[Physics] “white light” ? Uniform wavelengths or uniform frequencies

Question

Suppose you have a mixture of electromagnetic waves of wavelengths spreaded on the visible spectrum only (from $\lambda_{\text{min}} \sim 400 \, \text{nm}$ to $\lambda_{\text{max}} \sim 700 \, \text{nm}$). At some ideal detector, the light spectral distribution is described by a functional like this :
$$\tag{1}
I = \int_{\lambda_{\text{min}}}^{\lambda_{\text{max}}} L(\lambda) \, d\lambda.
$$
Since $d\omega \propto \lambda^{-2} \, d\lambda$, we could also define the spectral distribution with angular frequencies :
$$\tag{2}
I = \int_{\omega_{\text{min}}}^{\omega_{\text{max}}} F(\omega) \, d\omega,
$$
where $F(\omega) = \lambda^2 \, L(\lambda)$. So the two functions $L(\lambda)$ and $F(\omega)$ are two complementary ways of defining the spectral distribution.

Usually, "white light" is described or defined as an uniform mixture of waves. But on which distribution ? Wavelengths or frequencies ? i.e. $L(\lambda) = \textit{cste}$ or $F(\omega) = \textit{cste}$ ? It cannot be both at the same time ! Why favor one or another function ? A photon's energy depends on frequency ; $E = \hbar \, \omega$, but we could also say that it depends on the wavelength ; $E = h c \, / \lambda$ !

Answer 1

Your assertion that

Usually, "white light" is described or defined as an uniform mixture of waves

is pretty much completely incorrect: this is not how the term "white light" is treated in the literature. The meaning of the term is relatively well captured by this glossary at Plastic Optics:

light, white. Radiation having a spectral energy distribution that produces the same color sensation to the average human eye as average noon sunlight.

However, the term is not normally taken to have a strict technical meaning, a fact which is well reflected by the observation that in the first page of a search for "optics glossary" only a single resource has an entry for "white light".

The meaning of the term is even more complicated because it depends on who is using it:

• If it is a spectroscopist that needs a white-light source to obtain a reflectivity or absorptivity spectrum, they will usually require the light to have a broad bandwidth, with support over the entire visible-light range, to be called "white".

• However, if it's a manufacturer of light bulbs, they will only require that the light be perceived as white, even if it is produced e.g. by three-colour LEDs with narrow-band spectra like this one, and their use of the term will be completely justified.

In terms of its use within the physics literature, it is much more usual to require a broadband source, with a large continuum of wavelengths contributing significantly to the spectrum. However, there isn't a requirement that all the frequencies contribute equally (partly because, as you note, that doesn't even begin to make sense).

Thus, a flat wavelength spectrum (over a broad enough range) will normally be called "white", but so will a flat frequency spectrum over an equivalent range. Moreover, many of the standard models of white light do not have a flat spectrum in either representation, with the most famous model being, of course, blackbody radiation. This has frequency and wavelength spectral distributions of the form $$ P_\nu(\nu,T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/k_BT}-1} \quad\text{and}\quad P_\lambda(\lambda,T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_B T}-1} \quad \text{resp.}, $$ and at high enough temperature (i.e. $T\approx 5500\:\mathrm K$) it models white sunlight. At lower temperatures, such as those in incandescent light bulbs, it produces a rather different spectrum, which is still called white light in the literature.