Dipole Approximation for Electron-Photon Interaction

The dipole approximation assumes that the wavelength of the electromagnetic field is much larger than the spatial extent of the electron wavefunction. In this case, the interaction Hamiltonian simplifies significantly.

Interaction Hamiltonian

In the dipole approximation, the interaction term becomes:

Hint = -d·E(t),

where:

  • d = -er is the electric dipole moment of the electron,
  • E(t) is the electric field of the photon.

Simplified Schrödinger Equation

The time-dependent Schrödinger equation becomes:

iℏ∂ψ/∂t = [H0 - d·E(t)]ψ,

where H0 is the unperturbed Hamiltonian of the electron.

Solving for Energy States

Under the dipole approximation, solutions can be obtained using:

  1. Time-Dependent Perturbation Theory: To calculate transition probabilities between energy levels.
  2. Rabi Oscillations: For resonant interactions between two levels.
  3. Floquet Theory: For periodic electric fields (e.g., in laser interactions).