Planck’s Route to the Black Body Radiation Formula and Quantization

Wien’s Radiation Law

 Wien proved using classical thermodynamics that the shape of the black body curve didn’t change with temperature, the curve just grew and expanded.  However,  the thermodynamic methods didn’t specify the actual shape.  In 1893, Wien made a guess, based on the experimental data:

his Radiation Law.  α, β are constants.  In fact, this provided an excellent fit: it seemed that Kirchhoff’s challenge had at last been met!  But the long wavelength low frequency measurements were not very precise, and when improved infrared technology was introduced a few years later, in 1900, it became clear that at the lowest frequencies ρ(f) went as f ², not f ³, and furthermore the radiation intensity at these low frequencies was proportional to the temperature.  So Wien’s formula was wrong.  The challenge was still there.

Planck’s Thermodynamic Approach:  Oscillators in the Oven Wall

Photo Courtesy of the Clendening History of Medicine  Library, University of Kansas Medical Center.
After what’s happened in physics over the last century or so, it’s difficult to appreciate the mindset of a physicist like Planck in the late 1890’s.  He was forty years old and a well-established theorist at the University of Berlin. His earlier work had been in chemical physics, where applying thermodynamics had led to brilliant successes. He was convinced thermodynamics was the key to understanding nature at the deepest level. He spent years clarifying the subtleties of the Second Law (that entropy always increases).  He believed the Second Law was rigorously correct, and would eventually be proved so in a more fundamental theory. 
And now thermodynamics had made a good start in analyzing black body radiation, with proofs of Stefan’s Law and Wien’s Displacement Law.  It seemed very likely that thermodynamics would yield the whole black body radiation curve.  He felt this curve was the key to understanding just how electromagnetic radiation and matter exchanged energy.  This was one of the basic problems in physics, and of obvious technological importance.  And, in fact, just at that time experimentalists at his university were measuring the black body radiation curve to new levels of precision. 
(Boltzmann himself had gone on to a molecular analysis of the properties of gases, relating thermodynamic quantities to microscopic distributions of particles. Planck was not impressed by this approach, since it implied that the Second Law was only statistical, only valid in the limit of large systems.  However, he was fairly familiar with Boltzmann’s work—he’d taught it in some of his classes, to present all points of view.  Planck wasn’t sure, though, that atoms and molecules even existed.) 
But how to begin a thermodynamic analysis of black body radiation?  The oven used by the experimentalists was a dauntingly complex system: the hot oven walls contained many tiny oscillating electrical charges, the electromagnetic radiation from the acceleration of these charges being the heat and light radiation in the oven.  At the same time, the wall oscillators were supplied with energy by the oscillating electrical fields of the radiation.   In other words, at a steady temperature, the radiation inside the oven and the electrical oscillators in the walls were in thermal equilibrium, there was no net transfer of energy from one to the other over time, but small amounts of energy were constantly being traded back and forth for individual oscillators. 
Fortunately, Kirchhoff had long ago proved that the details of the oven don’t matter, if two ovens at the same temperature have different radiation intensity at some particular frequency, energy could flow from one to the other, violating the Second Law.  So Planck could consider as his “oven” the simplest possible material object that would interact with the radiation: he took a simple harmonic oscillator (one-dimensional, mass m, linear restoring force ).  As a preliminary exercise, he replaced the incoherent heat radiation with a monochromatic oscillating electric field  driving the oscillator.