Lecture 15 - Bose-Einstein Condensation Note that is valid...

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Bose-Einstein Condensation Note that is valid only if no state is macroscopically occupied. So when this inequality is broken, some state is macroscopically occupied. Lowest Temperature c TT when the equality is still satisfied (with 1 z ; actually 1 z for all c ) 2/3 3 2 1 2.612 2 2 2.612 c c hn T mk T mk     Which state is macroscopically occupied? Bosons can occupy any state macroscopically. Makes sense for 0 state though. Check: 1 1 1 i i n ze  When 0 1 1 0, as z 1 1 1 i z N z z   Also note that 1/2 ( ) ~ a  which means ( ) 0 a for 0 and we have not taken the 0 state into proper account. So what? What is one state worth in a continuum? Problem here is that this state is macroscopically occupied. So we better write 0 0 ( ) ( ) BE ex N a d f N N  0 3/2 3 1 () NN gz V  When c , 0 0 N , and when c , 0 0 N At c T , 3/2 3/2 ~ (1) c N Tg V (1) Below c T , 3/2 0 3/2 ~ V (2)
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Lecture 15 - Bose-Einstein Condensation Note that is valid...

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