Light quanta · Radiation spectrum and regime comparison
Where Wien's law holds,
and where it stops.
What does a measured radiation spectrum look like at a given temperature, in which regime is Wien's law or the classical law an accurate description, and what does a density plot actually measure?
LQ-03 · Radiation spectrum and regime comparison
The radiation spectrum and regime comparison
What does a measured radiation spectrum look like at a given temperature, in which regime is Wien's law or the classical law an accurate description, and what does a density plot actually measure?
Spectral densities at the probe frequency
| Law | uν (J·m⁻³·Hz⁻¹) | uλ (J·m⁻⁴) |
|---|---|---|
| Planck 1900 | 4.223940e-16 J/(m^3 Hz) | 5.072237e+5 J/(m^3 m) |
| Wien 1896 | 4.210617e-16 J/(m^3 Hz) | 5.056239e+5 J/(m^3 m) |
| Classical §1 | 2.318107e-14 J/(m^3 Hz) | 2.783655e+7 J/(m^3 m) |
Band energy: identical across representations
From the frequency integral: 1.286853e-1 J/m^3. From the wavelength integral over the same physical band: 1.286853e-1 J/m^3. These agree because the Jacobian is applied; a relabeled axis without it would not agree (see “Show the code”).
Peak locations: representation-dependent, and why
| Frequency-density peak νpeak | 2.939463e+14 Hz |
| Wavelength-density peak λpeak | 5.795544e-7 m |
| c / λpeak (NOT the frequency-density peak) | 5.172810e+14 Hz |
| Per-natural-log-interval peak (x = 3.9206904) | 4.084697e+14 Hz |
The frequency-density peak and c divided by the wavelength-density peak are different numbers on purpose: uλ and uν are different functions related by a Jacobian, so their maxima do not correspond under λ = c/ν.
Wien and classical regime verdict
At this temperature and probe frequency, x = hν/(kBT) = 5.759092. Wien's law's relative error here is 3.1540e-3; the classical law's relative error is 9.8178e-1. In the Wien regime at this temperature and frequency.
This pointwise error is not a certificate for the light paper's integrated entropy argument (§4), which holds only within its own stated limits.
Not modeled: emissivity and cavity imperfections; detector and spectrometer response; polarization; non-equilibrium radiation; any photon or quantum-statistical model beyond the displayed formulas.
Action contract: the same action without dragging, color, or a canvas
Every action here is typed text entry and a text result: type the band edges and probe frequency in hertz, choose the representation from the select lists above, and read the band-energy and peak tables and the regime verdict sentence. No control depends on dragging a handle, distinguishing color alone, or reading a canvas.
Open the derivation
Two printed limitations, made visible
The light paper states two regime limitations rather than one universal law. §2 concludes that the classical basis is suitable for large energy densities and wavelengths and fails completely for small wavelengths and low densities. §4 notes that Wien's law is not exactly valid, but that experiment fully confirms it for large ν/T, so results based on it hold only within those limits. This instrument shows the criteria for both limitations explicitly, at the temperature and frequency you choose, rather than asserting one law everywhere.
Two coordinate traps
A frequency-domain density uν and a wavelength-domain density uλ are related by a Jacobian, uλ(λ) = uν(c/λ)·c/λ², not by simple substitution. Their peaks do not correspond under λ = c/ν, and relabeling an axis without the Jacobian breaks the one invariant that does hold: the energy in a matching physical band, which agrees exactly between representations.