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---
title: "Probability current"
sort_title: "Probability current"
date: 2021-02-22
categories:
- Quantum mechanics
- Physics
layout: "concept"
---
In quantum mechanics, the **probability current** describes the movement
of the probability of finding a particle at given point in space.
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:
$$\begin{aligned}
P = \int_{V} | \psi |^2 \dd{\vb{r}}
\end{aligned}$$
As the system evolves in time, this probability may change,
so we take its derivative with respect to time $$t$$,
and substitute in the other side of the Schrödinger equation to get:
$$\begin{aligned}
\pdv{P}{t}
&= \int_{V} \bigg( \psi \pdv{\psi^*}{t} + \psi^* \pdv{\psi}{t} \bigg) \dd{\vb{r}}
\\
&= \frac{i}{\hbar} \int_{V} \bigg( \psi (\hat{H} \psi^*) - \psi^* (\hat{H} \psi) \bigg) \dd{\vb{r}}
\\
&= \frac{i}{\hbar} \int_{V} \bigg( \psi \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi^* + V(\vb{r}) \psi^* \Big)
- \psi^* \Big( \!-\! \frac{\hbar^2}{2 m} \nabla^2 \psi + V(\vb{r}) \psi \Big) \bigg) \dd{\vb{r}}
\\
&= \frac{i \hbar}{2 m} \int_{V} \bigg( \!-\! \psi \nabla^2 \psi^* + \psi^* \nabla^2 \psi \bigg) \dd{\vb{r}}
\\
&= - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}}
\end{aligned}$$
Where we have defined the probability current $$\vb{J}$$
as follows in the $$\vb{r}$$-basis:
$$\begin{aligned}
\vb{J}
&\equiv \frac{i \hbar}{2 m} (\psi \nabla \psi^* - \psi^* \nabla \psi)
= \Real\!\bigg\{ \psi \Big( \frac{i \hbar}{m} \Big) \nabla \psi^* \bigg\}
\end{aligned}$$
Let us rewrite this using the momentum operator
$$\vu{p} = -i \hbar \nabla = - \vu{p}^*$$
as follows, noting that $$\vu{p} / m$$ is simply the velocity operator $$\vu{v}$$:
$$\begin{aligned}
\boxed{
\vb{J}
= \Real\!\Big\{ \psi^* \frac{\vu{p}}{m} \psi \Big\}
= \Real\{ \psi^* \vu{v} \psi \}
}
\end{aligned}$$
Returning to the derivation of $$\vb{J}$$, we now have the following
equation:
$$\begin{aligned}
\pdv{P}{t}
= \int_{V} \pdv{|\psi|^2}{t} \dd{\vb{r}}
= - \int_{V} \nabla \cdot \vb{J} \dd{\vb{r}}
\end{aligned}$$
By removing the integrals, we thus arrive at the **continuity equation**
for $$\vb{J}$$:
$$\begin{aligned}
\boxed{
\nabla \cdot \vb{J}
= - \pdv{|\psi|^2}{t}
}
\end{aligned}$$
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, 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:
$$\begin{aligned}
\boxed{
\vb{J}
= \mathrm{Re} \Big\{ \psi^* \frac{\vu{p} - q \vb{A}}{m} \psi \Big\}
}
\end{aligned}$$
## References
1. L.E. Ballentine,
*Quantum mechanics: a modern development*, 2nd edition,
World Scientific.
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