ω is a chosen angular frequency of light.
τ is temperature, c the speed of light.
uω is energy per unit volume per unit frequency range.
Condensing
Matters:
A Summary Derivation of
the Planck Formula for Black Body Radiation
ω is a chosen angular frequency
of light.
τ is temperature, c the speed of light.
uω
is energy per unit volume per unit frequency range.
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Introduction
My experience of thermodynamics might have begun with Rudy Marcus if I
had managed to attend more of his lectures. Sleep contends with
ambition
in and out of the lecture hall I suppose. Instead my proper
introduction to thermodynamics began with a book (fairly remarkable):
Professors
Kittel and Kroemer's Thermal Physics published 1980. It was
such a good introduction that I feel it worth a revisit. The first 90
pages cover important
foundational ideas of thermodynamics and conclude with a derivation of
the Planck formula for black body
radiation. This derivation ends with the words "This
result is the Planck radiation law; it gives the frequency
distribution of thermal radiation. Quantum theory began here."
So
that's it: Where the river begins. All I need do is follow that 90-page
derivation to arrive at the PRL and from there continue downstream
into the rest of quantum mechanics. In fact these 90 pages include
other ideas; it's a matter of following the central thread based on a
fundamental premise: All accessible states of a closed system are
equally likely.
Historical note: Planck found the PRL
using ideas
about Harmonic Oscillators and it was
Einstein in 1905 who started the shift towards connecting oscillators
to quantized energy and hence light as energy corpuscles (photons). The
derivations have the same form but the physical
ideas had evolved in Einstein's 1905 contribution. In any event the
point of this page is to condense K&K's derivation
into a review-derivation of the PRL "the
easy/modern way".
Chapter 1 of Kittel and Kroemer:
States of a Model System
Examination of an idealized system of
many constitutive elements, all identical, all capable of occupying one
of two states.
The 'model system' states are taken to be spin-up or spin-down. The
system consists of N sites
where N is an even number for
convenience. Rather than start with all-up or all-down, start with
half-up and half-down and consider variations where s sites flip over from - to + giving
a spin excess of +2s . That
is, when s = 0 the total spin
is zero. When s = 1 the total
spin is 2, and when s=19 the total spin is 38. Suppose (N/2 + s)
magnets are up; how many ways can these be distributed? We use the
Combination idea "N-choose-(N/2 + s)"
from the binomial theorem to arrive at the multiplicity function g(N,s). This
tells us how many different states are available with spin excess s.
g(N, s) = (N N/2+s).
Chapter 2 of Kittel and
Kroemer: Entropy and Temperature
Summary
Text
Chapter 3 of Kittel and
Kroemer: Boltzmann Distribution and Helmholtz Free Energy
Summary
Text
Chapter 4 of Kittel and
Kroemer: Thermal Radiation and Planck Distribution
Summary
Text
Concluding Thoughts
Summary
Text
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