^2$. Now, the +probability of finding the particle within a volume $V$ is given by: + +$$\begin{aligned} + P = \int_{V} | \psi |^2 \dd[3]{\vec{r}} +\end{aligned}$$ + +As the system evolves in time, this probability may change, so we take +its derivative with respect to time $t$, and when necessary substitute +in the other side of the Schrödinger equation to get: + +$$\begin{aligned} + \pd{P}{t} + &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} + \\ + &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}$$ + +Where we have defined the probability current $\vec{J}$ as follows in +the $\vec{r}$-basis: + +$$\begin{aligned} + \vec{J} + = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi) + = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\} +\end{aligned}$$ + +Let us rewrite this using the momentum operator +$\hat{p} = -i \hbar \nabla$ as follows, noting that $\hat{p} / m$ is +simply the velocity operator $\hat{v}$: + +$$\begin{aligned} + \boxed{ + \vec{J} + = \frac{1}{2 m} ( \psi^* \hat{p} \psi - \psi \hat{p} \psi^*) + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p}}{m} \psi \Big\} + = \mathrm{Re} \{ \psi^* \hat{v} \psi \} + } +\end{aligned}$$ + +Returning to the derivation of $\vec{J}$, we now have the following +equation: + +$$\begin{aligned} + \pd{P}{t} + = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}$$ + +By removing the integrals, we thus arrive at the *continuity equation* +for $\vec{J}$: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vec{J} + = - \pd{|\psi|^2}{t} + } +\end{aligned}$$ + +This states that 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 +$\vec{J}$ represents the flow of probability, which is analogous to the +motion of a particle. + +As a bonus, this still holds for a particle in an electromagnetic vector +potential $\vec{A}$, thanks to the gauge invariance of the Schrödinger +equation. We can thus extend the definition to a particle with charge +$q$ in an SI-unit field, neglecting spin: + +$$\begin{aligned} + \boxed{ + \vec{J} + = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p} - q \vec{A}}{m} \psi \Big\} + } +\end{aligned}$$ diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html new file mode 100644 index 0000000..d736e49 --- /dev/null +++ b/static/know/concept/probability-current/index.html @@ -0,0 +1,108 @@ +<!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 | source</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> expresses the movement of the probability of finding a particle. Or 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 given by:</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">\[\begin{aligned} + \pd{P}{t} + &= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}} + = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}} + \\ + &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big) + - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}} + \\ + &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}\]</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">\[\begin{aligned} + \pd{P}{t} + = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}} + = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}} +\end{aligned}\]</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">\[\begin{aligned} + \boxed{ + \nabla \cdot \vec{J} + = - \pd{|\psi|^2}{t} + } +\end{aligned}\]</span></p> +<p>This states that 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> |