A A A. p mu E mc K mc E p c m c. = d /dk. c = 3.00 x 10 8 m/s e = 1.60 x C 1 ev = 1.60 x J 1 Å = m M Sun = 2 x kg

Transcription

1 Physics 9HE-Mod Physics Fial Examiatio Mach 1, 14 (1 poits total) You may ta off this sht Miscllaous data ad quatios: Miscllaous data ad quatios: c = 3. x 1 8 m/s = 1.6 x 1-19 C 1 V = 1.6 x 1-19 J 1 Å = 1-1 m M Su = x 1 3 kg 1 M Eath = 5.98 x 1 4 kg Eath = 6.38 x 1 6 m m = x 1-31 kg =.511 MV/c m p = x 1-7 kg = MV/c m = x 1-7 kg = MV/c 1 u = x 1-7 kg = MV/c m( 1 H) = 1.78 u G = 6.67 x 1-11 Nt-m /kg g = 9.81 m/s = 5.67 x 1-8 W-m - -K -4 h = 6.63 x 1-34 J-s = h/ = 1.5 x 1-34 J-s k B = 1.38 x 1-3 J-K -1 a =.59 Å N(t) = N o xp(-t/ O ) = N o xp(-.693t/ 1/ ) k C 1/(4 o ) = 8.98 x 1 9 N-m -C - R H = x 1 7 m if v c = v/c = [( -1)/ ] 1/ 1/ 1/ (1 v c ) (1 ) T= T o L = L o / 1/ (1 ) 1/ [1 ] fo 1 = c/ (lct.) = f(book) (1 ) p mu E mc K mc E p c m c 4 T GM 1 1 gh GM E pc h ( if 1) R Sch T c 1 c c ch 1 max T =.898 x 1-3 m-k I(, T) R(T) = T 4 5 hc kt 1 qq 1 mv h = K max + = dsi Fcoul kc F adial a = h/p p = k xp x / Et / v ph = /k v g = d /dk ix 1 ix ix 1 ix ix cos x i si x cos x si x i si t = si t cos t cos t = cos t si t = cos t 1 = 1 si t a ( x ) a cos( x ) b si( x ), with k, ad Å m a av. of ov, a ( x')cos( x')dx', b ( x')si( x')dx' 1 ikx 1 ikx' ( x ) c(k ) dk, with c(k ) ( x') dx' Ĥ i ˆpx i t x Hˆ Kˆ V m x ie t i t Ĥ E Kˆ A A A x A * Aˆ dx  a a a

2 m(e V ) ikx 1 m(v E ) k si kx,cos kx 1 / x si E L L ml x x / m 1 H ( x ) E m J( J 1) MM 1 Y (, ) ( ) ( ) E I R J J J I M M J JM JM M J V si (kl) V sih ( L) E E L T 1 T (wh L 1) 4E( E V ) 4E(V E ) V V 3 / 4 m TFE xp ISTM dv 3 dx KL f T ; T xp 4Z.993 MV 8 ZR (m) uc coll 15 E (MV ) qq 1 mv Fcoul kc F adial a m Z 13.6Z 4 E (V ) a Z 8 a m Z M.59 Å v Z RH m m M m u x si cos y si si z cos dv d sidd (,, ) R ( ) ( ) ( ) R ( )Y (, ) P ( ) R ( ) m m m m It. ; = 1, m, 1 L s ia E B L B s s B m m B m fial iitial J / Tsla ( ) c ( ) ( x ) u ( x ), wh u ( x ) u( x A) MO ikx j Ai, j Ai K K K Atoms A Obitals i V i1 ( o - ) q i 4πε i i λ 1 ρ 1 1 F Axp( E / k T ) F (E ) F (E ) B xp(e / k T ) 1 xp[(e E ) / k T ] 1 MB B BE FD B F B ch 1 max T =.898 x 1-3 m-k I(, T) R(T) = T 4 5 hc kt 1 kf (m ) 1/ 1/ 3 (3π val ) EF g(e ) E k F (3π 3 val ) v F= m π m 3/ 1/ 3 3 Z( Z 1) (N Z ) B( X ) [Nm Zm( H ) M( X )]c a A a A a A 1 A / 3 Z Z V A S 5 4Ruc A R (1. 1 m )A Q M M c 15 1 / 3 uc iitial fial ---Ta off this sht ad bgi xam---

3 3 Physics 9HE-Mod Physics Fial Examiatio Mach 1, 14 (1 poits total) Nam (pitd) Nam (sigatu) Studt ID No [1] (5 poits) Asw th followig qustios with bif statmts o calculatios. (a) Stat th xpimtally vifid cosqucs of Gal Rlativity. Ay th of ths: Pcssio of th pihlio of Mcuy Gavitatioal lsig Black hols Tim dilatio o cotactio fo clocks i difft pats of a gavitatioal fild Fam daggig (b) Th so-calld L x-ays a mittd fom a copp atom i which a iitial p vacacy is catd. What tasitios fom th = 3 shll to this vacacy a pmittd i gatig ths x-ays ad why?

4 4 (c) A d las is usd to poduc a pfct siusoidal tavlig light wav of fqucy o = 1 14 Hz. Howv, by spcial mas, th wav is abuptly cut off at ach d so that it has a fiit lgth i tim of 1-13 s. If this fiit wav is ow dscibd i a Foui pstatio, what would b th appoximat ag of fqucis ivolvd? (d) Dfi dgacy i quatum mchaics ad giv o xampl of it fom th systms w hav studid. () Th positio of a lcto alog th x dictio is masud with a uctaity of 1 Å. What ca you say about th uctaity i its x momtum? What ca you say about th uctaity i its y momtum? (f) I th las, what coditio o lvl populatios is quid fo succssful opatio? Explai bifly with a lvl diagam.

5 5 Th must b som kid of populatio ivsio, such that a xcitd stat is much mo populatd tha som low-lyig stat. Schmatically, this looks lik: (g) A ctai al solid is kow to hav a lctoic bad stuctu that is vy f-lcto lik. Sktch o th diagam blow what th lowst bad would look lik, idicatig th spcific quatio fo it: E E = k /m k [] (1 poits) Th missig Malaysia Ailis Flight 37 was fo a tim obsvd o ada to b flyig at a altitud of 45, ft = 13,7 m, ad its spd was 8 m/s. t 13,7 m t

6 6 (a) If a giv tim tim itval t was masud by a clock i th aipla as 1 hou, what would b th itval t masud o a goud-basd clock by a obsv dictly ud th pla. Nd to cosid both Spcial Rlativity tim dilatio which maks t < t, ad Gal Rlativity spdig up tim du to gavitatioal itactio with light, which maks t > t. SR ffct ca b calculatd i limit of v << c. GR ffct ca b calculatd i limit of costat gavitatioal attactio du to low altitud. So w hav fo SR, fo which tim is slowd dow du to dilatio.8x t t ' 1.5 t ' 1.5 t ' x1 t ' x1 t ' 3.x 1 o t 1 4.7x 1 t' 13 Ad fo GR, with + sig du to fallig light i gavitatioal fild so tim is spd up: Factioal chag i t lativ to t = t ' GM 1 1 gh 9.81(13,7) x1 1.49x1 t c c x1 So t ' x1 t So GR is about 3x SR, ad w gt a fial atio of 1 t' x1 13 t x1 1 9 = 36 sc of 36x1.49x1 5.36x1 s x x x1, so with a diffc ov 1 hou (b) If th ada statio was dictly ud th pla such that th vlocity vcto was ppdicula to th li coctig thm, would ay additioal coctio b dd du to th Doppl ffct? Explai why o why ot, ad how this lats to you asw to (a). Fo a objct movig ppdicula to th li coctig it to th obsv, th is o Doppl shift, but i SR, w do hav to accout fo tim dilatio, which is alady do i pat (a). So th asw is o, w d ot cosid if futh. [3]) (1 Poits) W hav cosidd i lctu ad laboatoy th tipl-wll pottial, ad som of Rady s wav fuctios fo th lowst th stats with E 1 < E < E 3 a show blow: (1) () (3)

7 7 As a mo alistic xampl of this typ of pottial, cosid th molcul C 3 H 4, with th C atoms i a li, ad fo which a fw of its molcula lctoic wav fuctio pobabilitiy distibutios a show blow (but ot i gy od) (A) (B) (C) Th blu ad g cotous pst opposit sigs fo th givig that pobability distibutio. (a) Nglctig th ifluc of th hydogs, idicat which of th wav fuctios A, B, o C fo this molcul a coctd i gal oigi with which of th wav fuctios 1,, o 3 of th tipl-wll pottial, ad povid via this also you stimat of thi od i gy. Thy must hav th sam od i gy ad also b of v-odd-v paity, sic th pottials of th th C atoms (glctig th ffct of H) a v alog th molcula axis. So th colatio is: C is latd to 1, A is latd to, ad B is latd to 3, ad thy a i that od of gy: E C < E A < E B. (b) Agai glctig th hydogs, Idicat th bodig chaact ad th appoximat atomic-obital makup of th wav fuctio dotd by (A), usig a sktch. This is a bodig wav fuctio (of chaact, but this was ot dd fo a full cdit asw) ad aythig lik th dawigs blow OK. + C 1 + C _ C 1 + C _ C 3 C 3 + [4] (15 poits) A paticl of gy E appoachs a bai of hight V ad width L ad E is much gat tha V. Tak th valu of k fo th paticl i th gio of th bai to b som umb k. E = ikx ik A x C k k B ikx = 1 D ik x ikx E k 1 3 V L L x

8 8 (a) Idicat th fom of th tim-idpdt wav fuctio i ach o gios 1,, ad 3, ad th bouday coditios that would hav to b satisfid to solv this poblm. Foms of wav fuctios idicatd i dawig abov. Bouday coditios o cotiuity of ad d/dx a: x : A B C D ad ika ikb ik C ik D k( A B) k ( C D) x L : A B C D ad ika ikb ik C ik D ikl ikl ik L ik L ikl ikl ik L ik L (b) Show that th flctio cofficit R i this high-gy limit is giv appoximatly by 1 [V si(k L) / E ] ad thus still shows quatum ffcts. Not that (1 x ) 1 x fo small x. 1 1 V si ( k L) V si ( k L) V si ( k L) T E( E V ) 4E( E / V 1) 4E (1 V / E ) V si ( kl) V si ( kl) 1 (1 V / E ) 1 4E 4E But T+R = 1, so V si ( kl) V si ( kl) R = 4E E (c) What is th coditio o th wavlgth of th paticl fo which R is a maximum? Illustat two o-tivial cass with a sktch o th diagam abov. Th coditio fo a maximum flctivity a fo si(k L) = 1, o k L = (/ )L= /, 3/, 5/, 7/, 9/, 11/, 13/, = / + (), as show blow: sik L / /+ /+4 /+6 / /+ /+3 /+5 /+7 /+9 k L with =,1,, o L/ = ½, 3/, 5/, 7/, 9/, 11/, 13/, = ½ +, o L = [ ¼ + ], i which cas th ikx ikx oudtip distac of th two compots isid th pottial stp gio C D maks thm ikx just out of phas at th flctig bouday at x = with th flctd wav pstd by B, thus miimizig tasmissio. Two cass fo = 1 ad = lik this a show abov. [5] (5 poits) O of th wavfuctios fo th hydog atom has th fom: 311 ao ao 3ao i,, C 6 si, with C = som costat. (a) What is th xpctatio valu fo th squa of agula momtum i this stat? Do this o th asist way you kow how; thik igfuctio!

9 9 Just us th kow igfuctio latioship fo th H-atom wavfuctios (you d to mmb this), which givs fo th = 3, = 1, m = 1 cas show abov: L ( 1) (b) What is th tim-dpdt wavfuctio fo this stat? Spcify th gy pcisly, but you d ot valuat it umically. Just add th gy xpotial, to giv ie t / 13.6(Z 1) 311(,,,t ) 311(,, ), with E V 3 (c) How would you dtmi th avag adius of a lcto i this stat? St up th itgal ivolvd, but you d ot valuat it. * (,, ) (,, ) d sidd ao i i 3 C 6 si d sidd ao ao with itgatio ov ( to ), ( to ), ad ( to ), so this givs fially 3ao 3 C 6 d si d a o a o (d) Wit dow th fomula fo th adial pobability dsity P ( ) R ( ) fo this stat, ad th qualitativly sktch it, icludig a idicatio of th pcis positios of ay ods. Th adial pobability dsity will b just 3ao ao ao P ( ) R ( ) C 6 which will hav a zo at th oigi, ad at = 6a, ad fially go to zo at ifiity du to th xpotial. So a qualitativ sktch is as blow., P 31 () 6 1 (i uits of a ) Ad this is also show i o of th lctu slids o th xt pag:

10 1 (f) If a wav fuctio with th sam quatum os. was ow modifid i fom so as to pst o of th lctos i a A atom with a cofiguatio of 1s s p 6 3s 3p 6, to what appoximat scd ucla chag would it cospod? Th A atoms th has a ucla chag Z qual to th total o. of lctos o 18. A 3p lcto is i th outmost shll h, ad so o ca oughly thik of all th lctos isid of it, o blow it i gy, as scig it. So Z ff Z-1 = 6. 3p lctos would also b patially scd by th oth 3p lctos, so th actual Z ff could b stimatd as a low o. fo full cdit. [6] (15 poits) Cosid th uclid 49 4C x of chomium, with atomic mass of u. (a) What is x h, ad what is th bidig gy of this uclid p uclo? (b) This uclid dcays by posito missio to fom a uclid of vaadium = V??????. Wit dow th ovall actio, icludig th w uclid that would b fomd. Posito missio ducs th atomic o. Z by o, ad ovall actio is: C V (c) Which o of th fou fudamtal itactios is sposibl fo th foc btw th quaks isid th uclos of this uclid, ad what is th paticl mdiatig this foc? Th stog itactio is sposibl fo quak itactio ad th mdiatig paticl is th gluo. (d) How may total quaks sid isid th uclid 49 4C x? Each uclo cotais th quaks, so th umb of quaks is just 3(4+5) = Ed of xamiatio-