Emission and absorption matrix elements: √n and √(n + 1)

SI units throughout, electron charge −e; no four-vector is needed here, and where one appears the metric is (+, −, −, −). Unit 11's Coulomb-gauge Hamiltonian, quoted, couples atom and field at first order in e through H₁ = (e/m)A⋅p, and one mode of its vector potential is A = √(ħ/(2ε₀ωV))[εa e^(ik⋅r) + ε*a† e^(−ik⋅r)]. Take product states |a, n⟩, the atom in its lower level a or upper level b with n photons in the mode. The a term lowers n by one and the a† term raises it, so each matrix element picks exactly one of them. Two pairings conserve energy at resonance: a with the atom going up, absorption, and a† with it coming down, emission. The other two, atom up while a photon is created or down while one is destroyed, are non-zero but miss energy by ħ(ω + ω_ba); they give no first-order rate and return at second order, in the level shift and in light scattering. The A² term, order e², moves n by 0 or 2 and cannot make a one-photon transition at first order, while the spin term −μ⋅B is linear in a and a† and inherits the same factors (12.04).

The norms do the work: ‖a|n⟩‖² = ⟨n|a†a|n⟩ = n, while ‖a†|n⟩‖² = ⟨n|aa†|n⟩ = n + 1, and the difference is [a, a†] = 1. With Quantum Mechanics I's phases, a|n⟩ = √n|n − 1⟩ and a†|n⟩ = √(n + 1)|n + 1⟩, so the absorption amplitude ⟨b, n − 1|H₁|a, n⟩ carries √n and the emission amplitude ⟨a, n + 1|H₁|b, n⟩ carries √(n + 1), derived. Because k⋅ε = 0, e^(ik⋅r) commutes with ε⋅p, and hermiticity then makes the two atomic factors exact complex conjugates. So |Mₑₘ|²/|M_abs|² = (n + 1)/n with no multipole approximation at all: the ratio is a statement about the field alone. Nor does it need a number state. Summed over final field states, an absorption probability goes as Σ|⟨f|a|ψ⟩|² = ⟨a†a⟩ = n̄ and an emission probability as ⟨aa†⟩ = n̄ + 1, whether the field is a laser's coherent state, a thermal mixture or anything else with mean occupation n̄. Topic 12.03 feeds the thermal n̄ into exactly these two factors.

Across an atom e^(ik⋅r) is close to 1: for Lyman α at 121.6 nm, ka₀ = 2.7 × 10⁻³. The first correction, i k⋅r, has the opposite parity to the dipole term and cannot connect the same pair of states, so setting e^(ik⋅r) = 1 errs at relative order (ka₀)² ≈ 10⁻⁵, unless ε⋅d_ba vanishes and the corrections are all that is left (12.04). Quantum Mechanics II's identity p_ba = i m ω_ba r_ba, quoted, from [r, Hₐₜₒₘ] = iħp/m, then gives (e/m)√(ħ/(2ε₀ωV))|ε⋅p_ba| = e ω_ba √(ħ/(2ε₀ωV))|ε⋅r_ba|. On resonance, ω = ω_ba, this is ℰ|ε⋅d_ba| with ℰ = √(ħω/(2ε₀V)) and d_ba = −e⟨b|r|a⟩, so |M| = √(n + s) ℰ|ε⋅d_ba|. Off resonance the A⋅p form keeps a factor ω_ba/ω, which is 1 on the energy shell, where every golden-rule rate evaluates it. Dimensions: V m⁻¹ × C m = J. For hydrogen 1s to 2p, |d_ba| = 6.32 × 10⁻³⁰ C m; in a 1.0 cm³ mode ℰ = 0.304 V m⁻¹, so the vacuum amplitude ℰ|d_ba| = 1.92 × 10⁻³⁰ J is ħ × 1.8 × 10⁴ s⁻¹.

Split the emission probability as |Mₑₘ|² = n ℰ²|ε⋅d_ba|² + ℰ²|ε⋅d_ba|². The first part grows with the photons already present, and its photon enters the same mode, with the same k and ε, as theirs: stimulated emission, the coherent amplification a laser runs on. The second part ignores what the mode holds and survives at n = 0. That is the result Quantum Mechanics II could not reach. Its semiclassical atom, with no applied field, has no perturbation at all, so an excited stationary state stays excited for ever, and Einstein's A had to be added through detailed balance. Here the excited atom has amplitude ℰ|ε⋅d_ba| to emit into every empty mode with ε⋅d_ba ≠ 0, written into the Hamiltonian. One mode alone does not make a decay, though: Unit 11's one-mode solution passes the energy back and forth at the vacuum Rabi frequency 2g, with g = ℰ|ε⋅d_ba|/ħ. An irreversible decay at a fixed rate needs the continuum of empty modes, which topic 12.02 sums with the golden rule and 12.05 treats exactly.

Give the mode the energy of a classical wave of amplitude E₀: nħω = ε₀E₀²V/2, so E₀ = 2√n ℰ, and a coherent state's mean field has exactly this amplitude with n replaced by n̄. Quantum Mechanics II's perturbation eE₀ ε⋅r cos ωt splits into halves rotating as e^(−iωt) and e^(+iωt), and each has a matrix element of size (E₀/2)|ε⋅d_ba| = √n ℰ|ε⋅d_ba|. The first is the quantised absorption amplitude exactly; the second is the stimulated part of emission, short only of the 1. So the semiclassical theory is not wrong but a limit: the emission probabilities differ by the factor (n + 1)/n, an excess of 1/n. For a 1.0 kV m⁻¹ wave at 780 nm in a 1.0 cm³ mode, n = 1.7 × 10⁷ and 1/n = 5.8 × 10⁻⁸. That 1/n counts only this mode; spontaneous emission into all the other, empty modes is a separate channel, summed in 12.02. At the other end, e → 0 sends every amplitude to zero, the free-field limit, because each is first order in e.

The cleanest test isolates one mode. The pair |b, n⟩ and |a, n + 1⟩ is coupled by Mₑₘ = ħg√(n + 1). Dropping the energy-violating couplings that lead out of the pair, the rotating-wave approximation, which costs only of order g/ω ≈ 5 × 10⁻⁷ here, the two-state solution, quoted from Unit 11, gives P_b(t) = cos²(g√(n + 1) t): the population oscillates at 2g√(n + 1). A classical wave of equal energy gives 2g√n instead. In 1996 Brune and co-workers in Paris sent circular Rydberg atoms, on a 51.1 GHz transition, through a superconducting cavity. With the cavity empty the atoms oscillated at 2g/2π ≈ 47 kHz, the vacuum Rabi frequency, where a semiclassical atom would not have moved. With a weak coherent field injected, the signal broke into discrete Fourier components at 47√2 = 66.5, 47√3 = 81.4 and 2 × 47 = 94 kHz, weighted by the field's photon-number distribution: the √(n + 1) of the emission amplitude, read one photon number at a time.