Phy5645/UV catastrophe problem2

(Submitted by Team 4-Yuhui Zhang) Try to use Boltzman-Maxwell statistics to deduce Plank Formula. (We have to consider quantum energy spectrum as Plank did.)

If the energy spectrum is: $0$, $h\upsilon$, $2h\upsilon$, $3h\upsilon$...

Then use Boltzman-Maxwell statistics:

${\overline {E}}={\frac {\sum \limits _{n}{E_{n}e^{-\beta E_{n}}}}{\sum \limits _{n}{e^{-\beta E_{n}}}}}={\frac {\sum \limits _{n}{E_{n}e^{-\beta nh\upsilon }}}{\sum \limits _{n}{e^{-\beta nh\upsilon }}}}={\frac {h\upsilon }{e^{\frac {h\upsilon }{kT}}-1}}$

So, the average particle number in $h\upsilon$ energy state is ${\frac {1}{e^{\frac {h\upsilon }{kT}}-1}}$. (This is just the result of Bose-Einstein statistics.) so:

$E(\upsilon )={\frac {8\pi }{h^{3}}}{\frac {1}{e^{\frac {h\upsilon }{kT}}-1}}p^{2}dp={\frac {1}{\pi ^{2}c^{3}}}{\frac {\hbar \omega ^{3}}{e^{\frac {h\upsilon }{kT}}-1}}d\omega =8\pi {\frac {h\upsilon ^{3}}{c^{3}}}{\frac {1}{e^{\frac {h\upsilon }{kT}}-1}}d\varepsilon$, this reflect the phenomenon of black body irradiation, which is called Plank Formula.