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Mathematics for Physics

102 Conservation of Charge

A principle that is used to keep the total electric charge of an isolated system constant, where electric charge is the source of the electromagnetic interaction.

Global charge conservation. The total charge of an isolated system does not change. Charge can be transferred but is neither created nor destroyed. This principle is used to balance charge in every quantum process.

The continuity equation. In quantum mechanics the same statement is local: the probability density of a charged particle obeys a continuity equation. This principle is used to identify \(|\Psi|^{2}\) with a conserved charge density for a single particle of charge \(q\).

The continuity equation is

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

where

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

Conservation from gauge invariance. Gauge invariance of the electromagnetic interaction requires a conserved current. This principle is used to couple the wavefunction to the electromagnetic potential without violating charge conservation.

102.1 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — conservation of charge.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — probability current and minimal coupling.