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05_01_28 - 12 Average energy in each mode for light quanta...

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[email protected] 12 Average energy in each mode for light quanta In statistical mechanics one learns that at temperature T, the probability P for a state to have energy E is: Since P is normalized to 1, Average energy in this mode, Energy of the mode is for light quanta, each with energy Average energy in each mode: K J kT H k e P 23 10 381 . 1 , - - = - - = H all kT H e kT H e P 1 n ) ( ν ε = n H ) ( ν ε 1 ) ( 1 / ) ( - = - = < - - kT e e e H ν ε βε βε ν ε ε - - = < H all kT H e H all kT H e H H 1 01/28/2005 , 1 1 ] [ ) ( 0 ) ( kT e e e n n kT H all kT H ν ε ν ε - = - - - = = 2 ) 1 ( βε βε β β ε β - - - - - = - = e e e d d He H all H H all H
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[email protected] 13 The energy of light quanta and Plank’s constant ν ν ν ν ν ε ν ε π d d u kT e c 1 ) ( 2 / ) ( 3 8 ) ( - = Number of modes in a frequency interval: Average energy density in that interval: ν ν π ν d dZ c L 2 3 ) ( 8 ) ( = Wien’s displacement law: only depends on and There is no longer an ultra-violet catastrophe 3 / ) ( T u ν T / ν ν ν ε h = ) ( 1 ) exp( ) / ( 2 3 4 3 2 ) ( - = = T k h T c h c T T u R ν ν π ν Js h 34 10 626 . 6 - = Plank’s constant:
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