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| -rw-r--r-- | latex/know/concept/probability-current/source.md | 13 | ||||
| -rw-r--r-- | static/know/concept/probability-current/index.html | 6 |
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diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md index 69faf0c..bffc599 100644 --- a/latex/know/concept/probability-current/source.md +++ b/latex/know/concept/probability-current/source.md @@ -1,9 +1,12 @@ +% Probability current + + # 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: +In quantum mechanics, the *probability current* 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 $|\psi|^2$. +Now, the probability of finding the particle within a volume $V$ is: $$\begin{aligned} P = \int_{V} | \psi |^2 \dd[3]{\vec{r}} @@ -66,7 +69,7 @@ $$\begin{aligned} } \end{aligned}$$ -This states that probability is conserved, and is reminiscent of charge +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 $\vec{J}$ represents the flow of probability, which is analogous to the diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html index 7b7ac32..b256e52 100644 --- a/static/know/concept/probability-current/index.html +++ b/static/know/concept/probability-current/index.html @@ -4,7 +4,7 @@ <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> + <title>Prefetch | Probability current</title> <link rel="icon" href="data:,"> <style> body { @@ -50,7 +50,7 @@ </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>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> @@ -94,7 +94,7 @@ = - \pdv{|\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>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{ |