Lecture 10 - Examples Photon gas an electromagnetic field in thermal equilibrium with its container ϖ ε = = → ∑ h i i In this case h harmonic

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Unformatted text preview: Examples: Photon gas, an electromagnetic field in thermal equilibrium with its container ϖ ε = = → ∑ h i i In this case h harmonic oscillator QM we know that i n i H = h To describe the state of the field, we need to know how many n are in each oscillator Photons are bosons x n=0,1,2,3….. ( 29 ( 29 ν ν β β ε ε ε ∞- + + +- = = ∑ ∑ 1 2 1 1 2 2 , ,..., ... Q N,V,T j j j n n n n n n E e e ( 29 ( 29 ( 29 ( 29 ( 29 β β β β ε ε ε ε ∞ ∞ =---- = = ∑ ∑ ∏ L 1 2 1 1 2 2 , ,..., first we want to show that Q N,V,T j j j j j j n n n n j n n n n e e e e ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 β β β β β β β β β ε ε ε ε ε ε ε ε ε- ⋅- ⋅- ⋅- ⋅- ⋅- ⋅- ⋅- ⋅- ⋅ = + + × + + × + + 1 1 1 2 2 2 3 3 3 1 2 1 2 1 2 Q N,V,T e e e e e e e e e ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 β β β β β β β β β β β β ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε + + + + + + + + +- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅- ⋅- ⋅ + + + + + + + + × + + = 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 3 3 1 2 1 2 1 2 1 1 1 2 2 1 1 2 e e e e e e e e e e e e ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 β β β β β β β β β ε ε ε β ε ε ε β ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε + + + + + + + + + + +- ⋅ ⋅ ⋅- ⋅ ⋅ ⋅- ⋅ ⋅- ⋅ ⋅- ⋅ ⋅ ⋅ ⋅- ⋅ ⋅ ⋅---- ⋅ + ⋅ + ⋅- ⋅ + ⋅ + ⋅ ⋅ + ⋅ + ⋅ ⋅ + ⋅ + ⋅ + ⋅ ⋅ + + + + + + + = + + + + + 1 2 3 1 2 3 1 2 1 2 3 1 2 1 2 3 3 1 1 1 2 3 1 2 3 1 2 3 1 2 1 1 2 3 1 2 3 1 1 2 2 1 2 2 1 2 ... e e e e e e e e e e e ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 β β β β β β β ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε + + ∞ + + + +-- ⋅ ⋅ ⋅ ⋅ ⋅ ⋅----- ⋅ + ⋅ + ⋅ ⋅ ⋅ + ⋅ + + ⋅ + ⋅ ⋅ + ⋅ + ⋅ ⋅ + + = + + + + ∑ 1 2 1 2 3 1 2 3 1 1 2 2 3 3 1 2 1 2 3 1 2 3 1 3 3 2 , ,..., 1 1 1 2 2 ... j n n n n n n e e e e e e e ( 29 ( 29 ( 29 β β ε ε ∞ = =--- = = ∑- ∑ ∏ ∏ j 1 1 where we used 1 Q N,V,T 1 j j j j j x n j j x n e e The average number of photons in a state j… ( 29 { } ν ν ν ν β β ε ε β-- + +- = = ∑ ∑ ∑ 1 1 2 2 j ......
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This note was uploaded on 05/29/2011 for the course CHM 6461 taught by Professor Bowers during the Spring '08 term at University of Florida.

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Lecture 10 - Examples Photon gas an electromagnetic field in thermal equilibrium with its container ϖ ε = = → ∑ h i i In this case h harmonic

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