From 05c61c6c96a72fdbcfbe6800519d2dc5b91db013 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 20 Feb 2021 16:01:15 +0100
Subject: Fix derivatives in "Probability current"

---
 latex/know/concept/probability-current/source.md   | 10 +++++-----
 static/know/concept/probability-current/index.html | 10 +++++-----
 2 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/latex/know/concept/probability-current/source.md b/latex/know/concept/probability-current/source.md
index a6780f7..69faf0c 100644
--- a/latex/know/concept/probability-current/source.md
+++ b/latex/know/concept/probability-current/source.md
@@ -14,8 +14,8 @@ 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}}
+    \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)
@@ -51,8 +51,8 @@ Returning to the derivation of $\vec{J}$, we now have the following
 equation:
 
 $$\begin{aligned}
-    \pd{P}{t}
-    = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}}
+    \pdv{P}{t}
+    = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}}
     = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}}
 \end{aligned}$$
 
@@ -62,7 +62,7 @@ for $\vec{J}$:
 $$\begin{aligned}
     \boxed{
         \nabla \cdot \vec{J}
-        = - \pd{|\psi|^2}{t}
+        = - \pdv{|\psi|^2}{t}
     }
 \end{aligned}$$
 
diff --git a/static/know/concept/probability-current/index.html b/static/know/concept/probability-current/index.html
index d736e49..7b7ac32 100644
--- a/static/know/concept/probability-current/index.html
+++ b/static/know/concept/probability-current/index.html
@@ -56,8 +56,8 @@
 \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">\[\begin{aligned}
-    \pd{P}{t}
-    &amp;= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}}
+    \pdv{P}{t}
+    &amp;= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \dd[3]{\vec{r}}
     = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}}
     \\
     &amp;= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big)
@@ -83,15 +83,15 @@
 \end{aligned}\]</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">\[\begin{aligned}
-    \pd{P}{t}
-    = \int_{V} \pd{|\psi|^2}{t} \dd[3]{\vec{r}}
+    \pdv{P}{t}
+    = \int_{V} \pdv{|\psi|^2}{t} \dd[3]{\vec{r}}
     = - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}}
 \end{aligned}\]</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">\[\begin{aligned}
     \boxed{
         \nabla \cdot \vec{J}
-        = - \pd{|\psi|^2}{t}
+        = - \pdv{|\psi|^2}{t}
     }
 \end{aligned}\]</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>
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