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Unformatted text preview: [-r: ., ~,J J ~ ~< b~S
~ c. o ~ C~ ~ Y3 (~:~J J -+~NsQ~~
11 3. Fermi-Dirac Particles (20 poin~)
We will see later in the course that two half-integer spin particles cannot be in the same
quantum state (Pauli exclusion principle). Therefore in one state s o f energy es per
particle with one particular spin orientation, we can have either zero or one particle. Let
consider such a state in equilibrium with a much larger system at temperature 't', with
which it can exchange energy and particles.
a) (3 points) Write down the grand partition function o f this state as a function o f
es, 't' and the chemical potentialp..
~ :..t< c e - t b) (3 points) What is the general expression o f the mean number o f particles in a
state in term o f a partial derivative o f the grand partition function? Justify
your answer (e.g. does not just copy it from your notes)
( ~s -f< J ~
1\[", _ (2.:> - l-<-1t.ls
- r a eo~ .J:l
~ os s" I '- ----G--d -;=. ..~ ( ) +-,-. < N~ > _ L S -' N S o~~
0 \-L c) (4 points) Deduce from this result that the mean number o f particles in state s
with energy Es per particle (and one spin orientation) is for half-integer spin
particles (Ns )= e1
exp ( e Ns":> ~ L...
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- Spring '06