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Mathematics for Physics

113 Electromagnetic Interaction

An interaction between charged matter and the electromagnetic field that is used to describe absorption and emission of radiation by quantum systems.

Minimal coupling. Coupling to the electromagnetic field is achieved by replacing \(\mathbf{p}\) with \(\mathbf{p}-q\mathbf{A}\) and adding \(qV\). This principle is used to write the Hamiltonian of a charged particle in a field.

The minimally coupled Hamiltonian is

\[ H = \dfrac{1}{2m}\bigl(\mathbf{p}-q\mathbf{A}\bigr)^{2} + qV \]

where

  • \(q\) is the charge.
  • \(\mathbf{A}\) is the vector potential.
  • \(V\) is the scalar potential.
  • \(m\) is the mass.
  • \(\mathbf{p}\) is the momentum operator.

The Lorentz force. The classical counterpart is the Lorentz force. This principle is used to recover the force on a charge from the same potentials.

The Lorentz force is

\[ \mathbf{F} = q\bigl(\mathbf{E}+\mathbf{v}\times\mathbf{B}\bigr) \]

where

  • \(\mathbf{F}\) is the force.
  • \(\mathbf{E}\) is the electric field.
  • \(\mathbf{B}\) is the magnetic field.
  • \(\mathbf{v}\) is the velocity.

Field-induced transitions. A time-dependent field induces transitions between energy eigenstates. This principle is used to compute absorption, emission, and Fermi’s golden rule.

Note: Linked from the Quantum Mechanics index as electromagnetic-interaction-1.

113.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — minimal coupling Hamiltonian.
  2. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(\mathbf{p}\mapsto\mathbf{p}-q\mathbf{A}\).
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — Lorentz force as classical electromagnetic interaction.