Categories: Physics, Quantum mechanics.

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 ψ2|\psi|^2. Now, the probability of finding the particle within a volume VV is:

P=Vψ2d3r\begin{aligned} P = \int_{V} | \psi |^2 \ddn{3}{\vb{r}} \end{aligned}

As the system evolves in time, this probability may change, so we take its derivative with respect to time tt, and when necessary substitute in the other side of the Schrödinger equation to get:

Pt=Vψψt+ψψtd3r=iVψ(H^ψ)ψ(H^ψ)d3r=iVψ( ⁣ ⁣22m2ψ+V(r)ψ)ψ( ⁣ ⁣22m2ψ+V(r)ψ)d3r=i2mVψ2ψ+ψ2ψd3r=VJd3r\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}} \\ &= \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}{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}} \end{aligned}

Where we have defined the probability current J\vb{J} as follows in the r\vb{r}-basis:

J=i2m(ψψψψ)=Re{ψimψ}\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\} \end{aligned}

Let us rewrite this using the momentum operator p^=i\vu{p} = -i \hbar \nabla as follows, noting that p^/m\vu{p} / m is simply the velocity operator v^\vu{v}:

J=12m(ψp^ψψp^ψ)=Re{ψp^mψ}=Re{ψv^ψ}\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 \} } \end{aligned}

Returning to the derivation of J\vb{J}, we now have the following equation:

Pt=Vψ2td3r=VJd3r\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}} \end{aligned}

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

J=ψ2t\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 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 J\vb{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 A\vb{A}, thanks to the gauge invariance of the Schrödinger equation. We can thus extend the definition to a particle with charge qq in an SI-unit field, neglecting spin:

J=Re{ψp^qAmψ}\begin{aligned} \boxed{ \vb{J} = \mathrm{Re} \Big\{ \psi^* \frac{\vu{p} - q \vb{A}}{m} \psi \Big\} } \end{aligned}


  1. L.E. Ballentine, Quantum mechanics: a modern development, 2nd edition, World Scientific.