Atom–field coupling and the dipole matrix element

Start from the gauge-invariant substitution p → p − qA, which for an electron of charge q = −e reads p → p + eA. The atom's Hamiltonian becomes H = (p + eA)²/2m + V(r). Choose the Coulomb gauge, ∇⋅A = 0: then A commutes with p, the two cross terms are equal, and the scalar potential of a free radiation field vanishes. Expanding gives H = H₀ + (e/m)A⋅p + e²A²/2m, with H₀ = p²/2m + V(r) the bare atomic Hamiltonian whose eigenkets |n l m⟩ are the basis you will work in. Nothing has been approximated yet. The whole of what follows is the business of deciding which of those two added operators can move an electron from one |n l m⟩ to another, and how big its matrix element is.

A plane wave carries the factor exp(ik⋅r) across the atom. Compare the two lengths: at hydrogen's Lyman-α line, λ = 121.6 nm, so k = 2π/λ = 5.17 × 10⁷ m⁻¹, and with a₀ = 5.29 × 10⁻¹¹ m the phase varies by k a₀ = 2.7 × 10⁻³ over one Bohr radius. Set exp(ik⋅r) → 1 and two things happen at once. The retained operator is the electric-dipole (E1) one; the discarded i k⋅r term supplies the magnetic-dipole and electric-quadrupole channels, smaller by k a in amplitude and (k a)² ≈ 7 × 10⁻⁶ in rate. And A(t) is now independent of r, so e²A²/2m is a multiple of the identity on the atomic Hilbert space: ⟨f|A²|i⟩ ∝ ⟨f|i⟩ = 0 for f ≠ i. It contributes a common time-dependent phase and drives nothing.

The velocity form (e/m)A⋅p is awkward: the operator is p, whose parity and angular-momentum content are less transparent than r's, and A depends on gauge. Apply the unitary transformation |ψ′⟩ = exp(−i q r⋅A(t)/ħ)|ψ⟩. Because the generator carries its own time dependence, the transformed Hamiltonian gains iħ(∂U/∂t)U† = q r⋅(∂A/∂t) = −q r⋅E, while U(p + eA)U† = p strips A out of the kinetic term altogether. What is left is H₁ = −d⋅E with d = q r = −e r. The A² term is absorbed in the same step. Two warnings. The transformation is unitary, so all observables agree, but |ψ⟩ and |ψ′⟩ are different kets while the field is on. And the gauge you compute in must be the gauge you interpret in.

The bridge is [H₀, r] = [p²/2m, r] = −iħp/m, which holds because V(r) commutes with r. Sandwich it between exact eigenkets: ⟨f|[H₀, r]|i⟩ = (E_f − Eᵢ)⟨f|r|i⟩, so ⟨f|p|i⟩ = (i m/ħ)(E_f − Eᵢ)⟨f|r|i⟩ = i m ω_fi ⟨f|r|i⟩. For hydrogen 1s → 2p₀ with ω_fi = 1.55 × 10¹⁶ rad s⁻¹ and ⟨2p₀|z|1s⟩ = 0.7449 a₀ = 3.94 × 10⁻¹¹ m, that gives |⟨2p₀|p_z|1s⟩| = 5.56 × 10⁻²⁵ kg m s⁻¹, about 0.28 of ħ/a₀. The identity used H₀|i⟩ = Eᵢ|i⟩ twice, so it is exact only for exact eigenkets. Run a Hartree-Fock or configuration-interaction calculation and the two gauges disagree; the size of the gap is a standard measure of how good the wavefunctions are.

Write the drive as H₁(t) = −d⋅ε E₀ cos ωt = −(d⋅ε E₀/2)[exp(iωt) + exp(−iωt)] and put it through first-order time-dependent perturbation theory. The amplitude c_f(t) = (1/iħ)∫₀ᵗ ⟨f|H₁|i⟩ exp(iω_fi t′) dt′ collects two denominators: δ = ω_fi − ω, which is small near resonance, and ω_fi + ω, which is not. Dropping the second is the rotating-wave approximation, and at Lyman-α, a detuning of 8 × 10⁹ rad s⁻¹ against ω = 1.55 × 10¹⁶ rad s⁻¹ makes the discarded amplitude smaller by δ/2ω = 3 × 10⁻⁷. What survives is |c_f(t)|² = Ω² sin²(δt/2)/δ², a resonance whose peak grows as Ω²t²/4 and whose width shrinks as 2π/t — energy conservation emerging from the clock, not assumed.

Turn the drive off and −d⋅E is identically zero, so an excited eigenket of H₀ is stationary and hydrogen 2p never decays. That is not a small blemish: the observed rate is A = 6.26 × 10⁸ s⁻¹, a 1.60 ns lifetime, and this model gives zero. The reason is structural — the field is a c-number, and a c-number has no vacuum fluctuation for the atom to emit into. Two repairs exist. Einstein's detailed balance against the Planck law fixes A/B = ħω³/π²c³ on the energy-density convention, taking A from the stimulated rate you already have; or quantise the field, where |0⟩ still supplies a non-zero ⟨1|a†|0⟩. Both give the same answer. Neither is contained in the classical calculation.