Light quanta · Capstone
Rebuild the heuristic viewpoint
What exactly does the entropy argument license, in which regime, and which step is the guess the paper's own title calls heuristic?
What to do with this page
Explain to a friend why an entropy formula made Einstein suspect that light comes in energy quanta, which step was a guess, and why the wave theory's successes remained.
Each claim below links to the passage it is read from. Follow the links and the argument is the paper's; read only this page and it is a summary of the paper, which is a different thing and says so.
The seven claims, in the order the paper makes them
The chain below fixes what must come before what, and 63 arrangements satisfy it. The paper prints one of them; the others are not mistakes.
A derivationNeeds nothing before it
The wave description of light succeeds, and it is kept. But sharing energy classically among the modes of radiation in equilibrium gives a density that rises with every higher frequency, so the total over all frequencies is not finite. The classical route does not reach a spectrum.
The wave description is not set aside here and is not set aside later. What fails is a particular way of sharing energy among the modes, and the paper says the optical successes stand because optical measurements are averages over time.
A derivationUses claim 1
Comparing Planck's formula where the wavelength is long and the density high with the classical result lets the constants be read off, and gives an estimate of the number of molecules in a gram-molecule from the printed constants. The formula is Planck's; the energy elements in his derivation belong to his oscillators, and this comparison does not adopt them as quanta of light.
Planck's formula is used and Planck's energy elements are not. In his derivation the elements are a property of the oscillators; treating them as quanta of light is the move this paper makes later and by a different route, and a reader who runs the two together has lost the paper's argument.
A derivationNeeds nothing before it
A measured spectrum determines the entropy of the radiation, because the rate at which the entropy density grows with the energy density at one frequency is the reciprocal of the temperature.
A derivationUses claim 3
In the regime where Wien's law holds, at fixed energy in a fixed narrow band, the entropy of the radiation exceeds its value at a reference volume by the energy divided by the product of Wien's constant and the frequency, times the logarithm of the volume ratio.
A derivationNeeds nothing before it
For a number of points moving independently in a volume, the chance that all are found in a sub-volume at one moment is the volume ratio raised to that number, so by Boltzmann's principle the entropy exceeds its reference value by the gas constant over the molecular number, times that number of points, times the same logarithm.
A heuristic stepUses claim 4 and claim 5
The two expressions have the same form, so dilute monochromatic radiation behaves thermodynamically as if it were made of independent energy quanta, each of magnitude the gas constant over the molecular number times Wien's constant times the frequency. The effective count that makes the two agree is a coefficient in an entropy law and is never rounded to a whole number of quanta. This is the step the paper's title calls heuristic.
This is the heuristic step, and it leans on independence and on the Wien regime. It concludes that dilute monochromatic radiation behaves thermodynamically as if made of independent quanta. It does not conclude that light is made of particles, and the count it produces is a coefficient rather than a tally.
A derivationUses claim 6
Under the further hypothesis that light is also emitted and absorbed in such quanta, three consequences follow and are offered for test: that fluorescent light cannot exceed the exciting light in frequency, that the greatest energy an escaping electron keeps rises with frequency and not with brightness, and that the work of ionizing a molecule cannot exceed one quantum of the absorbed light. The hypothesis is an extension of the thermodynamic result, not a consequence of it.
Everything here rests on the further hypothesis as well as on the step above it. Planck's position at the fork, that the agreement of the two entropy laws is a coincidence of mathematics and the elements belong to the oscillators, is coherent and this record does not mock it.
What the argument is granted
Every claim above names the assumptions it uses. These are the things the paper is given or asserts rather than establishes. Two of them are worth finding before the rest: the regime is a boundary the argument stays inside rather than a premise it leans on, and the further hypothesis that light is emitted and absorbed in quanta is an extension of the result above it rather than a consequence of it.
Approximation
The Wien regime: radiation dilute enough, and of high enough frequency against the temperature, for Wien's law to hold. Experiment confirmed it there and it is known not to hold generally.
Setup
A fixed narrow band of frequencies and a fixed energy within it, held the same on both sides of the comparison.
Premise
The counted elements are independent of one another, so the chance that all are in the sub-volume is the product of the separate chances.
Premise
Single-quantum absorption: one quantum gives the whole of its energy to one electron. The paper states this and says what becomes of the relation without it.
Idealization
A declared work of escape, the energy an electron spends leaving the surface, which the paper writes as a single quantity and does not derive.
Premise
The further hypothesis: that light is emitted and absorbed in such quanta, extending a result about the thermal behaviour of radiation to the emission and transformation of light. The paper introduces it as a question worth asking, not as something the entropy argument has established.
Premise
Classical equipartition applied to the modes of the radiation field, giving each the mean energy that collisions with a gas at the same temperature would give it.
Premise
Boltzmann's principle, connecting the entropy of a state to the probability of that state through the gas constant over the molecular number.
Premise
Radiation in equilibrium has a well-defined entropy per interval of frequency, and radiation of different frequencies is taken to be separable without doing work.
The displays this argument turns on
The entropy depends on the volume through a logarithm
How the entropy of radiation in a narrow band depends on the volume, inside the regime where Wien's law holds. The paper prints it with Einstein's own volume symbol and his logarithm.
The entropy difference is E over B frequency times the natural logarithm of the volume ratio.
Multiply one chance per independent point
The chance that independent points are all found in a sub-volume, which is the counting law the radiation result is compared with.
The probability W is f to the power n.
What the stopping potential measures
The photoelectric relation as the paper prints it, in the gas constant, the molecular number and Wien's constant. Einstein never writes Planck's constant, so the modern form with it is a concordance reading rather than the paper's own.
The electron's charge times the stopping potential equals the largest kinetic energy, which equals h times the frequency minus the escape work.
Where to watch the quantities move
Halve the volume at fixed energy in a fixed band and watch the entropy fall by the logarithm of the ratio, while the Wien-regime measure stays inside the regime.
Count the arrangements of independent points in half the volume, then lock them together and count again, and see what independence was doing.
Put the two volume laws side by side and read the effective count that makes them agree, which is a coefficient in an entropy law and not a number of things.
Follow the paper's own volt check through its printed representation, and see the greatest energy an escaping electron keeps move with frequency while the brightness moves only the rate.
What this does not claim
The paper calls its own standpoint heuristic, in its title, and this record keeps it that way. The inference is carried out where Wien's law holds and concludes that radiation there behaves thermodynamically as if it consisted of independent energy quanta. It does not say that light is not a wave, and it does not refute the wave description, whose interference successes stand. The photoelectric relation alone is not proof of quanta: it is a prediction drawn from the hypothesis and offered for test, and an instrument given the hypothesis returns it rather than confirming it. The effective count is a coefficient in an entropy law and is never a number of things counted.