Planck, radiation oscillators

The problem of specific heats, treated by Jeans from the classical point of view, as I said above, was discussed by Einstein in the case of solids, with special regard to the discrepancy observed at low temperature between the measured values and those deduced from the theory he had constructed in 1907 by quantizing the mechanical oscillators3 as Planck had quantized the radiation oscillators. 

In 1900, Max Planck (1858-1947) discovered a formula (now often called the Planck radiation law) that modeled curves like those shown in Figure 24-21 nearly perfectly. He followed this discoveiy by developing a theory that made two bold assumptions regarding the oscillating atoms or molecules in blackbody radiators. He assumed (1) that these species could have only discrete energies and (2) that they could absorb or emit energy in discrete units, or quanta. These assumptions, which are implicit in Equation 24-3, laid the foundation for the development of quantum theory and eventually won him the Nobel Prize in Physics in 1918. 

In 1907, EINSTEIN [1.3] proposed a simple model to account for the decrease of the specific heat at low temperatures. He took the atoms of a cys-tal to be independent oscillators, each having the same frequency and able to vibrate isotropically. He then quantized the energy of the oscillators in accordance with the results obtained by PLANCK for radiation oscillators. 

The explanation of the hydrogen atom spectmm and the photoelectric effect, together with other anomalous observations such as the behaviour of the molar heat capacity Q of a solid at temperatures close to 0 K and the frequency distribution of black body radiation, originated with Planck. In 1900 he proposed that the microscopic oscillators, of which a black body is made up, have an oscillation frequency v related to the energy E of the emitted radiation by... 

For radiation in equilibrium with electromagnetic oscillators consisting of the charges on material ions we can combine the formula for the mean energy of a resonator (Planck, Wcirme-strahluny, p. 124) ... 

The constant h, now called Planck s constant, has the value 6.626 X 10 i4 )-s. If the oscillating atom releases an energy E into the surroundings, then radiation of... 

When Planck used this relationship to calculate the spectrum of blackbody radiation, he came up with a result that agreed perfectly with experiment. More importantly, he had discovered quantum mechanics. Energy emitted by a blackbody is not continuous. Instead, it comes in tiny, irreducible packets or quanta (a word coined by Planck himself) that are proportional to the frequency of the oscillator that generated the radiation. 

Planck s constant (h) A universal constant of nature that relates the energy of a photon of radiation to the frequency of the emitting oscillator. Its numerical value is about 6.626 x ICh27 ergs/s. 

In 1901 Planck finally explained the frequency and temperature dependence of blackbody radiation, and ushered in the age of quantum physics, by introducing the quantization of the oscillators that Rayleigh had discussed. (Planck assumed that these oscillators were in the walls of the Hohlmum and that the radiation was in equilibrium with them.) The energy density (energy per unit volume) [m(v, T)/V] dv at the temperature i, in the frequency range between v and v + dv, is given by... 

This prediction was based on the notion that a blackbody is composed of tiny oscillators that produce a continuum of waves, like those you get when you pluck the strings of a violin. But the spectrum physicists predicted for blackbody radiation—an infinite amount of high-energy radiation—and the experimental data did not fit. They were not even close. And this was the problem that Max Planck was working on in 1900. 

This is reminiscent of Planck s formula for the energy of a photon. It comes as no surprise then that the quantum theory of radiation has the structure of an assembly of oscillators, with each oscillator representing a mode of electromagnetic waves of a specified frequency. 

Blackbody radiation was explained by Max Planck in 1901, but only by overthrowing the very foundations of classical mechanics. Planck reasoned that the very high-frequency oscillators must not be excited by the thermal energy of the hot body to the same degree as the lower frequency oscillators. This was a challenge to explain because classical mechanics allows an oscillator to have any energy. Planck s argument involved two steps, which are explained as follows. 

The only way that Planck could fit the experimental spectrum was to postulate that the oscillating charges responsible for the radiation were restricted to discrete energies, that an oscillator was either excited or not, and that the probability of an oscillator being excited depends on the temperature. 

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