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Mathematics for Physics

114 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. A charged particle couples to the electromagnetic potentials through the Hamiltonian. This principle is used to replace the free kinetic term by the minimally coupled kinetic term.

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.

The Bohr frequency condition. An oscillating field can raise or lower the energy of a bound system by one photon. This principle is used to describe absorption and emission.

The Bohr frequency condition is

\[ hf = \lvert E_{i}-E_{f}\rvert \]

where

  • \(h\) is Planck’s constant.
  • \(f\) is the frequency of the photon.
  • \(E_{i}\) and \(E_{f}\) are the initial and final energies.

Fermi’s golden rule for radiative rates. The rate of a weak transition is given by Fermi’s golden rule. This principle is used to compute Einstein coefficients and selection rules.