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:

\[\begin{aligned} \boxed{ \vec{J} = \mathrm{Re} \Big\{ \psi^* \frac{\hat{p} - q \vec{A}}{m} \psi \Big\} } \end{aligned}\]

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