Photoelectric Effect and the Photon

Every phone camera, every solar panel, every night-vision tube runs on a deal so simple it fits in four symbols: one photon in, one electron out. Yet when this effect was first mapped, between 1887 and 1902, it broke physics. Ultraviolet light falling on a clean metal plate knocks electrons loose — fine, waves carry energy. But the details refused to cooperate. A faint violet glimmer ejects electrons instantly, each with the same maximum energy no matter how dim the source; a blinding red floodlight, pouring in millions of times more power, ejects nothing at all. Classical electromagnetism, which had just triumphantly predicted radio waves, could not explain why color mattered and brightness did not. Einstein's 1905 answer was the first time anyone claimed the quantum was real physics rather than Planck's bookkeeping trick: light itself arrives in indivisible packets of energy $h\nu$ . That one postulate turns every photoelectric puzzle into arithmetic — and makes a prediction sharp enough to kill the theory: plot the electrons' stopping voltage against light frequency, and every metal on Earth must give a straight line with the same universal slope, $h/e$ . Millikan spent a decade trying to prove it wrong and measured Planck's constant to better than a percent instead. This article is where the quantum stops being about hot ovens (the previous article's territory) and becomes a claim about light in flight. Master the energy bookkeeping here and the photon carries you straight into wave–particle duality, Compton recoil, and de Broglie's matter waves.

Key takeaways

Light of frequency $\nu$ delivers energy in indivisible photons of energy $E = h\nu$ ; absorption is all-or-nothing, one photon per electron.
The escape cost from a metal is its work function $W$ ; electrons emerge with $K_{\max} = h\nu - W$ , and below the threshold frequency $\nu_0 = W/h$ nothing comes out at any intensity.
Intensity counts photons: it sets the photocurrent (how many electrons), never the energy of each one. Frequency sets the energy per electron.
The stopping potential converts energy to voltage: $eV_0 = K_{\max}$ , so plotting $V_0$ against $\nu$ gives a straight line of slope $h/e$ — the same for every material.
Emission is instantaneous (< ns) even in faint light: an electron grabs one whole photon, it never "soaks up" wave energy — classical accumulation would take minutes.
Key idea: the color of light sets what each photon is worth; the brightness only sets how many arrive.

The core insight

Light delivers energy to electrons in discrete lumps of size $h\nu$ , so the ejected electron's maximum kinetic energy depends on the light's color (frequency), not its brightness (intensity).

Where this lives today

The photoelectric effect is the working principle of most light detection on Earth. Every CMOS sensor and CCD — every phone camera — is Einstein's bookkeeping in silicon: one absorbed photon promotes one electron, with the semiconductor band gap playing the role of $W$ ; the threshold condition is why silicon cameras go blind beyond ~1100 nm and why infrared astronomy needs narrower-gap materials. Photomultiplier tubes push the original vacuum geometry to its extreme, turning a single cesium-cathode photoelectron into a measurable pulse — they watched neutrinos from Supernova 1987A arrive, and their successors line the walls of every neutrino observatory. X-ray photoelectron spectroscopy (XPS) runs equation (2) backwards: shine X-rays of known $h\nu$ , measure electron kinetic energies, and read off binding energies that identify elements and their chemical states on any surface. The frontier has moved to the time domain and the intensity extremes. Attosecond laser pulses now clock photoemission delays of tens of attoseconds ( $10^{-17}$ s) — the "instantaneous" of 1905, finally resolved into physics (2023 Nobel Prize). And at extreme intensities, electrons absorb several photons at once, $K_{\max} = nh\nu - W$ : multiphoton emission below threshold, a regime Einstein's postulate handles gracefully where the classical wave never got started.

The physical picture

Picture the metal's conduction electrons as marbles resting in a shallow tray whose rim is a height $W$ — the work function, the energy the least-bound electron must pay to escape the surface. Classical light is a steady drizzle: energy arrives continuously, spread over the whole wavefront, and any marble should eventually accumulate enough to climb out, given brightness or patience. The photon picture replaces the drizzle with hail. Each photon is a single hammer-blow worth exactly $h\nu$ : one blow either lifts a marble over the rim in one hit ( $h\nu > W$ ) or it does not ( $h\nu < W$ ) — and a million weak blows never add up, because each photon is absorbed whole by one electron, independently of all the others. In this picture the three experimental facts that ruined classical physics become three one-line consequences. Frequency is the hardness of each blow, so only color controls the exit speed. Intensity is the number of blows per second, so brightness controls only the count of ejected electrons — the photocurrent. And since an electron need not wait to soak up spread-out wave energy, emission is instantaneous even when photons arrive one per second. The graininess hid for a century for the same reason raindrops feel like a smooth stream: a desk lamp pours out some $10^{20}$ photons every second — as recalled in the Breakdown of Classical Physics article, quantum effects hide wherever $h$ is small compared to the action involved.

Figure

The photoelectric experiment — one photon in, one electron out, a voltage to weigh it

The instrument at the bottom of the figure is the whole trick of the quantitative experiment. Make the collector negative so it repels the electrons: only those with kinetic energy above $eV$ complete the crossing. Dial the retarding voltage up until the photocurrent just reaches zero — that reading, the stopping potential $V_0$ , catches exactly the fastest electrons, converting an energy measurement into a voltage reading: $eV_0 = K_{\max}$ .

Three facts no wave could survive

The experiments of Hertz (1887, accidental — his spark gaps sparked more readily under UV) and Lenard (1902, systematic) left classical physics three stubborn facts. First, a sharp threshold: below a frequency $\nu_0$ characteristic of the metal, no electrons come out — regardless of intensity. Second, frequency sets the energy: $K_{\max}$ rises with the light's frequency and is flatly indifferent to its brightness; intensity changes only the number ejected per second. Third, no time lag: even light so faint that classical accumulation would need minutes produces photoelectrons within nanoseconds. Each fact contradicts the wave picture, where energy is spread over the wavefront, grows with intensity, and must be soaked up gradually. Einstein's resolution begins with the quantum you met in Blackbody Radiation and the Planck Quantum — but with the crucial escalation. Planck had quantized only the exchange of energy between radiation and the cavity walls; the field itself he left classical. Einstein proposed that the light in flight is grainy:

E = h\nu = \hbar\omega \qquad (1)

where: $h = 6.626\times10^{-34}\ \mathrm{J\,s}$ — Planck's constant; $\nu$ — the light's frequency; $\hbar = h/2\pi = 1.055\times10^{-34}\ \mathrm{J\,s}$ and $\omega = 2\pi\nu$ — the angular versions. Read the structure: energy per photon scales with frequency and nothing else — intensity appears nowhere in (1). A convenient laboratory form: $E\,(\mathrm{eV}) = 1240/\lambda\,(\mathrm{nm})$ , so visible photons carry 1.8–3.1 eV, comparable to metallic work functions — which is exactly why this effect sits at the visible/UV boundary. Absorption is all-or-nothing: one electron swallows one photon whole. Energy bookkeeping for the luckiest electron — the one bound by only the minimum cost $W$ — gives the Einstein photoelectric equation:

K_{\max} = h\nu - W, \qquad \nu_0 = \frac{W}{h} \qquad (2)

where: $K_{\max}$ — maximum kinetic energy of the ejected electrons; $W$ — the work function (2–5 eV for clean metals: 2.28 eV for sodium, 2.30 for potassium, 4.34 for zinc); $\nu_0$ — the threshold frequency at which $K_{\max}$ hits zero. Electrons bound more deeply than $W$ emerge slower, filling the energy range below $K_{\max}$ ; below $\nu_0$ no photon carries enough for even the luckiest electron, and intensity — which only counts photons — cannot help.

Derivation AWhy no classical wave survives the photoelectric data

The strongest way to appreciate Einstein's postulate is to watch the classical picture fail quantitatively, not just qualitatively. We estimate how long a classical wave would need to deliver one electron's escape cost, then confront the observed emission time. The strategy is a physicist's order-of-magnitude argument: give the wave picture every benefit of the doubt, compute the soak time it predicts, and see it miss by eleven orders of magnitude. The failure is what forces energy to arrive in lumps.

Step 1. Set the classical energy-collection rate. A bound electron can plausibly collect wave energy from about one atomic cross-section, $\sigma \sim 10^{-19}\ \mathrm{m^2}$ . In faint light of intensity $I = 10^{-2}\ \mathrm{W/m^2}$ it gathers power $P = I\sigma \sim 10^{-21}\ \mathrm{W}$ . Why: a classical wave spreads its energy democratically over the wavefront; each electron can only sip from its own tiny patch.

Step 2. Compare with the escape cost. For sodium, $W = 2.28\ \mathrm{eV} = 3.65\times10^{-19}\ \mathrm{J}$ . The classical soak time is $t \sim W/I\sigma \approx 365\ \mathrm{s}$ — six minutes of darkness before the first electron could leave. Why: in the wave picture energy must accumulate to $W$ before escape; nothing can come out earlier.

Step 3. Confront experiment. Photoelectrons appear within nanoseconds of the light arriving, at this and far lower intensities. The discrepancy is a factor of $\sim 10^{11}$ . Why: minutes versus nanoseconds is not an error to patch with a better model of the atom — it falsifies continuous absorption outright.

Step 4. Let the energy arrive pre-packaged. If energy travels in indivisible units $E = h\nu$ absorbed whole by single electrons, the first photon to strike a lucky electron ejects it immediately — no accumulation, no lag, and no dependence of the exit energy on how many other photons are en route. Why: quantization converts "how long until enough energy gathers?" into "does one packet suffice?" — which is a question about frequency alone. *Verified: the soak-time estimate $W/I\sigma = 365\ \mathrm{s}$ recomputed numerically; observed photoemission delays are below $10^{-9}\ \mathrm{s}$ (Griffiths Ch. 1; Townsend Ch. 1).*

Go deeper · Derivation B · 1 minEnergy bookkeeping for one photon and one electron

Derivation BEnergy bookkeeping for one photon and one electron

Here is the entire derivation Einstein won the Nobel Prize for — deliberately short, because its power is in the postulate, not the algebra. We take the photon hypothesis seriously, apply conservation of energy to a single absorption event, and account honestly for the range of electron binding energies in the metal. The subtlety worth watching: the equation predicts a maximum, not a single energy — the spectrum of slower electrons below $K_{\max}$ is real and expected.

Step 1. One absorption event. An electron inside the metal absorbs exactly one photon, gaining energy $h\nu$ . Why: this is the postulate — photons are indivisible, and two-photon events are fantastically rare at ordinary intensities (they appear only in the modern high-intensity limit).

Step 2. Pay the escape cost. An electron bound with energy $E_b$ leaves the surface with $K = h\nu - E_b$ . Why: energy conservation, applied to one microscopic event rather than to beam averages — this per-event accounting is the conceptual break with classical thinking.

Step 3. Maximize over the electron population. The least-bound electrons have $E_b = W$ , the work function, so

K_{\max} = h\nu - W \qquad (3)

Why: deeper electrons emerge slower or not at all, so the measured quantity with a clean meaning is the maximum — precisely what the stopping potential catches.

Step 4. Read off the threshold and the meter. $K_{\max} = 0$ at $\nu_0 = W/h$ ; and since the stopping potential just turns back the fastest electrons, $eV_0 = K_{\max}$ , giving the measurable form

V_0 = \frac{h}{e}\,\nu - \frac{W}{e} \qquad (4)

Why: equation (4) is a straight line in the $(\nu, V_0)$ plane whose slope $h/e$ contains no property of the metal whatsoever — the most falsifiable statement quantum theory had yet made. *Verified: $h/e = 4.136\times10^{-15}\ \mathrm{V\,s}$ from CODATA constants; the linear form matches Griffiths Ch. 1 and Millikan (1916).*

Figure

Einstein's straight line — every metal, the same slope h/e

A "which knob does what" summary is worth pinning above the lab bench:

you turn up…	photocurrent	$K_{\max}$ (and $V_0$ )	why
intensity (fixed $\nu > \nu_0$ )	increases ∝ intensity	unchanged	more photons, same energy each
frequency (fixed intensity)	roughly steady	rises linearly	harder blow per photon
intensity (fixed $\nu < \nu_0$ )	stays zero	—	no photon suffices; count is irrelevant
work function (change metal)	—	drops by $\Delta W$	costlier escape; slope of $V_0(\nu)$ unchanged

Go deeper · Derivation C · 1 minReading Planck's constant off a stopping-potential plot

Derivation CReading Planck's constant off a stopping-potential plot

Equation (4) is not just a summary — it is an undergraduate lab and the experiment that settled the photon's reality. We show how the plot hands over three quantities at once, and why the slope is the theory's kill switch. The strategy: treat $V_0(\nu)$ as measured data, fit the line, and interrogate each feature. Watch which parts depend on the metal and which cannot.

Step 1. Slope. Measure $V_0$ at two frequencies and divide: $\text{slope} = (V_0' - V_0)/(\nu' - \nu) = h/e$ . Why: $W$ cancels in the difference — the slope is blind to the material. If any metal anywhere gave a different slope, equation (2) would be dead.

Step 2. Vertical intercept. Extrapolate to $\nu = 0$ : the intercept is $-W/e$ , the work function in volts. Why: the extrapolation is fictional (no emission occurs below threshold) but the fitted line stores $W$ in its offset all the same.

Step 3. Horizontal intercept. The line crosses $V_0 = 0$ at $\nu_0 = W/h$ : the threshold frequency, now read graphically. Why: the three readings — slope, intercept, crossing — overdetermine the physics, so the experiment checks itself.

Step 4. Convert to the constant. $h = e \times \text{slope}$ . Millikan's 1916 measurement gave $h$ to about 0.5%, agreeing with Planck's blackbody value — two utterly different phenomena, one constant. Why: the same $h$ appearing in a hot oven's spectrum and a photocell's voltage plot is the earliest, strongest evidence that quantization is universal. *Verified: with the mercury lines 546.1 nm and 435.8 nm on cesium ( $W = 2.14$ eV), $V_0$ = 0.130 V and 0.705 V; slope $= 0.575\ \mathrm{V} / 1.39\times10^{14}\ \mathrm{Hz} = 4.14\times10^{-15}\ \mathrm{V\,s}$ , giving $h = 6.63\times10^{-34}\ \mathrm{J\,s}$ — the worked example below.*

Limiting cases

Case	Result	Meaning
$\nu < \nu_0$	$K_{\max} = 0$ , no emission	no single photon suffices; intensity is powerless — the hallmark of quantization
$h\nu \gg W$	$K_{\max} \simeq h\nu$	escape cost negligible; photon energy converts wholly to electron energy (X-ray regime)
many photons: $N = P/h\nu \gg 1$	photocurrent ∝ intensity	the classical brightness scaling re-emerges for counts, while energy-per-electron stays quantized
extreme laser intensity	$K_{\max} = n h\nu - W$ , $n \ge 2$	multiphoton absorption: emission below single-photon threshold becomes possible — the modern, nonlinear extension
$V \to V_0$ from below	photocurrent $\to 0$ smoothly	only the fastest electrons survive the hill; the knee locates $K_{\max}$

Worked examples

1 · Ultraviolet on sodium (confidence builder)

Light of wavelength $300\ \mathrm{nm}$ falls on a sodium surface, work function $W = 2.28\ \mathrm{eV}$ . Find $K_{\max}$ , the stopping potential, and the longest wavelength that produces any photoemission.

Step 1. Photon energy. Using $hc = 1240\ \mathrm{eV\,nm}$ : $E = 1240/300 = 4.13\ \mathrm{eV}$ .

Step 2. Einstein equation (2): $K_{\max} = 4.13 - 2.28 = \boxed{1.85\ \mathrm{eV}}$ .

Step 3. Stopping potential from $eV_0 = K_{\max}$ : $V_0 = \boxed{1.85\ \mathrm{V}}$ — the same number, re-dressed in volts; that unit conversion is the meter.

Step 4. Threshold wavelength: $\lambda_0 = hc/W = 1240/2.28 = \boxed{544\ \mathrm{nm}}$ — green. Sodium responds to green and bluer light; a red laser pointer at $650\ \mathrm{nm}$ ( $1.91\ \mathrm{eV} < W$ ) does nothing at any power. Doubling the lamp's intensity doubles the ejected-electron count and changes neither answer above.

2 · Millikan's plot, reconstructed (exam level)

A photocell with a cesium cathode ( $W = 2.14\ \mathrm{eV}$ ) is illuminated by two mercury spectral lines: green $546.1\ \mathrm{nm}$ and blue $435.8\ \mathrm{nm}$ . Predict the two stopping potentials, then show how the pair alone yields Planck's constant.

Step 1. Photon energies: $E_1 = 1240/546.1 = 2.27\ \mathrm{eV}$ and $E_2 = 1240/435.8 = 2.84\ \mathrm{eV}$ (keeping an extra digit internally: 2.270 and 2.845 eV).

Step 2. Stopping potentials from (2): $V_{0,1} = 2.270 - 2.14 = \boxed{0.13\ \mathrm{V}}$ and $V_{0,2} = 2.845 - 2.14 = \boxed{0.70\ \mathrm{V}}$ . Note how close to threshold the green line sits — Millikan needed exquisitely clean surfaces because contamination shifts $W$ by tenths of a volt.

Step 3. Slope between the two points. Frequencies: $\nu_1 = c/\lambda_1 = 5.49\times10^{14}\ \mathrm{Hz}$ , $\nu_2 = 6.88\times10^{14}\ \mathrm{Hz}$ . Then $\text{slope} = (0.705 - 0.130)\,\mathrm{V} / (1.39\times10^{14}\ \mathrm{Hz}) = 4.14\times10^{-15}\ \mathrm{V\,s}$ .

Step 4. Planck's constant: $h = e\times\text{slope} = (1.602\times10^{-19})(4.14\times10^{-15}) = \boxed{6.63\times10^{-34}\ \mathrm{J\,s}}$ — within half a percent of the CODATA value, from two voltage readings. The work function itself is not needed for $h$ : it cancels in the slope, which is why the method is robust.

Step 5. Cross-check via the threshold: the fitted line crosses zero at $\nu_0 = W/h = 5.17\times10^{14}\ \mathrm{Hz}$ , i.e. $\lambda_0 = 580\ \mathrm{nm}$ — consistent with cesium responding across the visible, which is why photomultiplier cathodes are cesium-based.

Origins · 1 minRobert Millikan and how this was discovered

Robert Millikan

1868–1953 · Chicago, then Caltech

Millikan did not believe a word of it. Einstein's light quantum struck him as "reckless" — it seemed to throw away the wave optics that interference had proven beyond doubt — so in 1905 he set out to disprove the photoelectric equation, and kept at it for ten years. He built what he called a "machine shop in vacuo": surfaces shaved clean by a magnetically driven knife inside the vacuum, because a whiff of oxide shifts the work function and ruins the intercept. His 1916 paper is a masterpiece of reluctant conclusions — the straight line was there, the slope was $h/e$ for every surface, and his measurement of $h$ landed within half a percent of Planck's blackbody value. Millikan received the 1923 Nobel Prize largely for confirming, with the world's best data, an equation he had spent a decade trying to kill; he still declined, for years afterward, to accept that photons were real.

History

1887

— Hertz notices that ultraviolet light makes his spark gaps fire more readily — the effect discovered as a side note to the experiments confirming Maxwell's waves.

1902

— Lenard shows the ejected electrons' energy depends on the light's frequency, not its intensity — inexplicable classically.

1905

— Einstein's light-quantum paper:

E = h\nu

taken as a property of light itself, with the photoelectric equation as its sharpest test.

1916

— Millikan's decade-long precision campaign confirms

V_0 = (h/e)\nu - W/e

in every detail, measuring

h

to 0.5% while doubting the photon.

1922

— Einstein receives the 1921 Nobel Prize — cited specifically for the photoelectric law, not relativity.

1923

— Millikan's Nobel Prize; the same year, Compton's X-ray recoil experiments end serious resistance to the photon.

Going further

Texts. Griffiths, Introduction to Quantum Mechanics, 3rd ed., Afterword/Ch. 1 historical introduction · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 1 (The Failure of Classical Physics) · Cohen-Tannoudji, Diu & Laloë, Vol. 1, Ch. 1 · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 1 · Einstein, Annalen der Physik 17, 132 (1905); Millikan, Phys. Rev. 7, 355 (1916).

Provenance. All relations are standard: $E=h\nu$ , $K_{\max}=h\nu-W$ , $eV_0=K_{\max}$ , $\nu_0 = W/h$ . Constants: $h=6.626\times10^{-34}\,\mathrm{J\,s}$ , $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$ , $e=1.602\times10^{-19}\,\mathrm{C}$ , $hc = 1240\ \mathrm{eV\,nm}$ (CODATA). A verifier should recompute: the sodium worked example (4.13 eV, 1.85 eV, 544 nm), the cesium/mercury-line slope reconstruction ( $V_0$ = 0.130/0.705 V, slope $4.14\times10^{-15}$ V·s, $h = 6.63\times10^{-34}$ J·s), the classical soak-time estimate (365 s), and $h/e = 4.136\times10^{-15}$ V·s. Carries material relocated from the v3-wave-particle-duality pilot spine at Step-2 (chapter ownership: classical time-lag impossibility derivation).

Prerequisites. Breakdown of classical physics · blackbody radiation and the Planck quantum. Leads to. Wave–particle duality (photon momentum, Compton recoil) · de Broglie matter waves · double-slit experiment.

If one photon is absorbed by one electron, why do most photoelectrons come out slower than K max?

Because $W$ is the minimum escape cost, paid only by the luckiest electrons — those at the top of the conduction band, ejected straight out without collisions. Electrons bound more deeply pay more than $W$ ; electrons that scatter on their way to the surface lose energy to the lattice. Both effects populate the energy range below $K_{\max}$ . That is why the photoelectric equation is stated for the maximum: it is the one energy with a clean, loss-free interpretation, and it is exactly what the stopping potential measures — the voltage at which even the luckiest electron is turned back.

Where does the work function actually come from?

It is the binding of the conduction electrons to the lattice of positive ions, measured from the highest filled level (the Fermi level) to the vacuum just outside the surface. It is a surface property, not just a bulk one: adsorbed atoms, oxide layers, even the crystal face exposed change $W$ by tenths of an eV — which is why Millikan had to shave his metal surfaces inside the vacuum, and why lab values of $W$ come with wide error bars. Cesium's loosely held outer electron gives the lowest work function of the stable metals (~2 eV), which is why photocathodes are cesium-coated.

Doesn't the existence of interference prove light is a wave — so how can it be particles?

Both experimental facts are real, and the honest resolution is the subject of the next article. Light propagates — diffracts, interferes — according to a wave description, and exchanges energy — is emitted and absorbed — in indivisible quanta. Neither classical category fits alone. The photoelectric effect does not disprove wave optics; it proves wave optics is not the whole story. The synthesis, one amplitude whose square gives detection probabilities, is wave–particle duality, and it applies to electrons just as much as to light.

Why does increasing frequency not increase the photocurrent too?

At fixed intensity (power per area), higher frequency means each photon carries more energy — so there are fewer photons per second in the beam ( $N = P/h\nu$ ). Each photon still ejects at most one electron, so the current, which counts electrons, actually drifts slightly down as frequency rises at constant power. Current tracks photon count; energy per electron tracks photon frequency. Keeping these two bookkeeping columns separate is most of the skill in photoelectric problems.

Is the stopping potential a property of the light, the metal, or both?

Both, in a perfectly separable way — that separability is equation (4). Frequency (light) sets the position along the universal line of slope $h/e$ ; the work function (metal) slides the whole line horizontally. Change the lamp and you move along the line; change the cathode and you change which line you are on — but never its slope. Intensity, notably, appears nowhere: it is the one knob that does nothing to $V_0$ .

What happens to the photon's momentum in photoemission?

Excellent instinct — a photon carries momentum $p = h/\lambda$ as well as energy, and total momentum is conserved. In photoemission the crystal lattice absorbs the recoil, exactly as a wall absorbs the recoil of a ball bouncing off it, so the energy budget (2) is unaffected: the lattice is so massive that its recoil kinetic energy is negligible. The momentum ledger becomes interesting when the target is a free electron — then both energy and momentum conservation bind the collision, and you get Compton scattering, derived in the Wave–Particle Duality article.

Can two photons ever team up to eject one electron?

Yes — but only at intensities early experimenters could not dream of. With modern lasers the photon flux is high enough that an electron can absorb $n$ photons within its coherence time, giving $K_{\max} = nh\nu - W$ and emission below the single-photon threshold. At ordinary intensities the probability is fantastically small, which is why "below threshold means dark" was an exact law for a century. The multiphoton regime does not rescue classical physics: the emission still comes in electron-sized events whose energies sit on a ladder of photon numbers — quantization confirmed twice over.