W k ν x ν x ν x 29 thermal k b t v thus we obtain

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W k ν x ν x ν x ( 29 thermal = k B T V Thus, we obtain the famous Rayleigh-Jeans radiation law (which impels us to worry about an ultra-violet catastrophe .) W T ϖ ( 29 d ϖ = kT ϖ 2 d ϖ c 3 π 2 [ IV-6 ] Planck's quantization hypothesis: Following Planck we set W k ν x ν y ν z ( 29 = 1 16 ε 0 E 0 k ν x ν y ν z ( 29 2 = E n V = h ϖ n + 1 2 ( 29 [ ] V [ IV-7 ] where n = 0, 1, 2,…… Assuming that the resonator mode is in thermal equilibrium with its environment, we may use the Boltzmann probability factor exp E n ( 29 = exp - E n k B T ( 29 to find the probability P n that the mode (read oscillator) is thermally excited to an energy E n P n = exp E n ( 29 exp E n ( 29 n = exp n h ϖ ( 29 exp n h ϖ ( 29 n = exp n h ϖ ( 29 1 - exp h ϖ ( 29 { } [ IV-8 ] The thermal mean value of n is obtained in a similar manner 20 -- viz. 19 We justify this assertion in detail later: it suffices to note here, in anticipation of that later discussion, that a harmonic oscillator and an electromagnetic mode have analogous Hamiltonians. 20 It is also easy to show that thermal mean-square fluctuation in n is given by n T = n T 2 + n T .
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 42 R. Victor Jones, March 9, 2000 n T = n P n n = n exp n h ϖ ( 29 n exp n h ϖ ( 29 n = - ∂ ∂ β h ϖ ( 29 { } ln exp n h ϖ ( 29 n = - ∂ ∂ β h ϖ ( 29 { } ln 1 - exp h ϖ ( 29 { } - 1 = 1 exp β h ϖ ( 29 - 1 [ IV-9 ] which may be plotted
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 43 R. Victor Jones, March 9, 2000 Equation [ IV-9 ] taken in conjunction with Equations [ IV-5b ] and [ IV-7 ] leads then directly to the Planck radiation law W T ϖ ( 29 d ϖ = n + 1 2 thermal h ϖ 3 d ϖ c 3 π 2 = 1 exp β h ϖ ( 29 - 1 + 1 2 β h ϖ ( 29 3 h 2 π 2 c 3 β 3 d ϖ [ IV-10 ] From this expression we see that the Rayleigh-Jeans radiation law is correct in the limit k B T h ϖ , but, to our great relief, Planck has staved off the ultraviolet catastrophe. We may then plot this famous and an exceedingly important result. (From the particular point view of laser physics, the Planck radiation law gives a measure of the background radiative thermal noise spectrum. )
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 44 R. Victor Jones, March 9, 2000 E INSTEIN ' S P HENOMENOLOGICAL T HEORY OF R ADIATIVE P ROCESSES -- Einstein's "A" and "B" coefficients: We return to the two-level system and, with Einstein, make some physically reasonable (early quantum mechanical) postulates concern the absorption and emission of light.
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