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1 files changed, 34 insertions, 32 deletions

diff --git a/source/know/concept/probability-current/index.md b/source/know/concept/probability-current/index.md
index bd41dab..81ca586 100644
--- a/source/know/concept/probability-current/index.md
+++ b/source/know/concept/probability-current/index.md
@@ -10,48 +10,49 @@ layout: "concept"
 
 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:
+Basically, it treats the particle as a heterogeneous fluid with density $$|\psi|^2$$.
+Clearly, the probability of finding the particle within a volume $$V$$ is:
 
 $$\begin{aligned}
-    P = \int_{V} | \psi |^2 \ddn{3}{\vb{r}}
+    P = \int_{V} | \psi |^2 \dd{\vb{r}}
 \end{aligned}$$
 
-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:
+As the system evolves in time, this probability may change,
+so we take its derivative with respect to time $$t$$,
+and substitute in the other side of the Schrödinger equation to get:
 
 $$\begin{aligned}
     \pdv{P}{t}
-    &= \int_{V} \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \ddn{3}{\vb{r}}
-    = \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \ddn{3}{\vb{r}}
+    &= \int_{V} \bigg( \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \bigg) \dd{\vb{r}}
     \\
-    &= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big)
-    - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \ddn{3}{\vb{r}}
+    &= \frac{i}{\hbar} \int_{V} \bigg( \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \bigg) \dd{\vb{r}}
     \\
-    &= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \ddn{3}{\vb{r}}
-    = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}}
+    &= \frac{i}{\hbar} \int_{V} \bigg( \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big)
+    - \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \bigg) \dd{\vb{r}}
+    \\
+    &= \frac{i \hbar}{2 m} \int_{V} \bigg( \!-\! \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \bigg) \dd{\vb{r}}
+    \\
+    &= - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}}
 \end{aligned}$$
 
-Where we have defined the probability current $$\vb{J}$$ as follows in
-the $$\vb{r}$$-basis:
+Where we have defined the probability current $$\vb{J}$$
+as follows in the $$\vb{r}$$-basis:
 
 $$\begin{aligned}
     \vb{J}
-    = \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi)
-    = \mathrm{Re} \Big\{ \psi \frac{i \hbar}{m} \psi^* \Big\}
+    &\equiv \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi)
+    = \Real\!\bigg\{ \psi \Big( \frac{i \hbar}{m} \Big) \nabla \psi^* \bigg\}
 \end{aligned}$$
 
 Let us rewrite this using the momentum operator
-$$\vu{p} = -i \hbar \nabla$$ as follows, noting that $$\vu{p} / m$$ is
-simply the velocity operator $$\vu{v}$$:
+$$\vu{p} = -i \hbar \nabla = - \vu{p}^*$$
+as follows, noting that $$\vu{p} / m$$ is simply the velocity operator $$\vu{v}$$:
 
 $$\begin{aligned}
     \boxed{
         \vb{J}
-        = \frac{1}{2 m} ( \psi^* \vu{p} \psi - \psi \vu{p} \psi^*)
-        = \mathrm{Re} \Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\}
-        = \mathrm{Re} \{ \psi^* \vu{v} \psi \}
+        = \Real\!\Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\}
+        = \Real\{ \psi^* \vu{v} \psi \}
     }
 \end{aligned}$$
 
@@ -60,8 +61,8 @@ equation:
 
 $$\begin{aligned}
     \pdv{P}{t}
-    = \int_{V} \pdv{|\psi|^2}{t} \ddn{3}{\vb{r}}
-    = - \int_{V} \nabla \cdot \vb{J} \ddn{3}{\vb{r}}
+    = \int_{V} \pdv{|\psi|^2}{t} \dd{\vb{r}}
+    = - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}}
 \end{aligned}$$
 
 By removing the integrals, we thus arrive at the **continuity equation**
@@ -74,16 +75,17 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-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
-$$\vb{J}$$ represents the flow of probability, which is analogous to the
-motion of a particle.
+This states that the total probability is conserved,
+and is reminiscent of electric charge conservation.
+In other words, the probability at a point can only change
+by "flowing" towards or away from it.
+Thus $$\vb{J}$$ represents the flow of probability as if it were a fluid.
 
-As a bonus, this still holds for a particle in an electromagnetic vector
-potential $$\vb{A}$$, 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:
+As a bonus, the continuity relation still holds
+for a particle in an electromagnetic vector potential $$\vb{A}$$,
+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{