What the photon evidence establishes

An experiment supports a model when the model predicts what happens. It establishes the model only when every rival predicts something else. That second bar is far higher, and the history of the photon is where students most often miss the difference. The rival here is not classical physics wholesale — nobody defends a classical atom after 1913 — but semiclassical theory: quantum atoms obeying the Schrödinger equation, driven by a Maxwell field containing no photons. It is a real theory with real predictions, and nothing excluded it until Clauser's 1974 coincidence test and the antibunching measurements that followed. To retire it you need a quantity for which classical optics proves a bound. Three bounds matter here: g⁽²⁾(0) ≥ 1, Q ≥ 0, and α ≥ 1. Each is a theorem about any field whose intensity is a non-negative random variable, so a measurement on the wrong side of one is not merely evidence for quantisation — it is a refutation of the alternative.

Put the atom in a classical field E₀ cos ωt. The perturbation −(e/m)A⋅p̂ connects a bound state to the continuum, and Fermi's golden rule gives a transition rate R = (2π/ħ)|⟨f|H′|i⟩|²ρ(E_f). The photoelectric signatures fall out at once. The golden rule places the electron at Eᵢ + ħω, and ρ(E_f) vanishes unless that energy lies in the continuum, so there is a sharp threshold at ħω = φ and the fastest electrons leave with Kₘₐₓ = ħω − φ: the energy per electron tracks frequency because the atom's level structure does, not because the light arrives in lumps. The rate goes as |E₀|², hence as intensity, so brighter light gives more electrons rather than faster ones. And R is a rate, constant once the field has been on for a cycle or two, so the first count from the surface is expected within nanoseconds however dim the beam. Lamb and Scully put it bluntly in 1969: every textbook feature of the photoelectric effect is a statement about a quantised detector.

The argument that looked decisive runs like this. Let each sodium atom intercept only the light falling on its own geometric cross-section, and let that energy pile up until it reaches the work function. At a microwatt per square centimetre the formula card's estimate comes out at over an hour, whereas Lawrence and Beams in 1928 found the first electrons within 3 × 10⁻⁹ s of the light arriving. The conclusion drawn was "therefore photons". But look at what the estimate assumed: that energy accretes continuously into an atom-sized target until a threshold is crossed. The semiclassical calculation never says that. It gives a probability per unit time — small at low intensity, but a small rate still produces its first count almost immediately when you are watching some 10¹⁵ atoms per square centimetre. The null delay result kills classical accretion. It leaves the classical field standing.

Compton's 1923 result Δλ = (h/mₑ c)(1 − cos θ), with h/mₑ c = 2.426 pm, is derived by conserving relativistic energy and momentum between a lump carrying ħω and ħk and a free electron. It fits the data beautifully. But ħ enters that derivation through the electron's de Broglie relations as much as the field's, and in 1927 Schrödinger obtained the same shift by letting a classical wave Bragg-scatter off the moving density grating of an electron matter wave. A second experiment did more. Bohr, Kramers and Slater had proposed in 1924 that energy and momentum are conserved only statistically, so scattered photon and recoil electron need not appear together. Bothe and Geiger in 1925, and Compton and Simon the same year, found them in coincidence event by event, and the BKS model died. That is a real advance — but note precisely what it settled: strict conservation in individual events. The semiclassical account never gave up strict conservation, so the coincidence result left it untouched.

Here is the bound in four lines. Let the light be a classical field whose intensity I at the detector is some non-negative random variable. Then g⁽²⁾(0) = ⟨I²⟩/⟨I⟩², and since ⟨I²⟩ − ⟨I⟩² = ⟨(I − ⟨I⟩)²⟩ ≥ 0, it follows that g⁽²⁾(0) ≥ 1. Cauchy–Schwarz applied at two times gives a second bound, g⁽²⁾(0) ≥ g⁽²⁾(τ), so a classical correlation function can never climb out of a dip. Thermal light sits at g⁽²⁾(0) = 2, bunched; a coherent laser sits exactly at 1; nothing classical goes below. Now take a single atom or quantum dot. Having emitted, it is in its ground state and cannot emit again until re-excited, so two clicks at zero delay have probability zero: g⁽²⁾(0) = 0, recovering to 1 over the excited-state lifetime. Kimble, Dagenais and Mandel saw this in resonance fluorescence in 1977; Grangier, Roger and Aspect made it unambiguous in 1986, sending heralded photons onto a beam splitter and finding coincidences at a fifth of the classical minimum.

The counting statistics carry the same verdict in a different currency. Count photons in a gate and compare the variance with the Poisson value: the Mandel parameter Q = (⟨Δn²⟩ − ⟨n⟩)/⟨n⟩ is zero for a laser, equal to ⟨n⟩ for single-mode thermal light, and non-negative for any classical field, because a fluctuating positive intensity can only add noise to a Poisson process. Q < 0 — quieter than Poisson — is forbidden classically, and Short and Mandel reported it in atomic resonance fluorescence in 1983. The catch is fragility. A detector of efficiency η thins the count binomially, and the thinning maps Q → ηQ, so a source seen through a 10% detector looks only a tenth as sub-Poissonian; stray Poissonian light dilutes it in the same way, by the emitter's share of the total. Normalised correlations do not suffer this, because loss multiplies numerator and denominator of g⁽²⁾ by the same η² — which is why antibunching is the workhorse and Q the connoisseur's measurement. Hong–Ou–Mandel interference and Bell tests now do the same job in other coinage.