Add "Probability current"

4 files changed, 199 insertions, 0 deletions

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
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+++ b/static/know/concept/probability-current/index.html
@@ -0,0 +1,108 @@
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+<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">\[\begin{aligned}
+    \pd{P}{t}
+    &amp;= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}}
+    = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}}
+    \\
+    &amp;= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big)
+    - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}}
+    \\
+    &amp;= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}}
+    = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}}
+\end{aligned}\]</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>
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