Planck’s Radiation Law

Planck’s radiation law is the blackbody spectrum that fits the universal thermal radiation curve and contains the new constant $h$. Its historical importance is not merely that it matches data. It gives the first central appearance of an energy scale proportional to frequency:

$$\epsilon=h\nu.$$

That scale did not yet mean the full modern photon concept. In Planck’s December 1900 argument, finite energy elements entered the statistical count of assignments to material resonators. What those elements implied about individual absorption and emission is a separate historical question, discussed in Historical Cautions About Planck. Einstein’s light quantum, Compton scattering, and quantum field theory are later steps.

Statement of Planck’s Law

For spectral energy density per unit frequency, Planck’s law is

$$u_\nu(\nu,T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{\exp(h\nu/k_B T)-1}.$$

Here $u_\nu(\nu,T)\,d\nu$ is the radiation energy per unit volume in frequencies between $\nu$ and $\nu+d\nu$, $T$ is temperature, $c$ is the speed of light, $k_B$ is Boltzmann’s constant, and $h$ is Planck’s constant.

The same law can be written per unit wavelength as

$$u_\lambda(\lambda,T) = \frac{8\pi hc}{\lambda^5} \frac{1}{\exp(hc/\lambda k_B T)-1}.$$

These are not the same function with variables simply relabeled. Positive spectral intervals obey $u_\lambda\,|d\lambda|=u_\nu\,|d\nu|$. Since $\nu=c/\lambda$, the density transforms as $u_\lambda(\lambda,T)=u_\nu(c/\lambda,T)\,c/\lambda^2$. The absolute Jacobian is essential because frequency decreases as wavelength increases. This is why the peak wavelength and peak frequency do not obey the naive relation $\nu_{\max}=c/\lambda_{\max}$.

In angular-frequency notation, with $\omega=2\pi\nu$ and $\hbar=h/(2\pi)$,

$$u_\omega(\omega,T) = \frac{\hbar\omega^3}{\pi^2c^3} \frac{1}{\exp(\hbar\omega/k_B T)-1}.$$

The form changes with the spectral variable, but the physical content is the same.

Relation to Experimental Data

Planck’s law succeeds because it has the right behavior in the two regimes that earlier formulas captured separately. At low frequency, let

$$x=\frac{h\nu}{k_B T}.$$

When $x\ll 1$, $\exp(x)-1\approx x$, so

$$u_\nu(\nu,T) \approx \frac{8\pi\nu^2}{c^3}k_B T.$$

This is the Rayleigh–Jeans form. It agrees with the long-wavelength, low-frequency side of the blackbody spectrum.

At high frequency, $x\gg 1$, so $\exp(x)-1\approx\exp(x)$ and

$$u_\nu(\nu,T) \approx \frac{8\pi h\nu^3}{c^3} \exp\!\left(-\frac{h\nu}{k_B T}\right).$$

This is Wien-like exponential suppression. It gives the observed falloff and avoids the high-frequency divergence of the Rayleigh–Jeans law.

The formula therefore interpolates between a classical low-frequency limit and an exponentially suppressed high-frequency regime. That interpolation is exactly what the measured spectrum demanded.

Planck’s Oscillator Model

Planck modeled the cavity walls as containing microscopic resonators that exchange energy with the electromagnetic field. For resonators of frequency $\nu$, he introduced energy elements

$$\epsilon=h\nu.$$

Planck counted distributions of a total energy split into equal elements. A modern shorthand for the resulting assignments to one resonator is

$$E_n=nh\nu, \qquad n=0,1,2,\ldots .$$

Planck’s statistical counting led to the mean energy

$$\bar E(\nu,T) = \frac{h\nu}{\exp(h\nu/k_B T)-1}.$$

Multiplying this mean oscillator energy by the classical electromagnetic mode density

$$g(\nu)\,d\nu = \frac{8\pi\nu^2}{c^3}\,d\nu$$

gives Planck’s spectrum:

$$u_\nu(\nu,T)\,d\nu = g(\nu)\bar E(\nu,T)\,d\nu.$$

This reconstruction shows the hybrid character of the early result. The mode density is classical electromagnetic reasoning. The average energy contains the new discrete scale $h\nu$.

The Role of h

Planck’s constant has dimensions of action:

$$[h]=[\mathrm{energy}]\,[\mathrm{time}].$$

In the radiation law it appears through the dimensionless ratio

$$\frac{h\nu}{k_B T}.$$

This ratio compares the energy element $h\nu$ with the thermal energy scale $k_B T$. When $h\nu\ll k_B T$, the energy spacing is small relative to thermal energy and the classical Rayleigh–Jeans limit is recovered. When $h\nu\gg k_B T$, thermal excitation of that frequency is exponentially suppressed.

The significance of $h$ grew as the same constant appeared in other contexts: the photoelectric effect, atomic spectra, old quantum theory, commutation relations, and eventually the Schrödinger equation. In modern notation, $\hbar=h/(2\pi)$ is often more natural because angular frequency and generators of time evolution use $\omega$ rather than $\nu$.

Modern Derivation Versus Historical Derivation

A modern derivation usually starts from quantized harmonic oscillator modes of the electromagnetic field. A mode of angular frequency $\omega$ has energy levels

$$E_n = \hbar\omega\left(n+\frac{1}{2}\right), \qquad n=0,1,2,\ldots .$$

The thermal average excitation energy, excluding the temperature-independent zero-point part, is

$$\langle E\rangle_T = \frac{\hbar\omega}{\exp(\hbar\omega/k_B T)-1}.$$

Together with the density of photon modes, this gives Planck’s law. In this modern derivation, the result is connected to Bose occupation numbers and quantum field modes.

Historically, Planck did not begin from a completed quantum theory of fields. He used resonators, entropy, and statistical counting to obtain the equilibrium spectrum. The historical derivation and the modern derivation therefore have overlapping mathematics but different conceptual settings:

Question	Planck’s historical setting	Modern setting
Where does discreteness enter?	Finite energy elements in resonator counting	Field-mode excitations
What is the energy scale?	$h\nu$	$\hbar\omega=h\nu$
Is there a photon concept?	Not yet in the modern sense	Yes, in quantum electrodynamics
What supplies the statistics?	Resonator counting and entropy	Bose occupation of oscillator modes
What happens to zero-point energy?	Not part of the original law	Present in oscillator levels but not in the thermal spectral density

The modern derivation is cleaner for calculation. The historical derivation is essential for understanding how quantization entered physics.

How This Points Toward Quantization

Planck’s law exposes the failure of classical equipartition applied to the electromagnetic modes. If every mode has the classical quadratic energy variables and receives a mean energy $k_B T$, the high-frequency energy density diverges. In the modern oscillator account, the level spacing $h\nu$ instead suppresses thermal excitation when it is much larger than $k_B T$. This comparison diagnoses the classical counting and occupation assumptions; it does not turn the measured equilibrium spectrum into a direct observation of individual energy-transfer events.

The step is conceptually delicate. Planck’s law supports a discrete energy scale in radiation equilibrium, but it does not by itself establish that light consists of localized particles. The later Photoelectric Effect page explains why Einstein’s light-quantum hypothesis was a separate advance.

The enduring lesson is that $h$ changes the counting of possible states and the thermal occupation of modes. That is why blackbody radiation belongs at the entrance to quantum mechanics rather than as a mere historical curiosity.

Common Mistakes

• Treating Planck’s law as just curve fitting; it also has the right limiting laws and a statistical interpretation.
• Saying Planck discovered photons; his quantization was tied to oscillator energy elements.
• Forgetting that the Rayleigh–Jeans law is recovered when $h\nu\ll k_B T$.
• Confusing spectra per unit frequency and per unit wavelength.
• Adding zero-point energy directly to the thermal blackbody spectrum.
• Treating $h$ and $\hbar$ as different constants rather than different normalizations of the same scale.

The quantum harmonic oscillator derives the modern energy levels. Einstein coefficients give another route to the equilibrium spectrum through absorption and emission rates.

Exercises

1. Derive the Planck mean oscillator energy from the partition function.

Solution

For levels $E_n=nh\nu$, the partition function is

$$Z = \sum_{n=0}^\infty e^{-\beta nh\nu} = \frac{1}{1-e^{-\beta h\nu}},$$

where $\beta=1/k_B T$. The mean energy is

$$\bar E = -\frac{\partial}{\partial\beta}\ln Z = \frac{h\nu}{e^{\beta h\nu}-1}.$$

This is the energy factor in Planck’s law.

2. Use $x=h\nu/(k_BT)$ to show that the total radiation energy density is proportional to $T^4$. You may use $\int_0^\infty x^3/(e^x-1)\,dx=\pi^4/15$.

Solution

The substitution gives $d\nu=(k_BT/h)\,dx$, so

$$u(T)=\int_0^\infty u_\nu\,d\nu =\frac{8\pi(k_BT)^4}{h^3c^3} \int_0^\infty\frac{x^3}{e^x-1}\,dx =\frac{8\pi^5k_B^4}{15h^3c^3}T^4.$$

The emitted flux is $M=cu/4=\sigma T^4$, with $\sigma=2\pi^5k_B^4/(15h^3c^2)$. The factor $c/4$ comes from angular averaging of isotropic radiation, so energy density and surface flux are different quantities.

3. Why does the zero-point term not appear as an added temperature-dependent blackbody energy density?

Solution

The modern oscillator levels contain $\hbar\omega/2$, but this part is independent of temperature and is present even in the vacuum. The ordinary thermal blackbody spectrum describes the temperature-dependent energy above the vacuum contribution. Adding $\hbar\omega/2$ to every mode without renormalization would introduce a separate vacuum-energy problem, not the measured thermal radiation spectrum.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912, University of Chicago Press, 1987 (with a new Afterword).
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Academic Press, 2011.
• M. Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Annalen der Physik 309, 553-563 (1901), DOI: 10.1002/andp.19013090310.
• M. Planck, The Theory of Heat Radiation, translated by M. Masius, P. Blakiston’s Son & Co., 1914.