Read · Light quanta: from entropy to an energy scale

§6 · the heuristic correspondence

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A heuristic step, to an approximation

A coefficient suggests an energy element

What plays the role of the number of independent things?

The 7 printed paragraphs this passage explains

§4, paragraph 3: So the entropy of dilute monochromatic radiation depends on its volume by the same law as the entropy of an ideal gas or a dilute solution. Einstein will interpret this with Boltzmann's principle, that the entropy of a system is a function of the probability of its state.
§6, paragraph 1: Section 4 found how the entropy of monochromatic radiation depends on its volume. Rewritten as R/N times the logarithm of a power of the volume ratio, and compared with Boltzmann's principle, it leads to a conclusion.
§6, paragraph 2: For monochromatic radiation of energy E shut into a volume v₀ by mirrors, the probability that at a moment chosen at random all its energy is in the part v is the volume ratio raised to the power (N/R)(E/βν).
§6, paragraph 3: A connecting line: from that probability Einstein draws a further conclusion, stated next.
§6, paragraph 4: Monochromatic radiation of low density, where Wien's law holds, behaves in its thermal properties as if it consisted of independent energy quanta of size Rβν/N.
§6, paragraph 5: Einstein compares the mean size of these energy quanta in black radiation, which Wien's formula gives as 3(R/N)T, with the mean kinetic energy of a molecule's motion at the same temperature, (3/2)(R/N)T.
§6, paragraph 6: If radiation behaves, in how its entropy depends on volume, like a medium made of such energy quanta, it is natural to ask whether light is also produced and transformed as if it consisted of them. The rest of the paper takes up that question.

\Delta S_{\mathrm{gas}} = n\,k_B\,\ln\left(f\right)

\Delta S_{\mathrm{rad}} = \frac{E}{B\,\nu}\,\ln\left(f\right)

The gas entropy change has coefficient n times Boltzmann's constant; the radiation change has coefficient E divided by B times frequency.

If the radiation is interpreted through the independent-element analogy, matching the coefficients suggests $n_\text{eff} = E/(k_B B\nu)$ . Its corresponding energy per element is $k_B B\nu$ . The algebra identifies a scale conditional on the analogy; it does not prove the analogy.

n_{\mathrm{eff}} = \frac{E}{k_B\,B\,\nu}

\frac{E}{n_{\mathrm{eff}}} = k_B\,B\,\nu = h\,\nu

The effective coefficient is E divided by Boltzmann's constant times B times frequency. The implied energy scale is Boltzmann's constant times B times frequency, written h times frequency in modern notation.

Keep a non-integer effective coefficient as it stands. The entropy comparison concerns a macroscopic relation, not a directly observed list of individual packets. In the entropy-comparison laboratory, compare the coefficients before revealing their interpretation.

Open the foundation: Fractions and ratios

Show every step here: A coefficient suggests an energy element

Two results sit side by side. From §4: dilute radiation of energy E in a narrow band at ν changes its entropy by $\frac{E}{\beta\nu}\lg\frac{v}{v_0}$ when its volume goes from $v_0$ to v. From §5: n independent points change theirs by $\frac{R}{N}n\lg\frac{v}{v_0}$ . In this explanation f is $v/v_0$ , B is β, and $k_B$ is R/N.

\Delta S_{\mathrm{gas}} = n\,k_B\,\ln\left(f\right)

\Delta S_{\mathrm{rad}} = \frac{E}{B\,\nu}\,\ln\left(f\right)

The gas entropy change has coefficient n times Boltzmann's constant; the radiation change has coefficient E divided by B times frequency.

Both are a coefficient times ln f, the same function of the volume ratio. That shared form is what the comparison uses; letters that happen to look alike are not.
The paper rewrites the radiation result in the form of Boltzmann's principle, $S-S_0=\frac{R}{N}\lg W$ , by moving the coefficient into the exponent, since $a\ln f=\ln f^{a}$ . With $a=\frac{N}{R}\frac{E}{\beta\nu}$ this gives $S-S_0=\frac{R}{N}\lg\left[\left(\frac{v}{v_0}\right)^{\frac{N}{R}\frac{E}{\beta\nu}}\right]$ .
Read the bracket as a probability, as Boltzmann's principle invites: the probability that all the radiation energy is found, at a moment chosen at random, in the part v of the volume $v_0$ is $W=(v/v_0)^{\frac{N}{R}\frac{E}{\beta\nu}}$ .
Compare it with §5's $W=(v/v_0)^n$ . The exponent $\frac{N}{R}\frac{E}{\beta\nu}$ stands where the number of independent points stood.
Set the coefficients equal, only as the proposed correspondence: $n_{\mathrm{eff}}k_B=E/(B\nu)$ . Divide both sides by $k_B$ .

n_{\mathrm{eff}} = \frac{E}{k_B\,B\,\nu}

\frac{E}{n_{\mathrm{eff}}} = k_B\,B\,\nu = h\,\nu

Divide E by that number to find the energy of one independent element: $k_BB\nu$ , which the paper writes Rβν/N. The letter h for $k_BB$ , in the formula's last step, is a later name and is not the paper's.
Worked size, from the paper's inputs: $R\beta/N$ is $8.31\times10^{7}\times4.866\times10^{-11}/(6.17\times10^{23})=6.55\times10^{-27}$ erg-seconds, with R the standard value of 1905. At a frequency of $10^{15}$ per second, in the ultraviolet, one element would carry $6.55\times10^{-12}$ erg.
A second comparison in §6: using Wien's formula, the paper finds the mean size of an energy element in black radiation to be $3\frac{R}{N}T$ , twice the mean kinetic energy $\frac{3}{2}\frac{R}{N}T$ of a molecule's motion at the same temperature.
Keep $n_{\mathrm{eff}}$ as it comes out. It is a pure number that need not be whole, and nothing here rounds it.
Mark the status. The paper concludes that monochromatic radiation of low density, within the range of Wien's formula, behaves in thermodynamic respects as if it consisted of independent energy quanta of size Rβν/N. 'As if' is the claim: a heuristic reading of a matching form, not a proof of what light is.
What follows in the paper is a further question: whether the laws by which light is produced and transformed are also such as if light consisted of these quanta. §§7 to 9 test it on fluorescence, on electrons released by light, and on the ionization of gases.

A tool for this step

Fractions and ratios

What does dividing one quantity by another tell you?

The fraction 6/2 is 3, because 2 fits into 6 three times; it also says that 6 is three times 2. Every fraction a/b works the same way. The bottom number can never be zero, since no count of zeros adds up to 6.

When the two quantities have different units, their ratio is a new quantity with a unit of its own: 12 micrometres travelled in 4 seconds is 3 micrometres per second. When they have the same unit, the units cancel and the ratio is a plain number: 2 particles out of 8 is 0.25 of the group.

Worked example: Fourteen metres in ten seconds

A steady walk covers 14 metres in 10 seconds.
Divide the distance by the time: 14 divided by 10 is 1.4, and the unit divides with it, so the rate is 1.4 metres per second.
After 60 seconds the same walk has covered 84 metres, and 84 divided by 60 is 1.4 again. The walk is steady, so the ratio does not depend on how long it is watched.
Metres and seconds are different units, so neither cancels and what comes out is a new quantity that keeps a unit of its own.
The drawing below is the other case: 2 particles out of 8 has the same unit above and below, they cancel, and 0.25 is left as a plain number.

One in four and two in eight, drawn side by side

The filled dots are the ones that moved. Under each row, a bar with that share of its length shaded. The two groups have different sizes and the two shaded stretches are the same length.

The shaded stretch is the same length in both bars. A label that long does not fit inside a drawing, and a drawing's text does not reflow on a narrow screen, so it is written here.

What it shows, in words

The first row is a group of 4 with 1 filled, and the bar under it is shaded for a quarter of its length. The second row is a group of 8 with 2 filled, and the bar under it is shaded for the same length as the first. Doubling how many moved and doubling the size of the group together leave the share where it was, at 0.25.

The dots are counted and the bars are measured, so the two rows agree by two different routes. A fraction records how much of the group moved. The size of the group is a separate fact, and the fraction has already divided it out.

Where this lesson stops

A ratio of two quantities in the same unit is a plain number. A ratio of different units, such as micrometres per second, is a new quantity and keeps its unit.

Explore the equations in this step

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How many independent points the radiation behaves like

n_{\mathrm{eff}} = \frac{E}{k_B\,B\,\nu}

is divided by , and .

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The effective count equals the energy divided by k B times B times the frequency.

Matching the radiation's entropy change to a gas of independent points gives the number of points the gas would need. It is read off a comparison, not counted.

Model assumptions and every term’s meaning

Dilute radiation: the density is low enough, relative to temperature, that Wien's law describes the spectrum.
These are modern teaching equations in SI notation, with A and B for the paper's alpha and beta; they are not a transcription of the printed paper.
The gas comparison assumes n independent points whose chance of all lying in a fraction f of the volume is f to the n.

Effective count: The number of independent points a gas would need to show the same logarithmic entropy change. Nothing here observes points. Read the prerequisite
Energy in the band: The radiation's energy within the narrow band, held fixed while the volume changes. Read the prerequisite
Boltzmann's constant: The gas side of the comparison: n points change entropy by n times Boltzmann's constant times ln f. Read the prerequisite
Spectral constant B: From the Wien spectrum; B times the frequency divided by the temperature is a pure number. Read the prerequisite
Frequency of the band: The band's frequency. The count falls as the frequency rises for the same energy. Read the prerequisite
Two coefficients set equal: Both entropy changes are a coefficient times ln f. Setting the coefficients equal is the comparison; it is a correspondence, not a derivation that light is made of points. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

The energy each point would carry

\frac{E}{n_{\mathrm{eff}}} = k_B\,B\,\nu = h\,\nu

over is times times , which modern notation writes times the frequency.

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

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Each term and operation, in words

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The energy divided by the effective count equals k B times B times the frequency, which equals h times the frequency.

Dividing the band's energy among the effective points gives the same amount per point at every volume: Boltzmann's constant times B times ν, which modern notation writes hν.

Model assumptions and every term’s meaning

Dilute radiation: the density is low enough, relative to temperature, that Wien's law describes the spectrum.
These are modern teaching equations in SI notation, with A and B for the paper's alpha and beta; they are not a transcription of the printed paper.

Energy in the band: The radiation's energy in the band, the same energy the count was read from. Read the prerequisite
Effective count: The effective number of independent points from the entropy comparison. Read the prerequisite
Boltzmann's constant: R over N in the paper's notation. Read the prerequisite
Spectral constant B: Wien's β. With Boltzmann's constant it makes an energy per unit frequency. Read the prerequisite
Frequency of the band: The energy per point is proportional to the frequency. Read the prerequisite
Planck's constant: The later name for Boltzmann's constant times B, which the paper writes Rβ/N. Read the prerequisite
Frequency of the band: The same frequency, in the modern form of the product. Read the prerequisite
Energy per point: The step Einstein calls heuristic: radiation of low density behaves, for its entropy, as if it consisted of independent quanta of this size. Read the prerequisite
The same product in modern notation: Writing Boltzmann's constant times B as h renames the constant; it does not add a premise. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

The gas side of the comparison

\Delta S_{\mathrm{gas}} = n\,k_B\,\ln\left(f\right)

is times times the logarithm of .

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Each term and operation, in words

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The gas entropy change equals n times k B times the natural logarithm of f.

For n independent points, the chance of finding all of them in a fraction f of the volume is f to the n, so the entropy changes by n times Boltzmann's constant times ln f.

Model assumptions and every term’s meaning

The points move independently of one another.
These are modern teaching equations in SI notation, not a transcription of the printed paper.

Entropy change of the gas: A difference of entropies between the two volumes. Read the prerequisite
Number of independent points: The coefficient's count: each point contributes Boltzmann's constant times ln f. Read the prerequisite
Boltzmann's constant: R over N in the paper's notation. Read the prerequisite
Volume ratio: The factor by which the volume changes. Read the prerequisite
Independence gives the logarithm: The probability f to the n becomes n ln f once the logarithm is taken. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

The radiation side of the comparison

\Delta S_{\mathrm{rad}} = \frac{E}{B\,\nu}\,\ln\left(f\right)

is over times , times the logarithm of .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

The radiation entropy change equals E over B times the frequency, times the natural logarithm of f.

Section 4's result for dilute radiation of fixed energy and band: the same logarithm of the volume ratio, with the coefficient E over B nu.

Model assumptions and every term’s meaning

Dilute radiation of fixed energy in a fixed narrow band, described by Wien's law.
These are modern teaching equations in SI notation, not a transcription of the printed paper.

Entropy change of the radiation: From Wien's law, integrated at fixed energy and band. Read the prerequisite
Energy in the band: Held fixed while the volume changes. Read the prerequisite
Spectral constant B: Wien's beta. Read the prerequisite
Frequency of the band: The coefficient falls as the frequency rises. Read the prerequisite
Volume ratio: The same factor as on the gas side. Read the prerequisite
The same form as the gas: A coefficient times ln f, like the gas. That shared form is what the comparison rests on. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

Einstein calls his proposal a heuristic point of view, and the paper’s title says so. A matching mathematical form can suggest a model without forcing it: Planck’s energy elements of 1900 belong to the oscillators in the walls, Einstein’s quanta to the radiation itself, and the two agree on some numbers while saying different things about light.

Assumptions and limits: A coefficient suggests an energy element

Assumed here

Radiation in a narrow band obeys the entropy law derived for dilute (Wien) radiation.
Independent points obey the entropy law for n points, with k_B identified as R/N.
A matching dependence on volume is read as suggesting independent energy elements.

What this does not establish

E/(hν) is a pure number that need not be a whole number, and it is never rounded to one.
This does not derive Planck’s law or establish a general theory outside the Wien regime.
Extending the picture to individual emission and absorption processes is a further hypothesis.

Earlier step: Compare two states, not a compression movie Independence supplies the exponent

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

R. A. Millikan, A Direct Photoelectric Determination of Planck's "h". Physical Review (2), 7 (3), 355–388 (1916).

26 foundation readings sit behind this argument.

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§6 · the heuristic correspondence

Original journal scan

A coefficient suggests an energy element

How many independent points the radiation behaves like

The energy each point would carry

The gas side of the comparison

The radiation side of the comparison

Modern qualifications

What this relies on

Assumed here

What this does not establish

References

Reading a graph

Fractions and ratios

A sign records direction

A probability, a frequency and a single trial

Powers of ten and physical units

Squares and square roots

Adding and averaging

Rates of change and derivatives

Density is not probability

Charges, fields and light

Entropy and the number of ways

Entropy and temperature

Exponential change

Fields and waves

Functions and graphs

Adding continuously

Logarithms: turning products into sums

The push of light

Orders of magnitude

Partial derivatives and held-fixed quantities

Probability and independence

Quantities and units

Approximating a curve near a point

Temperature and thermal energy

The units of 1905

Energy of motion and inertia

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