# Probability current

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:

\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} \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) - \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} \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}

By removing the integrals, we thus arrive at the continuity equation for $$\vec{J}$$:

\begin{aligned} \boxed{ \nabla \cdot \vec{J} = - \pdv{|\psi|^2}{t} } \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 $$\vec{J}$$ 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}