Midterm 2 Solutions 09

# j j bs c o c y3 j j nsq 11 3

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Unformatted text preview: [-r: ., ~,J J ~ ~&lt; 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&lt;­ 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&lt; J ~ \rv.. O_ov-. ~_ ~ e =c ov-- ~ ' (' -0 S 1\[&quot;, _ (2.:&gt; - l-&lt;-1t.ls - r a eo~ .J:l t:':::' e or:; ~ os s&quot; I '- ----G--d -;=. ..~ ( ) +-,-. &lt; N~ &gt; _ 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&quot;:&gt; ~ L...
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