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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">\[\begin{aligned}
\pd{P}{t}
&= \int_{V} \psi \pd{\psi^*}{t} + \psi^* \pd{\psi}{t} \dd[3]{\vec{r}}
= \frac{i}{\hbar} \int_{V} \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \dd[3]{\vec{r}}
\\
&= \frac{i}{\hbar} \int_{V} \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vec{r}) \psi^* \Big)
- \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vec{r}) \psi \Big) \dd[3]{\vec{r}}
\\
&= \frac{i \hbar}{2 m} \int_{V} - \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \dd[3]{\vec{r}}
= - \int_{V} \nabla \cdot \vec{J} \dd[3]{\vec{r}}
\end{aligned}\]</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>
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