Schrödinger Equation for Electron-Photon Interaction

The system includes:

  • An electron with wavefunction ψe(r, t),
  • A photon field described by the vector potential A(r, t).

The total Hamiltonian includes:

  1. The electron’s kinetic energy, -ℏ2/2me∇2,
  2. The coupling between the electron and photon field through minimal coupling,
  3. The photon’s energy.

The Hamiltonian in SI units is:

H = (1/2me) [ -iℏ∇ - eA(r, t) ]2
    + eφ(r, t) 
    + (1/2)ε0 ∫ |E(r, t)|2 + (1/2μ0)|B(r, t)|2 d3r,

where:

  • φ(r, t) is the scalar potential,
  • E = -∂A/∂t - ∇φ is the electric field,
  • B = ∇×A is the magnetic field.

Simplifying for interaction only, the Schrödinger equation is:

iℏ∂ψe/∂t = Hψe.

Solving for Energy States

Solving the energy states requires quantizing the photon field. Using second quantization:

  • Represent the photon field as a superposition of modes: 
        A(r, t) = Σk,λ sqrt(ℏ/2ε0ωk) 
    [ ak,λeik·r + ak,λ†e-ik·r ].

Approach to Energy Levels:

  1. Electron in an Electromagnetic Field (Perturbation Theory): For weak coupling, perturbation theory gives corrections to the electron’s energy levels.
  2. Jaynes-Cummings Model: For resonant interactions (electron treated as a two-level system), one can use this model to calculate Rabi oscillations and energy splitting.
  3. Numerical Methods: For more general cases, computational methods are necessary.