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1 1. Consider a dilute gas of particles in the atmosphere. Near the earth's surface, the force on a particle of mass m may be taken as a constant, F = -mgj, where J is a unit vector in the vertical direction. a.) By enforcing the number constraint on the ideal gas ((N) = LsfCEsIJ1.1 r), with fcesi J1.,r) = Exp (~) Exp ( - ~ )), derive the expression /lint = r log (~) for the chemical 3/2 potential of an ideal gas. The quantum concentration is nq = ( 2::2) (5 points) b.) Write an expression for the total chemical potential for a gas particle in terms of the internal chemical potential and external chemical potential. (5 points) c.) What conditions must be satisfied for the total chemical potential to be uniform with height? (5 points) d.) The density of particles at height y = 0 is given as n(o). Calculate the density n(y) of the dilute gas of atoms of mass m at temperature r as a function of the height y above the earth's surface. (10 points) (' ( ) /A / '7-- -~. /rtv -=- 2..f.( 'i':j",...}{, 7 )..,... ~/~/l' ::. e e ~, 'I J 7'LL ~.-.:... ~ or-&.. C~'ca:.( PJ~~...-u ~.~ Irn(>~ em. ~ r,.rf6r...'r Je.L ~ eavo Ii. i-/-er.c.e. N ~ e:n- J: «':': =. e./.( ('f i. wh~ ~ I ir Jo- P(V~ -h 6A ~t"-c.4<'h..~ a. f (f1\j\4... rar-~c.ll I;' a. bcq( 0+ V"\~"'I~ V d"j.... L _ ~ + JJ _ ('no. f( M... I 1:. hi(l... ~.J..tL Sr "("' I.L ~...1,-. n V wk. n.n V\ J-4... Cl:.CA<f I\.~It\ CC~(~ ~ ~(- ~ I -{ 'J. )-<1'1 V AI:::. e. (\~.rutyil +v- /-"-; /ph- -::10lf (f\/''\g) -) ~::: r 1'1 (1\ If\ r.,) o.s. P. f\l ~ C-~I.\u.1 pc ~ v\ {f\(jj~ V\p c+ (i\.fe,.rj (t~j6j~ tf",j e,(tv"j (1@.vr-\z~t'~o.( po~-hd fb\ea7i) p~nh: ~:::. /"'''+ -+ ~J.-. 'It»: Ir,-\-6r~ Qo r J- J. j v./e,... In (t(j. 7l.L.- e»:+e.-.r ~ fa rj- C;] ~ du rje... (At\(~fJI. 5~vl~ kfejj ~ LA =- t'yi.ff-y:=~. ~~ -:: <f (~ (n(f\q) + 1Y\;J7 1k... d' (Y'\~f+ bll f~ +k:.r~ <t'\t:-t ~w~ ~"/lbrtcam. OJ<!.r t'ch Ae(fhij, ~ «(\~ -1- al"{'(~ 7' 7lL CC(l~a-\ ~ ~ r\ar~ -e, c:.. Y\ ( co) c~,('fj 1 't 0 w-i'-i{ ar-6 (cha""'{ l' :J Iv.e,f.i -r IdfJ (r-lc) / f\&() + Q sz r I~ (Y\(,J /I\q) +- fy\17 ~~/VL ~ V\ ("I) -:: f\(") e-rn~>,/'(-
2 2. Consider an ultra-relativistic dilute gas of non interacting Fermion particles contained in a three-dimensional cube of volume V = L 3. The particles have energy E» me", where m is the rest mass of the particle, so that the energy is given by E == pc, where p is the momentum and c is the speed oflight in vacuum. Note that the momentum p = :h.j nj.:+ n~ + n~, just as for the non-relativistic case..a.) Calculate the Fermi energy of this gas of N particles. (15 points) b.) Calculate the total energy of the ground state ofthis gas. (10 points) ~ d--l.l 1("IN. >f\ J... s~ (7 " ~) a (( S'~ are... ace f/111~ LfJ cj.o P q r-a. ~ll hvj~.j~ d~ v..f\-occvf('~. 7le... bch1 8-7evf LA(?- ~) -:;: 1 L e (1\'1-, fir J '~l: ) :1 -c ; 41i'" J J...~,...'- ( n) ~~/~/~~ 0 ~ n F fi r 1'..'\
3 3. Consider a one-dimensional transmission line of length L on which electromagnetic. fy h d. I. il2 E 2 il2 E he he electri waves sans t e one- imensiona wave equation ilt 2 = V ilx 2 ' were IS tee ectric field component and v is the speed of light. a.) Show that the nthmode of the line has energy hw n = mrv / L. Hint: The modes correspond to having an integer number a/half wavelengths spanning the line. (5 points) b.) Find the thermal average occupation number of a mode of energy iuo, assuming that the photons are in thermal equilibrium with a reservoir at temperature T. (5 points) co) do) Find the thermal average energy of photons on the line. Hint: Express the sum. J,00 U du 112 over states as an mtegral on energy, and note that 0 -u- = -. (10 points) Find the heat capacity of photons on the line. (5 points) e -1 6
4 G) Ik ~rr&. a.vewlct- enu9y ~ r~~ OV'- ~ f,~ ~ ISf ()=? <5(V,j)~{.Io wh"r<- <,(r,>.)-= ~W.I1-_1 kt I1tI lit:: "",w",::: ~ ~ V n -Y..,..., Lr JUt ::: 'i'1tv J" Lrr: C :::~I :::. L 61" L
5 4. The distribution function for fermions (+) or bosons (-) is given by 1 f(~) = exp[ (s - u) i't ] ± 1 a) State clearly in words the definition of the distribution function. (5 points) b) Give a physical explanation of why the + sign corresponds to fermions. (5 points) c) Particles confmed to a one-dimensional line of length L have orbitals labeled by n with energy e = ~(7r)2 n 2 n 2m L where the n are positive integers (1,2,3,..). The particles are spin-i12 fermions, so that each orbital can have at most two particles in it. IfN particles are placed into this system, what is the Fermi energy ~f? (5 points) d) Derive the density of states D(~) for this one-dimensional Fermi gas. (5 points) e) Write general finite-temperature expressions for <N>, U, and C, involving D(~); do not evaluate them. (5 points) w) c) ci 0-1k- c.u~r~, \1(.-h6.Z..f(r) V\ em-. JA~~ a.v~ OCC'1'adi4' z: (~~~ ct~ ~S!.M~ ~1il{6r/(),~ w# e-(. I are9- cl-evr(t;~ 0o+~+.;J /_ 41-?-::::6.J..& cv1<fnbvjf16a - l'c-f,~ ho '.-f11m- l~ (-F::;..1).~ 0. u.rckj<- 0 f.e" f?t rqervolr ~ ~R.~'r<. ~<2rT/e, be.jlj...i~ ~""(tj:l ~D~/ O{lt ~d; S'~ (f~) A- aff -rm uj/., &/ 2rf{ =: JhOI\ ~ C.JefNL~ ~Clde.-t~.r 'JA.i;.; ~ ~ - 'fev1i~~ f0 r('( ff2a (q,j. ~ ~e.rrr Jf.Jflc. h'yk't OC(~ NJ I'M 4 ~ F<2-f vr-i' V\~ r. 12.-e..- h~ k'01m o..~ ~ chjj2. ~ z.;?t!1r-hc:1l.s; FtOr ~ S~f"~ +0 h,,\t9-. rv ~ar-hc~ J Jk.- Ov-~t~ c.~{:.iwl la(i Jv \l\f,.me<'"".,..jl b~ Ai::= LY1p '""fk.. ~rm' ~tuf( if J tv~ 0>- f:f = ±."L t~)2jf\~ c:= ±:'(7r!1.):2 2...M I- r "?ffl L <~ ~f~ c2- ~ IA..if) [0" be.- ~u..j.. ~ ~ ~-hi,,,,",~~tzrc;-f ~ e, W7,. SoJu. -r J6L er-.dv.;.. ~ n I;' 1-e:rN\.J ~ '2: ~ ~O\VC-- y\::: ('2.;'1-L \((1- l::... TLe... J-;J.J. V\'"-VV\bu- 4 s~ t.>{j dv ~~;. ) ir \1 ~ 'IJ ~ 2..", - ~ [7..m:f..,-r,}:;~.. \)-J~,{h s~ ~re- a~ \,.>~ ~ ( s: ):;-.!:: J 1.11"\\...L 7T.-h'\.R ~ '-1(J P4r-+ ((;) wi(!..~ dftw' 4, c,f s J.zk tr d-k a;.:j.c.. J (I'\t:ru, II'» evtchf? ol r.:)::= JIJ(-Z)/ J ~
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