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# Statistical Mechanics I PHYS 6107

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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 6107 at Georgia Institute of Technology - Main Campus taught by Toan Nguyen in Fall. Since its upload, it has received 36 views. For similar materials see /class/234282/phys-6107-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.

## Reviews for Statistical Mechanics I

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Date Created: 11/02/15

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J K Q V NAVGLWW Q 7 ww ggt A Comm M Clo w 37 v A kw RKMUX A x w A 0 MS W af N k awkwh w mfg m werwa SW y 5 JSVJa2dz 6 ma JdeW5oy J2 E 1 ix 4 3 d56w 63 QQVLw WE 4y J 363 11 4 9 V g T a 1me K N CXE 5 Amva H at 50y arm 36 d quVgtooJ 5 waNt H w wwwij 2 Ava 3 9 x562 WVjY LQL BD W91 a H W Mfg av in M M 1 ax KWQMAVW amp9 W3 39 AWgtlt WW gt Aw Cr j Z lt2 W J3 5 W tug w L I 3 x a M Q EVE Q LM 39 2 L c V7 x5 ME gm om V 5 Qh L m WTM Luzml b q 00 K 3 X5 ff 2amp3 Q an x 55 3 17A 2 7 Y L b WW M L L h 3 x kE M 3m w 2 2 as L gW MQ 6km OQ 0a v9 u W ORB QMltTlt VA E Aw KNQVQ machhr wQkbg gm I Wm b 00 CVOVVW w CWW Mym J avaYMmW 2 I 1 WA Ely MmL3 TOWL122 3 W OVA k M WWqu gt wwm a W V BENSLIL EXERI c H 39 739 7 it Eduationst erVJOl and KOJOD the dens39 matrix reads 1 r quot x N 2 i 2 77 1 klr kNgt Egg V CXPI 2 k 5M 0103 b As sligg ted above we also w to transform Equation IQ03 into the coordinate 1 gt x 5 5 l Fl FN 2 12 10104 Son i g I Here the closure relation l 1Ntz 39 fl 7 it humkN was inserted twice Byinsgrtion of the wavefun 39on and Equation10103 EquationlOlO4 becomes 10105 f f N quot 55m 5 quot1 V l 123 lt15 3 In the bracket expression 1092 for p iclejappears Therefore the matrix lement i 10105 is also simply the product of the cyanide matrix elements Exercise 107 Density matrix at a harmonic oscillator Calculate the density matrix of a harmonic oscillator in the energy and coordinate representations Study the limiting cases T gt co and T gt 0 Hint The energy eigenfunctions in the coordinate representation are mm W Hx I 2 IJnq e pl Ex 10107 OPERATORS 10103 the coordinate 10104 luation 10104 10105 II as 21 1 a 1 a natrix element 10106 1d coordinate 10107 1O tXhRCISE 107 DENSITY MATRIX OF HARMONIC OSCILLATOR 281 Solution with x xmw q and the energy eigenvalues Equot 2 50 n Use the integral representation d n Hx quot exp x2 exp xz exp x2 I 2 21u quot ex u leu du 10108 m p The density matrix in the energy representation is trivial 1 1 pmn pnamn Wlth 10 E CXP hw n 1quot n O 1 2 where ZT v 1 Tr exp HD Zexp w 1 25011015014 was already calculated in Example 81 On the other hand the coordinate representation is somewhat more dif cult to obtain I31 1 Zlt q l n gtn lbl n nl q nn I q Here we have twice inserted the complete set of energy eigenfunctions 10107 439 l l I Z 11Iq pnnl 1 7q 1 1 2 exp hw n 5 WTq 11nq 1 ma 12 1 2 Ta Pl 5 x2 3 Z exp n Hl1xHllxl n0 Here one now inserts the integral representation 10108 of the H L m quot2 1 2 a f f lt 2uvgtquot q Zn Tm expl2 x x we du 3900 do I quot0 n39 exp h w n exp u2 2ixu exp 12 2ix v 10109 DENSITY OPERATORS The summation over n can be carried out since 392uvquot l PlM 5 l I exp 5 Mm z 2uv exp B w n l exp 5 lm exp Zuv exp hw Thus Equation 10109 becomes 1 12 1 00 00 q39 q Z exp x2x392 ww dill0 dv exp u2 2ixu v2 Zz39x v Zuv exp w The argument of the exponent is a general quadratic form which can be also written in the form 1 u2 2ixu v2 2ix v 2uvexp w i wTAw 117 w ifwe put I Ho ho expl w 1 Now the general formula a 1 s fdquotwexp wTAwibw 10110 holds if 1 is an invertible symmetric matrix One proves this as follows At rst one substitutes th the vector 33 is de ned by 53 21 5 The Jacobian deter m absolute value I so that no additional factors appear in th 1 E e new variableE 112 13937 where inant of this transformation has the e integrand Then it holds that s1 wTAwzbw zzyTAzzyzbz1y H I 39ms1s EETAZ yTAzzTAy yTAyllbZ by 1 07210 diagAIt 10 itten in 0110 where as the 0 1 quotlt1 Z cond while gona EXERCISE 107 DENSITY MATRIX OF HARMONIC OSCILLATOR is a diagonal matrix with the eigenvalues on the diagonal With the new variable 3 0 one obtains 1 1 quot quot 2 dquotzexpl 27Az fdquotsexp 213 n 2 2 1 i1 i The J acobian determinant of the transformation 2 gt E is just the determinant of 0 which has however the absolute value 1 so that again no additional factors appear The product of the eigenvalues however is equal to the determinant of A since det A det 2160quot det 3quot10 det OTAOI det diagoth 1 hence Equation 10110 is proven With 2r 1 1 ax131 1351 21 CKPFZMM expi hw 1 deu i 41 exp 2 w one obtains ETA lb l exp 2 3iia 1 x2 x 2 2xx exp w Al 1 mw12 exp B w qlpq T mm x exp x2 x39z 1 exp 213ia1 x2 x392 2xx exp hw A 1 mm W 1 2 2 xx39 1 lplq E 2nnsinhwnm exPE x x thWwH sinh ha If one here exploits the identity 1 l sinh w tanh i w cosh hw 1smh3Ew 1 coshwhw one nally gets A 1 mm U2 1 p q 2 2 27m mumsm x exp q q392 tanh hw q q 2 coth 10111 The diagonal elements of the density matrix in the coordinate representation yield directly the average density distribution of a quantum mechanical oscillator of temperature T 12 pq tanh wgt exp ta h1350q2

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