Light quanta: from entropy to an energy scale

Wave optics describes propagation and interference successfully. Einstein asks whether energy exchange with matter needs an additional, explicitly heuristic description.

A wave can spread through space and interfere with another wave. Those successes are not withdrawn by the light-quantum proposal. The question changes: what happens when radiation is generated, absorbed, or converted into another form?

The introduction contrasts a continuously distributed field with matter described through a finite number of moving constituents. Einstein proposes considering independently moving energy elements whose energy remains localized during propagation and is exchanged as a whole. In this paper that is a heuristic viewpoint: something to investigate through consequences, not a conclusion supplied by an animation.

In the wave-optics laboratory, double the field amplitude while holding the wave shape and medium fixed. The model intensity becomes four times as large. That establishes a consequence of the wave model; it does not settle how an individual energy-transfer event occurs.

Open the foundation: Fields and waves

At a positive temperature, classical allocation assigns more energy as the frequency cutoff rises. Its all-frequency total has no finite value; the plotting cutoff is not a physical explanation.

Counting cavity modes and giving each mode the classical mean energy $k_B T$ produces a spectral energy density $\rho_\nu$. A density must be multiplied by a frequency width to give energy per unit volume.

At fixed positive temperature, doubling the cutoff multiplies this finite-range total by eight. Extending the model to arbitrarily high frequencies produces an unbounded total. The resonator laboratory keeps that failure distinct from the finite portion displayed on screen.

Open the foundation: Adding continuously

The result of §1 depends on two assumptions spelled out in its footnotes: electrons and gas molecules have the same average energy of motion, and the radiation is completely disordered, with nothing linking the strengths and timings of its waves.

§1 puts gas molecules, free electrons and bound electrons in a box with mirror walls, and lets the free electrons collide like the molecules of a gas. A footnote says what that amounts to: at equilibrium, gas molecules and electrons have equal mean kinetic energies. Drude, Einstein notes, used the same assumption to derive the ratio of heat conduction to electrical conduction in metals.

That equality is what fixes the mean energy of each straight-line vibration of a resonator. A resonator whose average energy were higher or lower would, through the collisions, keep giving energy to the gas or taking it, so equilibrium needs:

The second assumption belongs to Planck's relation between a resonator and the radiation around it, which §1 uses. Planck derived it for radiation that is 'as disordered as possible', and a footnote defines the phrase. At one point of the space, expand one component Z of the electric field over a long time T in a Fourier series of sine waves with amplitudes A and phases α. Repeat the expansion from starting moments chosen at random: the amplitudes and phases come out differently each time, with a statistical probability for each combination.

The radiation is as disordered as possible when that probability splits into separate factors, one for each amplitude and one for each phase:

Then the value of any one amplitude or phase tells nothing about the others. Einstein adds that the radiation in his box comes closer to this the more each pair of an amplitude and its phase depends on emission and absorption by its own particular group of resonators. Neither assumption is derived in §1. The footnotes state them, and the equilibrium, with the unbounded total energy it leads to at the end of §1, rests on both.

Open the foundation: Probability and independence

Matching a spectral coefficient to classical thermal allocation constrains R/N. Quantized resonator energies and independently propagating light quanta are different claims.

Write the historical spectral constants as $A$ and $B$ in this explanatory notation. At low $\nu/T$, expanding the exponential denominator gives a term proportional to $\nu^2 T$. Its coefficient can be compared to the classical allocation from §1.

This comparison is about coefficients and their experimental provenance. In the modern SI, $R$ equals $N_A k_B$. Inserting that identity and the modern spectral constants recovers the defined $N_A$; it is not independent evidence for a molecular count.

Planck’s resonators belong to the matter that emits or absorbs radiation. Einstein’s proposed energy elements concern the radiation itself. Moving between those statements requires an argument.

Open the foundation: Fractions and ratios

At fixed volume and frequency, the entropy increase per added energy is 1/T. Inverting an admitted spectrum therefore determines an entropy derivative, but not yet its integration constant.

Let $\rho_\nu$ be energy per volume per frequency interval and $s_\nu$ the corresponding entropy density. At fixed volume, equilibrium maximizes entropy while keeping total energy fixed. Moving a small amount of energy from one spectral interval to another must give no first-order entropy gain at equilibrium.

Consequently the derivatives of entropy with respect to spectral energy density have the same value across the equilibrium spectrum. The thermodynamic identity dS/dE = 1/T, with other constraints fixed, identifies that common value.

Einstein attributes this thermodynamic reasoning to Wien. The next step is to express 1/T as a function of density by using Wien’s spectral law, not by assuming particles of light in advance.

Open the foundation: Entropy and temperature

Inverting Wien’s law and integrating gives the entropy density plus a frequency-dependent constant. Requiring zero entropy density at zero radiation fixes that constant; it does not cancel automatically.

Divide by $A\nu^3$ and take the natural logarithm. Its argument is dimensionless. Solving for inverse temperature turns the entropy derivative from §3 into an explicit function of density.

As positive density approaches zero, $\rho\ln\rho$ tends to zero. The remaining limit is $C(\nu)$. The boundary condition $s_\nu \to 0$ therefore sets $C(\nu) = 0$. Without this condition, the volume comparison in the next step has an extra term.

Open the foundation: Logarithms: turning products into sums

For the same energy and narrow frequency band, Wien radiation gains entropy logarithmically with accessible volume. Both endpoint states must remain in the admitted regime.

Within a narrow band, write $E = V\,\Delta\nu\,\rho_\nu$ and $S = V\,\Delta\nu\,s_\nu$. Substitute $\rho_\nu = E/(V\,\Delta\nu)$ into the integrated entropy density. Comparing $V$ with $V_0$ at the same $E$ and band removes terms independent of volume.

$E/(B\nu)$ has entropy units. Only after identifying $B = h/k_B$ can it be written $k_B$ times the dimensionless coefficient $E/(h\nu)$. This distinction matters when interpreting a displayed coefficient as a count.

Doubling the accessible volume adds $(E/(B\nu))\ln 2$ under the stated constraints. Doubling it again adds the same amount. The radiation-entropy laboratory compares those constrained states and exposes the otherwise hidden constant.

Open the foundation: Partial derivatives and held-fixed quantities

Einstein says the word probability is often used loosely when entropy is calculated from molecules. He means one thing by it: how likely it is to find the system in a given state at a moment chosen at random, and he holds that this is enough for Boltzmann's principle.

§5 opens with a warning about a word. When entropy is calculated from molecular theory, Einstein writes, 'probability' is often used in a sense that does not match its definition in the theory of probability. In particular, which cases count as equally probable is often fixed by hypothesis, even where the theoretical picture is definite enough to deduce them instead.

He announces that he will show in a separate paper that the statistical probability is all that thermal questions need, and hopes this removes a logical difficulty that still stands in the way of applying Boltzmann's principle. Here he states the principle only in general and applies it to special cases. The first special case shows what statistical probability means: the probability that, at a moment chosen at random, the system is found in the state considered.

Open the foundation: A probability, a frequency and a single trial

Here W is the probability of the state with entropy S relative to a starting state with entropy $S_0$, R is the gas constant, N the number of real molecules in a gram-equivalent, and lg is the natural logarithm as the paper writes it. Read with a statistical W, the formula needs no list of equally likely cases: a state the system is found in at 1 moment in 16, compared with a state it is always in, has W = 1/16, and its entropy is lower by R/N times ln 16.

The reading matters for everything after it. The probability that n independent points are all in part of a volume, and in §6 the probability that all the energy of monochromatic radiation is in part of its volume, are statistical probabilities in exactly this sense. The step from entropy to energy quanta is only as good as that reading of W.

For n independent uniformly distributed points, the chance that all lie in a fraction f of a volume is fⁿ. Perfectly locked positions instead give f; the exponent is a statement about independence.

Each independent point has probability f of being inside the chosen fraction. For all n to be inside, multiply n independent probabilities. Taking the logarithm makes the exponent n a coefficient.

Open the foundation: Entropy and the number of ways

With four independent points and half the volume, the probability is 1/16. If all four positions are exactly the same uniformly distributed position, the probability is 1/2 instead. Having four labels is not enough to justify four independent factors.

The concentration from V₀ to V smaller than V₀ has a negative entropy difference. Reversing the constrained comparison reverses the sign. The independent-configurations laboratory makes the probability and entropy readouts separate.

Open the foundation: Probability and independence

In the Wien regime, the radiation entropy coefficient has the form associated with independent elements. Reading an energy scale from that match is a heuristic inference, not a proof of every property of light.

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.

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

With one absorbed quantum, no extra energy, and a single emitted quantum, the outgoing energy cannot exceed the incoming energy. Extra thermal energy or multiple-quantum input changes those premises.

If an event receives only $h\nu_\text{in}$, emits $h\nu_\text{out}$, and retains a nonnegative remainder, energy conservation gives a bound on the outgoing frequency. It is a consequence of the assumed event budget, not evidence that all fluorescent events have that budget.

Supply extra thermal energy or allow more than one absorbed quantum and the bound changes. The fluorescence laboratory's controls expose those extra-energy and channel assumptions rather than quietly creating energy.

Proportional fluorescence intensity requires further assumptions about absorption and conversion yield. Holding those assumptions fixed can make a weak-illumination rate proportional to the incident rate; the energy bound alone does not determine the yield.

Open the foundation: Energy of motion and inertia

In the one-quantum model, frequency sets the maximum electron energy after paying the escape cost. Increasing power at fixed frequency can increase the number emitted without increasing that maximum.

The most energetic electron receives a complete quantum and escapes with the least modeled energy loss. Subtracting the escape work $\Phi$ gives the maximum kinetic energy, provided emission is energetically allowed.

For a hypothetical work function of 2 electron volts and a frequency of 600 terahertz, the maximum is about 0.4814 electron volts using modern SI constants. Doubling optical power while retaining that frequency and escape work leaves the maximum unchanged.

Power is energy arriving per second. At fixed frequency, dividing it by $h\nu$ gives the incoming quantum rate. Under a fixed yield, twice the power can therefore produce twice the electron count per second. Below threshold, negative $h\nu - \Phi$ is an energy deficit, not a negative kinetic energy of an emitted electron.

Open the foundation: Functions and graphs

Einstein adds a caution to his photoelectric laws: they probably have limits, of the same kind as the exceptions he allowed to Stokes's rule for fluorescence.

After working out how the escaping electrons should behave, Einstein adds one sentence of caution: about the presumable limits of validity of these regularities, remarks would have to be made like those about the presumable deviations from Stokes's rule. His word, 'mutmaßlich', means presumable: he expects limits, and he does not locate them.

The regularities are those of the paragraph before. If each energy quantum gives its energy to electrons independently of all the others, the speeds of the escaping electrons do not depend on the intensity of the exciting light, while their number is proportional to it. The energy relation behind them assumes one quantum for one electron:

The remarks on Stokes's rule, in §7, named two conditions. First, so many quanta are being transformed at once in each unit of volume that one produced quantum can take its energy from several exciting quanta. Second, the light is not of the kind black radiation has within the range of Wien's law, for instance light from a body so hot that Wien's law no longer holds at the wavelength concerned.

Carried over, as the sentence invites: in very intense light an electron might gather energy from more than one quantum, and then its energy is no longer bounded by one quantum's (R/N)βν less the work P it spends leaving. For light outside Wien's range, the entropy argument of §6 that gave quanta of size Rβν/N does not apply, so it does not secure the size of the energy portions. Einstein spells out neither case for the photoelectric effect; the caution stays a caution.

Open the foundation: Energy of motion and inertia

The ideal stopping-potential magnitude is Kmax divided by the positive elementary-charge magnitude. Partial transfer or internal losses produce an inequality rather than a larger endpoint energy.

$V_s$ is a nonnegative magnitude. The sign of a laboratory electrode voltage must also specify which electrode is the reference. Using $e$ for a positive magnitude does not change the negative charge of the electron.

If the electron receives only part of the available energy or loses energy inside matter, its outgoing kinetic energy is at most $h\nu - \Phi$. These losses do not reverse the frequency dependence or turn extra intensity into extra energy for one electron.

A footnote to the stopping condition allows that the light may also have to free each electron from a neutral molecule, at the cost of some work, before the electron can leave. The paper says nothing in the derived relation changes: its $P'$, which it defines as the potential of a gram-equivalent of negative electricity with respect to the body, is then read as a sum of two terms, the work to free the electron and the work to leave the body. In the letters used here, $\Phi$ stands for that sum, and the stopping potential still rises in a straight line with frequency, with the same slope.

An inverse energy-budget question asks for the smallest accelerating-potential magnitude capable of supplying a quantum $h\nu$. Within that idealized conversion, $eV$ must be at least $h\nu$. This is an energetic lower bound, not a promise that every collision produces light.

Open the foundation: Charges, fields and light

Open the foundation: The units of 1905

Under a one-quantum ionization assumption, a quantum must meet the required ionization energy. Absorbed energy bounds the count; equality requires that every absorbed quantum causes exactly one counted ionization.

Let $I$ be the required energy for the specified ionization channel. A single quantum can pay this cost only when $h\nu$ is at least $I$. A smaller quantum leaves an energy deficit; a larger quantum makes the event energetically possible, not certain.

For monochromatic absorbed energy $E_\text{abs}$, the number of absorbed quanta is represented by $E_\text{abs}/(h\nu)$. With at most one counted ionization per quantum this gives an upper bound. It becomes an equality only when all absorbed quanta each produce one counted ionization.

The paper compares two upper limits for the work of ionizing a gram-equivalent of air. Lenard's longest ionizing wavelength gives $6.4\times10^{12}$ erg, and Stark's smallest measured ionizing voltage, about 10 volts at platinum anodes, gives $9.6\times10^{12}$, nearly the same. A footnote concedes that inside the gas the ionizing voltage for negative ions is five times larger. Each measured voltage caps the work, so the smallest gives the tightest cap; the fivefold figure caps it only at about $4.8\times10^{13}$ erg, and the near agreement the paper notes holds for the smallest value.

The ionization laboratory separates incident energy, absorbed fraction, threshold, and conversion assumptions. Its historical checks use their own stated units; this explanation does not infer a new gas species or a real-material yield from a threshold slider.

Open the foundation: Powers of ten and physical units