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Mathematics for Physics

135 Probability Current

A vector field built from a wavefunction that is used to express the flow of probability density through space.

The continuity equation. Probability is locally conserved: any decrease of \(\rho\) in a volume equals the outward flux of the probability current. This principle is used to write a continuity equation for \(|\Psi|^{2}\).

The continuity equation is

\[ \dfrac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} = 0 \]

where

  • \(\rho = \lvert\Psi\rvert^{2}\) is the probability density.
  • \(\mathbf{j}\) is the probability current.
  • \(t\) is time.

The formula for \(\mathbf{j}\). The probability current is built from \(\Psi\) and its gradient. This principle is used to compute the flow from a known wavefunction.

The probability current is

\[ \mathbf{j} = \dfrac{\hbar}{m}\operatorname{Im}\bigl(\Psi^{*}\nabla\Psi\bigr) \]

where

  • \(\mathbf{j}\) is the probability current.
  • \(\Psi\) is the wavefunction.
  • \(m\) is the particle mass.
  • \(\hbar\) is the reduced Planck constant.

Vanishing current for a real wavefunction. A purely real stationary wavefunction has vanishing current. This principle is used to identify bound standing waves with no net flow.

Note: Also called probability flux. Also called probability current density.

135.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — probability flux \(\mathbf{j}\) and continuity equation \(\partial\rho/\partial t+\nabla\cdot\mathbf{j}=0\).