Waves explain the light you see.
What about the moment it is made?

The 1904 shelf

Everything this route uses was available to a careful reader at the end of 1904. There is no imported later result: unlike the mass-energy route, which has to borrow from the June relativity paper, this argument stands on its own shelf.

Two results the argument is often told with are not here: Jeans’s constant for Rayleigh’s law is from July 1905, after the paper was received, and the name “ultraviolet catastrophe” is Ehrenfest’s, from 1911.

The nagging fact

Turn up the lamp and the electrons come out in greater numbers, but not with greater energy. A stronger wave should shake them harder.

The first honest question

Is there any regime of light where you can do honest thermodynamics without knowing what light is made of?

01 / Start with what works

The wave theory is not the problem

Send light through two slits and the bright and dark bands land where the wave account says they will. Diffraction, interference, polarisation, the whole of optics: the wave description predicts them quantitatively and measurement agrees. Nothing in this route takes any of that away, and any replacement that cannot reproduce it is not a candidate.

The ending is often told as though waves were overthrown. They were not. What follows narrows to a region the optical evidence never covered.

02 / Find the edge of the evidence

Optics measures averages

The gap is easy to miss, because it is a statement about instruments rather than about light. Every optical measurement confirming the wave theory is an average over an enormous number of cycles. A fringe pattern is a time-averaged intensity. Nothing in that evidence says what happens in a single act of emission or absorption, because no optical measurement of the period resolved one.

So there is a region where the wave theory is untested rather than confirmed: the moment light is produced, and the moment a body takes it up. And there is already an observation sitting in that region that does not sit comfortably. Make ultraviolet light brighter and more electrons come out of a metal, but they do not come out faster. Brightness buys you more of them, not more energetic ones.

03 / Make a prediction

Give every vibration its fair share. What is the total?

Before going further, commit to an answer. Thermal equilibrium hands every vibrating degree of freedom the same average energy, fixed by the temperature. A box of radiation has vibrations at every frequency. Add up the energy that rule assigns across all of them. Is the total a modest amount, an enormous but definite amount, or no definite amount at all?

04 / Work where a law is solid

Pick the regime you can trust, then take its entropy

Rather than argue about the whole spectrum, retreat to ground that holds. Wien’s law describes the measured distribution well when the frequency is high and the temperature low, and in 1904 that is not in dispute. Work only there.

In that regime you can write down the entropy of a quantity of monochromatic radiation, and then ask a single question about it: if you keep the energy and the frequency band fixed and let the radiation occupy a different volume, how does the entropy change? That is a question about counting, not about mechanism, and it can be answered.

05 / Do the same sum for something you understand

How a gas of independent things behaves

Now put the radiation aside and take a container of gas with a definite number of particles that do not interact. Ask exactly the same question: keep everything else fixed, let it occupy a different volume, and see how the entropy changes. The answer comes from counting how many ways the particles can be arranged, and it depends on the volume logarithmically, with the number of particles as a multiplier out front.

You now have two answers to one question, derived independently: one for radiation in a regime where a measured law holds, one for a gas of a known number of independent things.

06 / Compare the two, and read the exponent

What plays the part of the number of molecules?

The two expressions have the same shape. Set them beside each other and the multiplier that counts particles in the gas is matched, in the radiation case, by the total energy divided by a quantity built from the frequency. The paper states the conclusion in the careful form its title promises: radiation of low density behaves, as far as this thermodynamic question goes, as if it consisted of mutually independent energy quanta each of size

R is the gas constant and N the number of molecules in a gram-molecule; β here is Wien’s constant. The relativity paper uses the same letter for 1/√(1 − v²/V²), a different quantity.

07 / Demand consequences

Three predictions it did not have to get right

A resemblance found in one thermodynamic calculation earns nothing until it says something about a different experiment. Take the picture seriously for a moment and three consequences follow, each testable against work already done by 1904.

Light re-emitted by a fluorescing substance should not have a higher frequency than the light that excited it, because one incoming piece cannot pay for a larger outgoing one. Stokes had recorded exactly that in 1852. The energy of the electrons driven out of a metal should not depend on the brightness, which is what Lenard measured in 1902 and what the wave account could not place; that it should rise in a straight line with the frequency was not tested until Millikan’s measurements of 1916. And ultraviolet light should only ionise a gas when its frequency is high enough, whatever its intensity.

First write the rule about the electrons as a formula.

Then try it on numbers. It needs one constant, and here it takes today’s value.

08 / Check it against the world

A number the paper set down in advance

Section 8 does not stop at a rule. It gives a figure for light of one frequency, worked from the paper’s own constants, and sets it beside Lenard’s results. The laboratory below computes the same quantity from the same rule; change the frequency or the exit cost and read what the rule gives.

/**
 * Stopping potential magnitude: V_s = K_max / e = (h * nu - Phi) / e (in Volts).
 *
 * Epistemic rule:
 * When nu < nu_0, stopping potential is `not-applicable` with reason
 * "no emitted electron in this model".
 * Under partial transfer, returns `underdetermined` with upperBound = (h * nu - Phi) / e.
 */
export function stoppingPotentialMagnitude(
  nu: number,
  workFunctionJoules: number,
  set?: ConstantSet,
  transferModelOrOptions:
    | EnergyTransferMode
    | Readonly<{ transferModel?: EnergyTransferMode }> = "complete",
): PhotoelectricResult<number> {
  const transferModel: EnergyTransferMode =
    typeof transferModelOrOptions === "object" && transferModelOrOptions !== null
      ? (transferModelOrOptions.transferModel ?? "complete")
      : transferModelOrOptions;
  const kResult = kMax(nu, workFunctionJoules, set, transferModel);
  if (kResult.status === "underdetermined") {
    const e = getElementaryCharge(set);
    const extra: {
      upperBound?: number;
      unit?: string;
      citation?: string;
      reason?: string;
      bounds?: readonly [number, number];
      quantityId?: string;
    } = {
      unit: "V",
      quantityId: "stoppingPotentialMagnitude",
    };
    if (kResult.upperBound !== undefined) extra.upperBound = kResult.upperBound / e;
    if (kResult.citation !== undefined) extra.citation = kResult.citation;
    if (kResult.reason !== undefined) extra.reason = kResult.reason;
    if (kResult.bounds !== undefined) extra.bounds = [kResult.bounds[0] / e, kResult.bounds[1] / e];
    return underdetermined(kResult.compatibleFamily, kResult.neededInformation, extra);
  }
  if (kResult.status !== "value") return kResult;
  const e = getElementaryCharge(set);
  return ok(kResult.value / e, "V", {
    ...(kResult.note ? { note: kResult.note } : {}),
    quantityId: "stoppingPotentialMagnitude",
  });
}

The later card tested the straight line itself, in 1916. It is evidence the paper did not have, and it is cited here rather than plotted: this edition’s table of Millikan’s points is withdrawn.

09 / Try it yourself

Five pieces of the argument to work by hand

The first three put the rule to numbers under a modern lens, with today’s value of Planck’s constant h, which the paper never writes: it has R, β and N, and h is their later shorthand, Rβ/N. The checker converts your unit and compares numbers, not text.

First, read the constant off a line, as the photoelectric effect lets you.

Next, the lowest frequency that frees anything from a made-up surface.

Then how many quanta an ordinary green source gives off.

Two to put in your own words: one from step 07, and one you can try first in the counting laboratory’s locked counterexample.

Last, a prediction to make before you change anything, in the laboratory at step 08.

Where this enters the paper

The paper runs to seventeen pages and nine numbered sections. The divergence is section 1, the entropy of radiation is sections 3 and 4, the gas comparison is section 5, the conclusion this route builds towards is section 6, and the three checks are sections 7 to 9. Its German text and an English translation are on this site.

Two doors lead to the same place, the effective count at the heart of section 6: the paper’s own argument, and a counting loop a programmer can write.

The chain can also be rebuilt from its claims rather than followed through them. The capstone holds the regime and the independence assumption apart from the conclusion they license, so the heuristic stays a heuristic while you put it in order: Rebuild the heuristic viewpoint. It is optional, and nothing above depends on it.