The photoelectric effect and what it does not prove

From Lenard's 1902 experiments onward the facts hardened. A maximum kinetic energy that grows linearly with frequency and ignores intensity; a threshold frequency below which nothing leaves however bright the lamp; and emission that begins promptly — Lawrence and Beams bounded the delay below about 3 ns in 1928. A classical field feeding energy continuously into a classical electron fails on the timetable spectacularly. At 1.0 μW m⁻² on sodium, an atom of cross-section ~1 × 10⁻¹⁹ m² collects ~1 × 10⁻²⁵ W, and gathering φ = 2.28 eV = 3.65 × 10⁻¹⁹ J takes 3.7 × 10⁶ s — six weeks against the measured nanoseconds. Note what that estimate assumed, though: an electron absorbing continuously, which is to say a classical atom. The failure convicts the pairing; it does not yet say which partner must be quantised.

Postulate energy exchanged in lumps hν and the bookkeeping is one line: the fastest electron carries Kₘₐₓ = hν − φ, so the voltage that just stops it obeys eVₛ = hν − φ. Plot Vₛ against ν and every metal gives a straight line of the same slope h/e = 4.136 × 10⁻¹⁵ V s; the work function only slides the intercept −φ/e. Two mercury lines suffice: caesium stopped by 0.32 V at 546.1 nm and by 1.45 V at 365.0 nm gives slope 1.13 V / 2.72 × 10¹⁴ Hz = 4.15 × 10⁻¹⁵ V s, hence h = 6.65 × 10⁻³⁴ J s, within a third of a percent. That is exactly Millikan's 1916 programme — he ran it hoping to break Einstein's equation and instead confirmed it to half a percent, while writing that the theory behind it seemed untenable. His instinct that the equation outran its interpretation is the actual subject of this topic.

Now run the calculation Lamb and Scully made famous in 1969. Let the atom have a bound state |i⟩ at energy Eᵢ and a continuum of states |f⟩, and couple it to the unquantised wave A(t) = A₀ cos ωt through Ĥ′(t) = (e/m) A(t)⋅p̂ = (e/2m) A₀⋅p̂ (e^(−iωt) + e^(iωt)). First-order time-dependent perturbation theory gives an amplitude c_f(t) whose sinc² factor collapses, after a few optical periods, onto energy conservation E_f = Eᵢ + ħω — only the e^(−iωt) term can reach the continuum; its partner would need a state ħω below |i⟩ — and Fermi's golden rule turns it into the rate Γ = (2π/ħ)|⟨f|(e/2m) A₀⋅p̂|i⟩|² ρ(E_f). Read the three signatures off. The rate is constant from t = 0, so the first electrons appear immediately: promptness without accumulation. It is proportional to |A₀|², so the count rate tracks intensity while each electron's energy does not. And the continuum is only reachable when ħω exceeds the binding energy: threshold, and with it the h/e slope. The ħ entered through the atom's stationary-state phases e^(−iEt/ħ) beating against the drive e^(−iωt) — nowhere was the field cut into lumps.

Hold the two calculations side by side. Einstein's photon ledger predicts the facts; the semiclassical golden rule predicts the same facts. When two models agree on every measured number, the experiment cannot separate them — it establishes only what both require and merely permits what one of them adds. Both require quantised atomic energy exchange: stationary states, with transitions bookkept in units set by ħ. Only one requires a granular field, so the photoelectric effect is evidence for the quantum atom and silent on the photon. Keep this discipline for the rest of the course: Compton scattering meets the same audit in the next topic (a Doppler model of a recoiling electron reproduces the shift, and only Bothe–Geiger coincidences force quantum-by-quantum transfer), and the duality bounds of UPV-02.07 are theorems about a two-state space, not about clumsy kicks. To convict the field itself you must measure a quantity for which classical and quantum field theories give different answers — not the energy of one electron, but the statistics of many detections.

Split the beam at a 50:50 mirror, put a detector on each output, and count coincidences inside a short window: g²(0) is the coincidence rate divided by what chance alone would give. Model the light classically as a fluctuating intensity I(t) ≥ 0, each detector firing in proportion to I; then g²(0) = ⟨I²⟩/⟨I⟩², and Cauchy–Schwarz forces ⟨I²⟩ ≥ ⟨I⟩². Every classical field has g²(0) ≥ 1: thermal light gives 2, an ideal laser exactly 1 — even a perfectly steady wave splits, so both halves fire together at least at chance. Quantum mechanically the same measurement reads g²(0) = ⟨n̂(n̂−1)⟩/⟨n̂⟩², and the one-photon state gives zero, since n̂(n̂−1)|1⟩ = 0: a photon delivered whole to one output cannot also fire the other. Kimble, Dagenais and Mandel saw g²(0) < 1 in the resonance fluorescence of single sodium atoms in 1977; Grangier, Roger and Aspect measured 0.18 ± 0.06 on a heralded cascade source in 1986, thirteen standard deviations below the classical floor. That — not 1905 — is where field quantisation became an experimental fact.

In the laboratory g²(0) is an estimator built from three numbers: singles rates R₁ and R₂, a coincidence window τ, and the measured coincidence rate R_c. Uncorrelated detectors coincide by accident at R₁R₂τ, so g²(0) ≈ R_c/(R₁R₂τ) — chance sits at 1 by construction. With R₁ = 9000 s⁻¹, R₂ = 8000 s⁻¹ and τ = 10 ns, chance predicts 0.72 coincidences per second; a true single-photon emitter should deliver almost none. The systematic floor comes from whatever mimics classical light: background counts, detector afterpulsing, and a second emitter in the focal volume — two independent single photons give g²(0) = 1/2. The statistical floor is Poisson: with N_c recorded coincidences the fractional uncertainty is 1/√N_c, so quoting g²(0) = 0.10 ± 0.02 needs about 25 of them. And the claim worth making is never 'g² is small' but 'g² sits below 1 by k standard deviations' — the error propagation is two lines of NumPy; the physics is knowing which floor you are standing on.