Crib_Prob_Set_2

Crib_Prob_Set_2 - CH M 3 7 o Finthhe-real and imaginary...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CH M 3 7 o Finthhe-real' and imaginary parts Qflhé'ffiflbifiihg quantities: iii; {2‘03- bi eff/:25 C. e-‘lHn/Z' d. fi+2064nfl a. — £93 2 8 — 121' + 6? ,~ :23- =_:%-2;.;._ 1415;. 116(2) 2 2;‘Im(z) :2 — 11 b. = nosey/2) + i mar/2) =1: '%@=mmm=1 c; e—Z—h‘mjl 1:. _—2 Reta): 0; 131(2) Z g—z 6‘ (‘5 +-353€":"’2 = 1N5 +- zii-):-sih-'(‘—‘€rr/;~2) -= 2 — Mi 2‘ Im(z) = elf-5&2! {fez = .x + Qiy, then find em? - €743 33' Rea-‘5) b- Refit?) '9‘; ImC'zZ) d... Reta?)- e. Im(zz*‘) 1.167(6) = Rc(x-— 2in r: x 33(222) = Rc[(x + 20):] :Refixz fxéfiscykrflayz) 2 x3 — 4y? 1111(3) = Im[(x + 2:502} 4x94 - "Ram =-.R.e. {(x + 2mm“~2=ay3:1;;-?-Rzew +4321) = x2 + 4),:- ImEz-z"): Im'{;rc2 — 4})2) =..Q;;,—.__ujQ-. >?# «:99 Show-{hag and that. -Usi5n_grEquaL.ion A6, a“; --: 60529 i sin 6' ,f‘” = cos-79 — i sin 9 Adding-these two exPrcssions gives: ejg +~efw = 2 (:08 E9 eff? +6 a“? ._ V 2 7 and subtracting the first two exprcssigfisigivasi == {30.5.9 "9“" ~ g" '*‘-§?:;-i€ . '——-—-' 4" ‘ =:' -s;1n 9 m ‘9 ,= 21' sin ‘9 A‘—9. Consider the get 6f'ffine'tions 1 . . . . ‘ wm : “HG-viii. j 0 S; 05 271" 3 First Show that _‘22r dafiqlm Eqfi} = 0 for all valfieflfm :35 0 IO. = 231' m=0.. - Now show that 2n f d¢¢;(¢)¢n(¢i)=0 mail-"n -o '='I' mun (but?) =' “b m = 0.. stile-3&2, . 0 s 95 s 2;: Let m = 0. Then '2” l 27: a .- —d = -'— =' ' 27:: [0 «HI: G5 «121? ' Ifm. 7‘ O, . II: I. __ 2n 1 I .12: r _ - 6""?d'q5: f _ cosm¢d~¢+ , si'nmqfidrb (l) [0‘ «J-"Z-zrf’ ' 'u s/er _jp «#27: Each integral. in i— 'i'Sié qua! to zero, because they are evaluated over one full cycle of the funcfion.'Now Consider ' I 2x L l / d¢¢;(¢)¢n(¢) 2 — dgse-werw 0' 2.}; .0 l 121 = _ _ 5(h-mj‘ifir 23 , ¢e If :1 :,£ m, we-carj define: 11: a k and the resulting integral-i's}iidefi.tieal'to that in Equation l.,_-afid so has a value-of 0‘. IE): : m, 1 2” f d¢¢;;(¢)¢m-(¢)=2— ] d¢=1 .0 7? 0- .- "Consider a particle to beéconstrained to lie along a one-dEmensiongl‘segmentO to a. We will learn in the next chapter: that the:probahiglity'ithat the particle is' found to 'lié'bet-ween x and x + dx is given. by i 2 ._ _ _p(x')'dx = — sin2 373-4; . a a Where I: = l, 2. 3, .. Firg'l Sliow'thatj 30:) is normalized. Now calciulate the average position 'of the particie along- the: line segment. The integrals Lhatyou naediate' (The CRC Handbook of Chemistry and Physié‘s of Standard MathematicaiIab‘fés,iCRC Press) - _ s‘n-Ea-x fSi=n2ticxdx = E — 14a ‘ and ' ‘ .l d,,.._________ [x51] WINK—4 If {10:} is normalized, thong infjdx = l.. 2‘" 7, “6.2 , pixj'dx 2 f 1* sin‘ ’1de in: " in at a _ 2: J_c _ sin2n'n'ag'lx, u W Va 2 4mrg‘l' ’0- 2‘[a sin2rm . Sin-'0 ] 0+ u, 2 4111151" aim-4% =§9m Thus, p(x) is normalized. Toffind the-average position of ihepartic‘le along theline segment, u'se Equation 3.12: r a. £2 a 'tf' a 9 ‘ (act; were: f xieinz'W—xdx D- ' 0 [95'2 x ’s‘i‘nflmm ‘1 .1: cos 2rzrc'a‘1'xf:|“ M6: in .I' 1:: H9 ‘2; IN firm-a” B'n'zrrgd7 o 2, 1 + i :4 gfl'z.1§2£1_2 iiiizntzgz'“2 Fla. B'—-2. Calcm'ate the variancfij assaciazed with the j' nacessary in‘tg‘gral- _i_s'_ (CR6 30518;) 1- 2 - - - . x: x 1 7, , xcosla; [4'2 SL112 Elf-161x: -6— — (—M - Sl.n.23;2:,_;;. — . ' 2 USE: Equation B..1:_3: ,6 2 u {x2} = x2p(x)fdx = .—.f x2 3m2 mdx i 7n " a o H 2 1x3 ‘ f I -S__, 2 L, x‘c0812n2f0"x “ 2 =-— -,——e m mm xr-«u -, ; a 6 = énrjqfl 8:1‘12r3c1"2 ' ,_ 43-2314 0 ‘ g3 ' a2 l , aCQS}2n7r =7 "7*- _l— 114 sm2n — 21:2_2 ! a - 6 .--4n-nra 8n Jr ‘ 4n 3: a i _ .. : ._- . #1 ) * “a I 67 ting]? 612‘ 0-2 = 3— — 22:17:71? The variance 02 is given by (72 = (x2) —. (9;)2 (333' ; Using the result of'Problbm B—~"l' and the above result for {x25 gives a a 2/1!” .. zflflx' . = — 5111 ~— x a 0 a _ ' . _-j .11]? __ 2 x .stmrm 'x -‘ m d . 2 I - 4n‘ma'."" w; 2 a sin 2m: sin-0 _- a 4 Enact“ ’ AHJPdL' .2 - where m is the mass of the pa‘rtiie‘leé EB divided. by the AVOgfldi‘O'COnSEEiI-IU; and." mo.lecu1ar'speed_s is called the .Ma.x.well=B.b and _Lhern.-‘d6termine the average speed " (CR-C -.table'.§) where 1‘1"! is nfactor’i'al, or :11 First, -‘ = “(in -—~ 2139:.- — .23 . .(1) W¢=d.€m0n-strat.e: that 1700 is. nqrimalizedtby showing is the Boltzmann constant (the molar- gatis Con‘Stah-t- .R ‘» ".isv‘th‘e- Kielyjin temperature. The probability- distribut-io’n of , "itgzmann distribution Firstshow. that p§1j)sis nonnaIE-zedi as a function. of temperature... The necessary integrals are that jhm P (0)111): -.: 1.: - [27.2) “M I Wm 13% (R72) ,we 9% ‘ 3/" k, ._ v2; ' <V2>ZLILM£KRT> LV+€MAK9T01V __ m yag-lKgTzlfiKng/l uq—fiLlfikeT) \m m ) __ 2m “ W J: I___,__..—- 30 <\!2 "::: 3m 2 Wm When TEZWK v V " (Vii/9': l 3HT Z 3XI‘33’ilo'fijk’ixzoOK (mg ‘- 2 3" 0 1m“ 7L {‘66)lean K3 ' amwi ’" 5 ._ zflbflo «ii a S”; ._ W Since 3 :KJ m ‘— ' mg _ K3 :— Lflzm-fl ‘/h '__ :— >< 2:1,, .. w M. , WW3“ T: [000 K 3 “we +31% Same Mei/Twig 1/4” Wm 3<V15 : 1H0 mfg/Ii] . [3—5) F6“ at“, €553 2 We “£61470 6(6de Gm @Q'Prdjflm Hr {/511 ' wine-re I New r- M/‘l W] § I r— Nécoc) 1 A“ [ a‘bfi] : AMP The 0130119“ (A; {3 “Hat/n ’51 5U} ’ 3W +90§MJ __ 0&7) 9'“) — +M£lo§<¥+gmf§g§+o< Afiflt 9mm 3 xéfi—Mfiqu’lfir) 3’s 3’ w www 00 31: ROE: +19<§§~ +( +7? 5? Clj‘l‘f'cx) :- CE‘f’GX) - I M '0‘ awn-1m :LKCOSCRM) M*-‘E”005( A“ 01 0. 0 0L ' a ‘ (mmm 6‘ Z: 1 51-} (“M “fix I H l _7 5m 0L o 6k D a I " ‘ ._ I S’L‘nerm ‘ mummmm‘m)" mmm m I 0 19%“ finNfi =0 £mr sentejeY vms oz? A! » Naive flfWE 13f ncm ,aL R {L infiac r { fifimficnUA/M : .Z‘JDAOC «— 0:25 a M D mi 99 «the 77mfidfh-‘Cfl—a- b626- Ww’f cure armoflo'rmal , 33x2me PWLW m flamed mwmodflmcs (A) Acmddnj {:0 Jam defiame oO[ ymfiwéem Mfi‘m 1C7) : 621:? exp? _. 350 '3'— QXP[*%]+ZG7CP[-fi] +3 fl “Fr” _ . _, . , ,2; 24'» 3Q“ G} i”) “ exfl' K. "('ZW’); 4&3??? .__ -\ “9- -—3 -—€ +9.6 +?€ =- mm I _ ~~ . e” o 0 n) K . . __ (30v 1 ‘ C $chng cg 31W SW __ guzélfig? % / {+6 97% ...),_ 0 meim cfi I;t M’ififl. 9W : 26 7¢ID°% :1 3435/” €+2e‘1+3€"3 «3 g ficwfiimvc? 2ML mike-0L SW76 :_ 36 “374(an z; l-‘Kt‘l 5'/ waffle LC) when T ’>"’° get, 0 ~ , 42 ,2}: WW“ r" = 65 am flaw WNW H} U") Exam :15; giownak 9cm : rig—axle??? 2 NMHA erczcetw 05' Pt wfiwkwic I Z )Llsz: 3233?; BMW“ 0g? 1“ (am/Heck -:. fifxlwz :_ EWWKI ...d- ..._- 6 ...
View Full Document

  • Fall '08
  • OldSleepyMan
  • K3, imaginary parts Qflhé'ffiflbifiihg, 2‘03- bi eff/, Standard MathematicaiIab‘fés,iCRC Press, nacessary in‘tg‘gral- _i_s, Toffind the-average position

Page1 / 10

Crib_Prob_Set_2 - CH M 3 7 o Finthhe-real and imaginary...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online