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Final(3)

# Final(3) - STATISTICAL MECHANICS Final Exam — Spring 2003...

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Unformatted text preview: STATISTICAL MECHANICS Final Exam — Spring 2003 PROBLEM #1 : (10 points) Use the grand canonical ensemble to ﬁnd the expression of the entropy for a gas of non—interacting particles obeying (i) Bose-Einstein statistics, and (ii) Fermi-Dirac statistics in terms of the mean occupation number of a single—particle state with energy 6, (n6). Hint: In the grand canonical ensemble the entropy is given by 61n(Z) 8T s = kT( >zy — (N)kln(z) + 1.3111(2) where Z and (N) are the grand partition function and the expectation value of the total number of particles, respectively. PROBLEM #2 : (12 points) Consider a two—dimensional ideal Bose gas. Let V = L2 be the area available to the system. The number of particles (which is conserved) is given by a N = 25 ln[Z(z, V, T)] = Z[z_1eccp(ﬁep)— 1] _1 , where Z is the grand partition function. (a) Consider the system in the thermodynamic limit and discuss the importance of the p = 0 state. (b) Show that there is no Bose-Einstein condensation at T 75 0 for a two—dimensional ideal gas. (c) What is the groundstate of the system? PROBLEM #3 : (8 points) The atom H 63 consists of three nucleons, has spin 1 / 2 and is a fermion. The density of liquid H e3 is 0.0819/ cm3 near absolute zero. Calculate the Fermi energy GP (in meV) and the Fermi temperature TF (in kelvin). h = 1.055 X 10‘27erg/s, mnucleon = 1.67 X 10—24g7 k3 = 1.38 X 10—16ergK'1, 16V 2 1.6022 x 10—12erg. PM Exam ~ Sew/m L__mLeem#l Expom 6 W112?) wwng £142 = —Zﬂw (Hug/‘5) S=L§§E§Faeh0+a2521 V‘<”€>ﬂh2+;£u {Macaw} =LZg§£ - 's “MOI“:‘L é£“('+”£ﬂ£)] LT 213," +a (“0 - ’ I—a<n 2EF5+a "9 PE-A? :2" <h€>p> l+ﬂ3€~P£= I [—0019 ’IBE : a= —l S = L 26 2 (010% I». ((+049) ~<we>juzng>} P1) : a=+l ’8»?an #1: ’ 2 I _ La. 0° (0:) N ~51? \$311027?) ——r—'8P?/2h_l + .3 2 ' 9 T :2 %_ ZWMLT dx_______ +_'______ :ZFI‘J 31(2)+ET:2_ =n - 1‘2 o ‘1?”qu L3 1-2 L1 If. 2<| 4h Woglawm :‘nﬂd’a‘r—oﬁtﬁ Isaac/wﬂﬂm- Ja/wwvicw V° . ~x de i'ei—I =—ktxe'f-z" =45”? :— n = lwkzLTpLo-apﬁrg 1 (\1 a ()A‘l’ T=0 a,” Wthﬂa=0daih7t¢ MM :wsivh a: a“ M'drzf/‘n ﬂu. molwml‘cf Hum, T; J36! ...
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