Quantum Mechanics Series -5 : Thermal Equilibrium and Fluctuations

The Thirty Year War (1900−30) – Quantum Physics Versus Classical Physics

There were three critical experiments in the pre-quantum era which could not be explained by a straightforward application of classical physics.
Each involved the interaction of radiation and matter as reported by reliable, experimental scientists.

The measurements were accurate and reproducible, yet paradoxical… the kind of situation a good theoretical physicist would die for.
We will describe each experiment step-by-step, pointing out the crisis engendered and the solution advanced by Max Planck, Albert Einstein and Niels Bohr respectively.

In putting forward their solution, these scientists made the first fundamental contributions to a new understanding of nature. Today the combined work of these three men, culminating in the Bohr model of the atom in 1913, is known as the Old Quantum Theory.

Black-Body Radiation
When an object is heated, it emits radiation consisting of electromagnetic waves, i.e. light with a broad range of frequencies.

1. Measurements made on the radiation escaping from a small hole in a closed heated oven – which in Germany we call a cavity – shows that the intensity of the radiation varies very stronger with the frequency of the radiation.

The dominant frequency shifts to a higher value as the temperature is increased, as shown in the graph drawn from measurements made in the late 19th century.

A black-body is a body that completely absorbs all the electromagnetic radiation failing on it.

Inside a cavity the radiation has nowhere to go and is continuously being absorbed and re-emitted by the walls. Thus, a small opening will give off radiation emitted by the walls, not reflected, and thus is characteristic of the black body.
When the oven is only just warm, radiation is present but we can’t see it because it does not stimulate the eye. As it gets hotter and hotter, the frequencies reach the visible range and the cavity glows red like a heating ring on an electric cooker.

This is how early potters determined the temperature inside their kilns. They would notice the color of fire where pots are heated and thr color gave them idea of temperature. Already in 1792, the famous porcelain maker Josiah Wedgwood had noted that all bodies become red at the same temperature.

In 1896, a friend of Planck’s Wilhelm Wien, and others in the Berlin Reichsanstalt (Bureau of Standards) physics department put together an expensive empty cylinder of porcelain and platinum.
At Berlin’s Technische Hochschule, another of Planck’s close associates, Heinrich Rubens, operated a different oven.
These radiation curves – one of the central problem of theoretical physics in the late 1890s – were shown to be very similar to those calculated by Maxwell for the velocity (i.e. energy) distribution of heated gas molecules in a closed container.

Paradoxical Results
Could this black-body radiation problem be studied in the same way as Maxwell’s ideal gas… electromagnetic waves (instead of gas molecules) bouncing around in equilibrium with the walls of a closed container?
Wien derived a formula, based on some dubious theoretical arguments which agreed well with published experiments, but only at the high frequency part of the spectrum.
The English classical physicist Lord Rayleigh (1842−1919) and Sir James Jeans (1877−1946) used the same theoretical assumptions as Maxwell had done with his kinetics theory of gases.
The equation of Rayleigh and Jeans agreed well at low frequencies but they got a real shock at the high frequency region. The classical theory predicted an infinite intensity for the ultraviolet region and beyond, as shown in the graph. This was dubbed the ultraviolet catastrophe.
What does this experimental result actually mean and What Went Wrong ?
The Rayleigh-Jeans result is clearly wrong, otherwise anyone who looked into the cavity would have eyeballs burned out.

This ultraviolet Catastrophe became a serious Paradox for classical physics.
If Rayleingh and Jeans were right, it would be dangerous for us even to sit in front of a fireplace.

If classical physicists had their way, the romantic glow of the embers would soon turn into life-threatening radiation. Something had to be done!

The Ultraviolet Catastrophe
Everyone agreed that Rayleigh and Jeans’ method was sound, so it is instructive to examine what they actually did and why it didn’t work.
1. We applied the statistical physics method to the waves by Analogy with Maxwell’s gas particles using the equipartition of energy, i.e. we assumed that the total energy of radiation is distributed equally among all possible vibration frequencies.
2. But there is one big difference in the case of waves. There is no limit on the number of modes of vibration that can be excited…
3. …Because It’s easy to fit more and more waves into the container at higher and higher frequencies (i.e. the wavelengths get smaller and smaller).
4. Consequently, the amount of radiation predicated by the theory is unlimited and should keep getting stronger and stronger as the temperature is raised and the frequency increases.
5. No wonder it was known as the ultraviolet catastrophe.

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