Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on the blan sheets provded, wrtng your name clearly. (You may eep ths exam.) The varables have ther usual meanngs: E = energy, S = entropy, V = volume, N = number of partcles, T = temperature, P = pressure, µ = chemcal potental, B = appled magnetc feld, C V = heat capacty at constant volume, C P = heat capacty at constant pressure, F = Helmholtz free energy, G = Gbbs free energy, = Boltzmann constant, h = Planc constant. Also, represents an average. e x2 dx = π, x 2 e x2 dx = 1 2 Γ 1 2 = π, Γ 1 ( ) = 1, Γ( z +1) = zγ( z) π, x n e x2 dx = 1 0 2 Γ n + 1, ln N! N ln N N, e x = 1+ x + x2 2 +... 2 E = TS PV + µn, E = T 2 ln Z, F = T ln Z, T x H x j = δ j T, p = H q, q = H p
1. In the Ensten model, a sold conssts of N atoms, each vbratng wth an angular frequency ω. They are regarded as dstngushable, so that the partton functon s Z = z N, where z s gven by the sum over the states n of a sngle oscllator wth energes ε n = n + 1 2!ω. (a) (5) Obtan z n a smple form, usng 1+ x + x 2 +... = ( 1 x) 1, and then the partton functon Z. (Is the functon you are summng wthn the radus of convergence?) (b) (5) Calculate the thermodynamc energy E n a smple form. (c) (5) Calculate the specfc heat at constant volume, C V, n a smple form. (d) (5) Assumng that ω does not depend on the volume V, calculate the pressure and the heat capacty at constant pressure, C P. (e) (5) Show that E = NT + O!ω T 2 for T >>!ω (wth no frst-order correcton n!ω / T ).
2. (a) (6) Wth V and T treated as the ndependent varables (and N held fxed) show that ds = S V dv + C V T dt. (b) (6) Use the expresson for df (where F = E TS ) to obtan the Maxwell relaton nvolvng S P V and T. (c) (6) Usng the fundamental expresson for de n terms of ds and dv, show that de = T P T P V dv + C V dt. (d) (7) For a van der Waals gas, wth the equaton of state P + a v 2 V Nb E V T n terms of a and V. T T V ( ) = NT, v = V N, obtan
3. Suppose that the entropy of an deal quantum gas s gven by S = n ln n ± ( 1 n )ln 1 n ( ) where n s the average number of partcles n state. Here the upper sgn holds for fermons and the lower sgn for bosons. Also suppose that the partcles n all the other states act as a reservor for the specfc state, so that and n ε are both fxed constants, whch can be treated wth Lagrange multplers γ and β respectvely. (a) (5) Demonstrate wth clear arguments that the entropy per partcle goes to zero for fermons n the ground state. (b) (5) Demonstrate wth clear arguments that the entropy per partcle goes to zero for bosons n the ground state. [Show that S / N 0 as N n the ground state.] (c) (5) By maxmzng the entropy, subject to the constrants above, obtan the equlbrum value of n for fermons n terms of β, ε, and γ. n (d) (5) By maxmzng the entropy, subject to the constrants above, obtan the equlbrum value of n for bosons n terms of β, ε, and γ. Do these results mae sense?
4. Consder the average of the quantty dg over a tme τ, wth τ and dt G q p. The q are a set of coordnates, and the p are the correspondng momenta. We assume a classcal system n whch the magntude of G s bounded. (a) (5) Show that u p = q F where u s the velocty and F s the force correspondng to the coordnate q. (Here, and n all cases where you are gven the answer, your arguments must be clear, complete, and convncng.) Use ths result n obtanng the results of parts (b) and (c) below. (b) (5) For a planet revolvng around a star (subject to only the Newtonan gravtatonal force of the star), obtan the relaton between the average netc energy K planet and the average potental energy U planet. The moton s nonrelatvstc but the orbt can be hghly ellptcal. (c) (5) Consder a star revolvng at a dstance r around the center of a dar matter halo wth a sphercally symmetrc matter densty ρ r ( ). The matter enclosed n the sphercal volume of radus r s r ( ) = dr '4πr' 2 ρ ( r' ) M r. 0 The observed average velocty of the stars around a gven halo s found to be equal to a constant v 0 wth respect to r, out to large dstances from the center. Determne M ( r) as a functon of r (nvolvng v 0 ). In ths part you can assume crcular orbts for smplcty. (d) (5) Now let us swtch to a dfferent applcaton of the vral theorem: Consder N partcles that are confned to a volume V by a pressure P. We frst wsh to obtan q F = q F ' + 3PV (1) where the forces F ' are due to nteractons among the partcles (and represents an average over partcles as well as a tme average). Let F!" walls be the total force on the partcles due to ther collsons wth the walls enclosng the volume. Start by assumng that N r! F "! walls = P r! ds! where S s the surface enclosng V and r! s the poston vector of a partcle. Show that ths relaton leads to the equaton labeled (1) above (gvng a clear mathematcal argument). S [next page for parts (e) and (f)]
(e) (5) Now suppose that F ' = U q and that the potental energy U s a homogeneous functon of order n of all the partcle coordnates: For nonrelatvstc partcles, show that U ( λq 1,λq 2,... ) = λ n U ( q 1,q 2,... ). K = constant U + dfferent constant PV whle at the same tme obtanng the constants. Here K s the mean value of the total netc energy and U s the mean value of the total potental energy. (f) (5) Fnally, consder an deal gas (wth F ' = 0 ). Usng the equpartton theorem, plus Hamlton s equatons, obtan the equaton of state relatng P to N, V, and T.