For Equation 6.6 Nelson briefly mentions a very important constant: the Planck Constant. The text states vaguely that it comes from quantum mechanics and it is used in the Sakur-Tetrode formula because it has the correct units. Here is the brief but real story.
When a young scientist Max Planck was working on developing a formula for predicting black-body radiation he happened across the concept of quantised energy. He found that heat radiation was produced in multiples of a discrete and very small number. The number is now known as the Planck constant and it has the value of ~6.63 x 10^-34 Js.
Planck derived the equation of E = hv by assuming that radiated heat was produced by atoms oscillating in harmonic motion between discrete energies. The difference between the energies were characterised by the frequency v and the energy constant h. Planck’s work was clear evidence against the previous assumption that energy was a continuous variable. His work led to Einstein’s proposal of light existing as photons as well as Schrodinger’s quantum wave mechanics.
The Planck constant also applies to energies concerning angular frequency. The reduced Planck constant (or h-bar) is equal is h/2π and is used in the equation E = ħω.
For more details see:
Radiation physicse for medical physicists
Planck Constant Wiki
Planck's constant has the units of action; this means that it has the same units of as the "volume element" of the phase space associated with a single direction of motion (this would be an "area" to most, but I mean "volume" in a generalized sense...)
ReplyDeleteThe idea that the phase space comes in "bins" of minimal volume h is fortuitous, because it was already being used without apparent justification long before quantum mechanics arose. Even Boltzmann found it impossible to derive meaningful expressions for the entropy of phase space densities without using a discrete binning and counting procedure.
For this reason and others, it is nearly impossible to find basic texts on statistical mechanics that do not begin with the statement that quantum mechanics allows us to discretize the problem.
What are you referring to when you say units of action? Is it the per second part of J.s = m^2 kg / s. And the volume element of phase space is that just a measure of how much of something is moving in this single direction of motion?
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