Discover · Light quanta · An explanatory investigation

Can a volume law suggest what light is made of?

Recover an entropy dependence, compare independent and locked configurations, then test what a further energy-transfer hypothesis predicts.

A route you could take, not a transcript of Einstein’s private thoughts. This worked preview does not publish the reviewed historical journey, its knowledge shelf, or a new translation. Editorial and physics review remain pending.

Begin with what the wave description preserves

The question is not whether a particle animation can replace interference. A successful description of wave propagation and a hypothesis about energy exchange address different obstacles. The calculations below do not model interference or claim that it disappears.

Investigate prescribed waves and interference · Inspect classical energy allocation and its cutoff · Compare spectral laws and their admitted regimes

Notation and what is held fixed

In these source-style formulas β is Wien’s spectral constant, not a speed or a Lorentz factor. R is the molar gas constant, N the molecular number per mole, ν the cyclic frequency, E the stored radiation energy, and P the electron’s exit cost. In the explicitly modern numerical model, R/N corresponds to k_B and Rβ/N to h. The source uses ε for elementary charge; the emission formula here uses the editorial symbol e to distinguish it from energy per quantum.

Open the notation concordance · Take the no-algebra counting entrance

Static worked example. Modern SI constants are used throughout these numerical examples. They are declared model calculations, not measurements available in 1904. No result here is labeled as a FrankenSim calculation.

One frequency, three linked questions

The cavity comparison, toy counting problem and emission bench are different model systems. They share a frequency and volume fraction where relevant, not an experimental apparatus. Optical power is not the cavity’s stored energy. The efficiency and exit cost describe a hypothetical surface, not a measured material.

Inspect every accepted setting

Frequency: 600 THz
Reference temperature: 3000 K
Band width: 1 THz
Reference volume: 1 litres
Volume fraction V/V₀: 0.5 1
Independent points: 3 count
Incident optical power: 1 mW
Exit cost: 2 eV
Declared quantum efficiency: 0.1 fraction
Collector potential: 0 V

Step 1 · A thermodynamic calculation

Keep the energy. Change the room.

Compare two narrow-band radiation states with the same energy, frequency and band width. The reference temperature fixes the initial state; it is not held fixed during this volume comparison. This is not a simulation of pushing a piston or preparing those states.

S-S_0=\frac{E}{\beta\nu}\ln\frac{V}{V_0}

The entropy change equals a coefficient multiplied by the logarithm of the final volume divided by the initial volume.

Fixed radiation energy: 9.0556 nJ
Entropy change: -2.1798e-13 J/K
Initial implied temperature: 3000 K
Final implied temperature: 3233.5 K

Why the regime and integration constant are part of the argument

Invert Wien’s law to express reciprocal temperature in terms of spectral density, then integrate the entropy derivative with frequency fixed. An arbitrary integration constant would leave an additional volume-dependent term. The zero-radiation entropy boundary condition removes that ambiguity; simply forgetting the constant does not.

Both endpoint states must be dilute and the band must be narrow. This calculation uses the LQ-04 owner’s one-percent Wien criterion and one-percent maximum relative band width. Compress far enough and the inference becomes unavailable rather than a false number.

Entropy and temperature · Integration and its constant · Inspect the entropy workbench and its unfixed-constant variant

Step 2 · A simpler problem with an explicit premise

What does independence change?

Place 3 points independently in the larger volume. Ask for the probability that all of them lie in the fraction 0.5. Then remove independence: perfectly locked positions move as one uniformly placed unit. These are toy configuration models, not pictures of photons.

W_{\mathrm{independent}}=f^n,\quad W_{\mathrm{locked}}=f,\quad f=\frac{V}{V_0}

Independent placements multiply their probabilities. Perfectly locked placements have only one placement probability, regardless of how many points move together.

Independent probability: 0.125
Perfectly locked probability: 0.5
Logarithm of independent probability: -2.0794

Start without algebra, then try sixty points

In half a space, a single uniformly placed object has one chance in two of being inside. With three independent placements, exactly one of the eight equally likely inside/outside assignments has all three inside. Lock all three placements together and only the two collective assignments remain. This counterexample identifies the independence premise.

The logarithmic calculation stays finite for sixty points without enumerating an enormous list of configurations. Neither the point count nor this toy probability is inferred from light measurements.

Probability and independence · Why logarithms turn products into sums · Enumerate and sample configurations in LQ-05

Step 3 · The move is a heuristic inference

Match the coefficient, not the picture.

The two entropy laws share a volume logarithm. Match its coefficients, including the thermal factor R/N. E/(βν) has units of entropy; the dimensionless effective count is NE/(Rβν). It is not the small integer entered for the separate counting example.

\frac{E}{\beta\nu}=n_{\mathrm{eff}}\frac{R}{N},\qquad n_{\mathrm{eff}}=\frac{NE}{R\beta\nu},\qquad \epsilon=\frac{R\beta\nu}{N}

The radiation coefficient suggests an effective number of independent things and an energy per thing proportional to frequency. Matching this form does not prove what light is made of.

Entropy coefficient: 3.1448e-13 J/K
Effective count, never forced to an integer: 22778000000
Suggested energy per thing: 2.4814 eV

Which claim are you willing to make? This choice does not change any numbers. Keep the thermodynamic correspondence without choosing a microscopic account. Investigate independent radiation quanta as a further physical hypothesis.

The shared entropy dependence survives without a unique microscopic interpretation. A simulator programmed with quanta cannot settle that choice by reproducing its own assumptions.

Why this does not derive a full theory of light

The correspondence is restricted to the admitted Wien regime. It neither derives Planck’s full spectrum nor removes the successes of wave propagation and interference. Modern numerical constants construct these illustrative states; they are not independent historical measurements from which a new physical constant has been discovered.

Inspect the coefficient correspondence in LQ-06 · Read the argument and its qualifications

Step 4 · Add a physical hypothesis, then demand consequences

More light: more electrons, or more energy per electron?

This branch explicitly assumes complete single-quantum transfer to one electron, an exit cost, and the declared emission efficiency. It does not follow from coefficient matching alone. Its results remain conditional calculations even when the entropy model above is outside its domain.

K_{\max}=\frac{R\beta\nu}{N}-P,\qquad \Pi=\frac{K_{\max}}{e}\quad(K_{\max}\geq 0)

The maximum electron kinetic energy is the quantum energy minus the exit cost. Below threshold there is no emitted electron in this model, so its kinetic energy and stopping potential are not applicable, rather than negative or zero values.

Optional prediction: what will change, and why?

Your note stays on this page and does not control any calculation. No answer is required.

Each button pins the currently accepted result, changes only the named input, and calculates a new comparison. Raising frequency also creates a new cavity reference state at the chosen reference temperature; only a volume change keeps that cavity energy fixed. Power means intensity at a fixed illuminated area.

Read the explanations without making a prediction

Doubling optical power at fixed frequency and surface assumptions doubles the arrival and emission rates, not the maximum kinetic energy. Raising frequency at fixed power raises the energy per quantum but lowers the incident quantum rate. Raising the exit cost lowers the energy left to an escaping electron and may stop emission altogether.

A positive predicted current also depends on the declared yield and collection model. At a retarding potential between zero and the stopping point, the current is underdetermined without an electron energy distribution. The workbench must show that missing information.

Partial transfer would instead leave an upper bound on electron energy. Real surface states, contact potentials, space charge, thermionic emission and multiphoton processes are not modeled.

Open the photoelectric bench and its model limits

Carry this investigation forward

Share the current accepted settings or save both completed comparison columns. Unapplied form edits, predictions and interpretation choices are never included. A settings link opens a draft for explicit application; it does not claim to preserve results across source revisions.

Continue in LQ-06 with this exact accepted radiation energy →

The specialist page asks before applying linked settings. Without JavaScript, its prepared example remains visible, not a claimed calculation of the linked state.

Pinned baseline and current accepted result

No settings differ from the pinned baseline.

One completed calculation per column; missing values retain their scientific meaning.
Quantity	Pinned baseline	Current result
Radiation energy held fixed during a volume comparison	9.0556 nJ	9.0556 nJ
Radiation entropy change	-2.1798e-13 J/K	-2.1798e-13 J/K
Coefficient of the volume logarithm	3.1448e-13 J/K	3.1448e-13 J/K
Effective count (not rounded to an integer)	22778000000 1	22778000000 1
Probability: independent points	0.125 1	0.125 1
Probability: perfectly locked points	0.5 1	0.5 1
Energy scale suggested by matching	2.4814 eV	2.4814 eV
Maximum emitted-electron kinetic energy	7.7129e-20 J	7.7129e-20 J
Stopping potential magnitude	0.4814 V	0.4814 V
Incident quantum rate under the assumed model	2.5153e15 s⁻¹	2.5153e15 s⁻¹
Emission rate at the declared efficiency	251530000000000 s⁻¹	251530000000000 s⁻¹
Collected current	40.3 μA	40.3 μA

Baseline snapshot 1; current snapshot 1.

Inspect calculation ownership and reproducibility

Model: light-quanta-investigation-v1. Constant set: modern-si-2019.

Source digest: source:sha256:de04818516747bd7968c409ac8d725afbda19e28eef44815d8e9a5856e2117d0

Existing owners produce the physical laws; this investigation assembles their accepted results. Source identity is not evidence of a correct theory or a reviewed translation.

lq04.acceptedInputs
radiation.entropy
radiation.quanta
radiation.independentPointsProbability
radiation.lockedPositionsProbability
photoelectric.kMax
photoelectric.stoppingPotentialMagnitude
photoelectric.quantumRate
photoelectric.emissionRate
photoelectric.photocurrent

What survives, and what still needs observations?

The entropy correspondence suggests an interpretation in a restricted regime. Extending it to emission, fluorescence and ionization adds hypotheses. A programmed consequence is not an empirical confirmation, and a curve generated from the assumed law is not a historical dataset.

Two further consequences, with their limits

Fluorescence asks how one absorbed energy budget constrains emitted light, including thermal qualifications. Ionization asks which thresholds and count bounds follow without inventing an absorption cross-section or an exact yield. Both retain the distinction between a bound and a measurement.

Inspect fluorescence budgets and thermal qualifications · Inspect ionization thresholds and count bounds

Before leaving, explain why locked positions change the counting law, which inference fails outside the Wien regime, and why doubling power is not the same experiment as raising frequency.