29 exp i ϖ ν x ν y ν z t 29 e z ν x ν y ν z r

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( 29 exp i ϖ ν x ν y ν z t ( 29 E z ν x , ν y , ν z r r , t ( 29 = E z 0 k ν x , ν y , ν z ( 29 sin π ν x x L ( 29 sin π ν y y L ( 29 cos π ν z z L ( 29 exp i ϖ ν x ν y ν z t ( 29 [ IV-1 ] where ν x , ν y , ν z { } is a set of positive integers. To satisfy the homogeneous Helmhotz equation we must have k ν x ν y ν z 2 = π ν x L ( 29 2 + π ν y L ( 29 2 + π ν z L ( 29 2 = ϖ ν x ν y ν z c ( 29 2 [ IV-2 ] The cycle-averaged value of the stored energy density associated with the particular mode is given by W k ν x ν y ν z ( 29 = 1 4 V ε 0 r E ν x , ν y , ν z r r , t ( 29 2 0 r H ν x , ν y , ν z r r , t ( 29 2 dV cavity = 1 2 V ε 0 r E ν x , ν y , ν z r r , t ( 29 2 dV cavity = 1 16 ε 0 r E 0 k ν x ν y ν z ( 29 2 [ IV-3 ] For a thermal source, the most significant experimentally measurable object is the noise spectrum -- i.e., the frequency distribution of the stored energy density. To obtain this distribution, we take the energy density in the frequency range between ϖ and ϖ+ d ϖ -- viz. W ϖ ( 29 d ϖ = W k ν x ν y ν z ( 29 ν x , ν y , ν z { } with k ν x ν y ν z between ϖ c and ϖ+ d ϖ ( 29 c { } = W k ν x ν y ν z ( 29 × Number of modes with frequencies between ϖ and ϖ+ d ϖ [ IV-4 ]
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 40 R. Victor Jones, March 9, 2000 Thus the density of states is defined as ρ k dk Number of modes with frequencies between ϖ and ϖ+ d ϖ [ IV-5a ] In the frequency range where λ << 2 L , the following k -space argument holds: ρ k dk Two polarization states per r k state Fraction of shell with vaid states Volume in shell Volume of state = Number of r k states within shell 2 × 1 8 4 π k 2 dk π L ( 29 3 = V ϖ 2 d ϖ c 3 π 2 ≡ ρ ϖ d ϖ [ IV-5b ] This extremely important density-of-states construction may be visualized most elegantly in two dimension. 18 18 For use in later discussions, we note here that by this same argument the 2D density-of-states is found to be ρ ϖ (2 D ) = A ϖ π c 2 and the 1D value ρ ϖ 1 D ( 29 = L π c .
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T HE I NTERACTION OF R ADIATION AND M ATTER : S EMICLASSICAL T HEORY P AGE 41 R. Victor Jones, March 9, 2000 Following the traditional (Rayleigh-Jeans) argument, we identify a resonator mode as a harmonic oscillator 19 and take its average energy to be the classical thermodynamic value for a system with two degrees of freedom -- i.e.
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