Quantum Physics ANSWERS TO QUESTIONS CHAPTER OUTLINE

Transcription

1 CHAPTER OUTNE 8 Blackbody Radiation and Planck s Tory 8 T Potolctric Effct 83 T Compton Effct 84 Potons and Elctromagntic Wavs 85 T Wav Proprtis of Particls 86 T Quantum Particl 87 T Doubl-Slit Eprimnt Rvisitd 88 T Uncrtainty Principl 89 An ntrprtation of Quantum Mcanics 8 A Particl in a Bo 8 T Quantum Particl Undr Boundary Conditions 8 T Scrödingr Equation 83 Tunnling Troug a Potntial Enrgy Barrir 84 Contt Connction T Cosmic Tmpratur Quantum Pysics ANSWERS TO QUESTONS Q8 Planck mad two nw assumptions: () molcular nrgy is quantid and () molculs mit or absorb nrgy in discrt irrducibl packts Ts assumptions contradict t classical ida of nrgy as continuously divisibl Ty also imply tat an atom must av a dfinit structur it cannot ust b a soup of lctrons orbiting t nuclus Q8 (c) UV ligt as t igst frquncy of t tr, and nc ac poton dlivrs mor nrgy to a skin cll Tis plains wy you can bcom sunburnd on a cloudy day: clouds block visibl ligt and infrard, but not muc ultraviolt You usually do not bcom sunburnd troug window glass, vn toug you can s t visibl ligt from t Sun coming troug t window, bcaus t glass absorbs muc of t ultraviolt and rmits it as infrard Q83 No T scond mtal may av a largr work function tan t first, in wic cas t incidnt potons may not av noug nrgy to ct potolctrons Q84 T Compton ffct dscribs t scattring of potons from lctrons, wil t potolctric ffct prdicts t ction of lctrons du to t absorption of potons by a matrial Q85 Wav tory prdicts tat t potolctric ffct sould occur at any frquncy, providd t ligt intnsity is ig noug Howvr, as sn in t potolctric primnts, t ligt must av a sufficintly ig frquncy for t ffct to occur Q86 A fw potons would only giv a fw dots of posur, apparntly randomly scattrd Q87 T -ray poton transfrs som of its nrgy to t lctron Tus, its frquncy must dcras Q88 igt as bot classical-wav and classical-particl caractristics n singl- and doubl-slit primnts ligt bavs lik a wav n t potolctric ffct ligt bavs lik a particl igt may b caractrid as an lctromagntic wav wit a particular wavlngt or frquncy, yt at t sam tim ligt may b caractrid as a stram of potons, ac carrying a discrt nrgy, f Sinc ligt displays bot wav and particl caractristics, praps it would b fair to call ligt a wavicl t is customary to call a poton a quantum particl, diffrnt from a classical particl 77

2 77 Quantum Pysics Q89 An lctron as bot classical-wav and classical-particl caractristics n singl- and doubl-slit diffraction and intrfrnc primnts, lctrons bav lik classical wavs An lctron as mass and carg t carris kintic nrgy and momntum in parcls of dfinit si, as classical particls do At t sam tim it as a particular wavlngt and frquncy Sinc an lctron displays caractristics of bot classical wavs and classical particls, it is nitr a classical wav nor a classical particl t is customary to call it a quantum particl, but anotr invntd trm, suc as wavicl, could srv qually wll Q8 T discovry of lctron diffraction by Davisson and Grmr was a fundamntal advanc in our undrstanding of t motion of matrial particls Nwton s laws fail to proprly dscrib t motion of an obct wit small mass t movs as a wav, not as a classical particl Procding from tis rcognition, t dvlopmnt of quantum mcanics mad possibl dscribing t motion of lctrons in atoms; undrstanding molcular structur and t bavior of mattr at t atomic scal, including lctronics, potonics, and nginrd matrials; accounting for t motion of nuclons in nucli; and studying lmntary particls Q8 Any obct of macroscopic si including a grain of dust as an undtctably small wavlngt and dos not ibit quantum bavior Q8 f w st p m q V, wic is t sam for bot particls, tn w s tat t lctron as t smallr momntum and trfor t longr wavlngt p Q83 T intnsity of lctron wavs in som small rgion of spac dtrmins t probability tat an lctron will b found in tat rgion Q84 (a) T slot is blackr tan any black matrial or pigmnt Any radiation going in troug t ol will b absorbd by t walls or t contnts of t bo, praps aftr svral rflctions Essntially non of tat nrgy will com out troug t ol again igur 8 in t tt sows tis ffct if you imagin t bam gtting wakr at ac rflction KJ T opn slots btwn t glowing tubs ar brigtst Wn you look into a slot, you rciv dirct radiation mittd by t wall on t far sid of a cavity nclosd by t fitur; and you also rciv radiation tat was mittd by otr sctions of t cavity wall and as bouncd around a fw or many tims bfor scaping troug t slot n igur 8 in t tt, rvrs all of t arrowads and imagin t bam gtting strongr at ac rflction Tn t figur sows t tra fficincy of a cavity radiator Hr is t conclusion of Kircoff s trmodynamic argumnt: nrgy radiatd A poor rflctor a good absorbr avoids rising in tmpratur by bing an fficint mittr ts missivity is qual to its absorptivity: a T slot in t bo in part (a) of t qustion is a black body wit rflctivity ro and absorptivity, so it must also b t most fficint possibl radiator, to avoid rising in tmpratur abov its surroundings in trmal quilibrium ts missivity in Stfan s law is %, igr tan praps 9 for black papr, for ligt-colord paint, or 4 for siny mtal Only in tis way can t matrial obcts undrnat ts diffrnt surfacs maintain qual tmpraturs aftr ty com to trmal quilibrium and continu to cang nrgy by lctromagntic radiation By considring on blackbody facing anotr, Kircoff provd logically tat t matrial forming t walls of t cavity mad no diffrnc to t radiation By tinking about insrting color filtrs btwn two cavity radiators, provd tat t spctral distribution of blackbody radiation must b a univrsal function of wavlngt, t sam for all matrials and dpnding only on t tmpratur Blackbody radiation is a fundamntal connction btwn t mattr and t nrgy tat pysicists ad prviously studid sparatly

3 Captr Q85 T motion of t quantum particl dos not consist of moving troug succssiv points T particl as no dfinit position t can somtims b found on on sid of a nod and somtims on t otr sid, but nvr at t nod itslf Tr is no contradiction r, for t quantum particl is moving as a wav t is not a classical particl n particular, t particl dos not spd up to infinit spd to cross t nod SOUTONS TO PROBEMS Sction 8 Blackbody Radiation and Planck s Tory *P8 Tis is an ampl of Stfan s law At t lowr tmpratur, P 4 σ AT At t igr, P 4 σ AT Tn P P H G T K J T a 4 9f K J a f, four tims largr tan t T prcntag incras in powr is % prcntag incras in tmpratur P8 (a) P Aσ T 4, so T H G P A K J σ NM NM O 8 4 QP W O 8 4π 696 m 567 W m K QP K mk 898 mk 7 ma 5 m 5 nm 3 T 578 K *P83 T pak radiation occurs at approimatly 56 nm wavlngt rom Win s displacmnt law, 89 8 mk 89 8 mk T 9 56 m ma 5K Clarly, a firfly is not at tis tmpratur, so tis is not blackbody radiation P84 (a) E f 6 66 Js 6 s E f 6 66 Js 3 s (c) E f 6 66 Js 46 s continud on nt pag V 9 6 J KJ 57 V V KJ J V KJ J 7 V V

4 774 Quantum Pysics c 3 (d) f 6 8 c 3 f 3 c 3 f ms m 484nm, visibl ligt blu H ms 968 m 968 cm, radio wav H ms H 65 m, radio wav J P85 Eac poton as an nrgy E f a f Tis implis tat tr ar 3 5 Js 3 7 potons s 6 66 Jpoton P86 Enrgy of a singl 5-nm poton: 34 8 c 6 66 Js 3 ms Eγ f m T nrgy ntring t y ac scond π 3 5 E P t A t 4 Wm 85 m s 7 J 4 QP 9 O NM a f T numbr of potons rquird to yild tis nrgy J 5 E 7 J n 57 9 E 398 J poton γ 3 potons P87 W tak θ 3 radians Tn t pndulum s total nrgy is a f b g b g E mg mg cosθ E kg 9 8 m s J g T frquncy of oscillation is f ω π π T nrgy is quantid, E nf Trfor, E n f H 3 44 J J s 498 s 3 m r g G P87 a f *P88 (a) V πr π m 3 35 m m ρ V 7 86 kg m 3 35 m 63 kg continud on nt pag

5 a f 3 A 4πr 4π m 5 3 m P σ AT W m K m K 4 8 W 4 a f Captr (c) t mits but dos not absorb radiation, so its tmpratur must drop according to (d) ma T mk dq Q mc T mc T T mc dt f f i dt dt dq dtf dt P 8 Js dt mc mc Cs 99 kg J kg C Cmin d i mk ma K 6 m infrard 34 8 c 663 Js 3 () E f m m s J (f) T nrgy output ac scond is carrid by potons according to N P t E N P 8 Js 898 t E J poton H G K J 9 poton s Mattr is coupld to radiation, quit strongly, in trms of poton numbrs Sction 8 T Potolctric Effct c P89 (a) c φ Js 3 ms 9 4 V 6 J V a f 96 nm f c c 3 96 c 8 9 ms m 5 H c φ + V S : a4 V f J V V S 8 Trfor, V S 7 V

6 776 Quantum Pysics J V P8 Kma mvma V nm (a) φ E K ma 6 nm 38 V 65 nm fc φ 38 V J s V 9 J KJ H c 4 nm V P8 (a) VS φ φ 376 V 9 V 546 nm V S c 4nm V φ 9 V VS 6 V 5875 nm P8 T nrgy ndd is E V 6 9 J T nrgy absorbd in tim intrval t is E P t A t so 9 E 6 J t A O NM QP 7 8 s 48days 5 5 J s m π8 m T gross failur of t classical tory of t potolctric ffct contrasts wit t succss of quantum mcanics P83 Ultraviolt potons will b absorbd to knock lctrons out of t spr wit maimum kintic nrgy Kma f φ, Js 3 ms V or K ma KJ 47V 5V 9 9 m 6 J T spr is lft wit positiv carg and so wit positiv potntial rlativ to V at r As its potntial approacs 5 V, no furtr lctrons will b abl to scap, but will fall back onto t spr ts carg is tn givn by V k Q or Q rv r k b g 5 m 5 N m C N m C 84 C Sction 83 T Compton Effct P84 E c Js 3 ms 9 84 J 78 V 9 7 m Js 8 p 947 kg m s 9 7 m

7 P85 (a) acosθf: mc E cos a f m c : 3 V 6 J V 44 m and m c E (c) K E E 3 kv 68 5 kv 3 5 kv P86 Tis is Compton scattring troug 8 : E 34 8 cos θ 43 m cos m mc Js 3 ms 9 9 c m 6 J V a f a f Captr ms 6 66 Js 3 ms 4 43 J 68kV 463 m 3 kv c + 5 nm so E 8 kv By consrvation of momntum for t poton-lctron systm, and p + c p 3 ms Js + G 9 J JV m 5 m G P86 $ i $ i + p $ i KJ 8 kv c H K K J By consrvation of systm nrgy, 3 kv 8 kv +K so tat K 478 V 4 bg Cck: E p c + m c or mc + K pc + mc a f a f a f 5 kv kv kv + 5 kv 6 6

8 778 Quantum Pysics P87 Wit K E, K E E givs E E E E E and c E + cosθ C P88 (a) K m v c c E E a f Ca + cosθ f 6 cosθ θ 7 C : K 9 kg 4 m s 8 93 J 5 58 V c E E K and E b E c 4 V nm 55V 8 nm 4 V nm 83 nm 55V 558 V 88 nm 88 pm 88 nm C cos θg: cos θ 89, 43 nm so θ C Sction 84 Potons and Elctromagntic Wavs *P89 Wit poton nrgy V f J f Js H Any lctromagntic wav wit frquncy igr tan 4 5 H counts as ioniing radiation Tis includs far ultraviolt ligt, -rays, and gamma rays Sction 85 T Wav Proprtis of Particls 6 66 Js P8 397 p mv kg m s 34 3 m

9 P8 (a) Elctron: p m v p and K mv m m Captr so p m K and mk Js 3 9 a f 9 kg 3 6 J 7 9 m 79 nm Poton: c f c and E E and E f so f Js 3 ms m 44 nm J P8 rom t condition for Bragg rflction, φ m dsinθ dcos But d a H G sin φ K J H G K J wr a is t lattic spacing Tus, wit m, φ φ asin cos asin φ G P8 H G K J H G K J p mk Js kg 54 6 J 67 m 67 m Trfor, t lattic spacing is a 8 8 nm sinφ sin 5 P83 (a) ~ 4 m or lss p ~ 66 Js 9 kg m s or mor 4 m T nrgy of t lctron is E p c + m c or E ~ J ~ V or mor, ~ or mor so tat K Em c ~ V 5 V ~ V T lctric potntial nrgy of t lctron-nuclus systm would b U kqq r 9 9 a f 9 N m C C ~ ~ 4 m 5 V Wit its K + >>, t lctron would immdiatly scap t nuclus U

10 78 Quantum Pysics P84 (a) T wavlngt of t studnt is p tn w nd so tat w f w is t widt of t diffracting aprtur, mv H G mv K J Js mw b8 kg ga 75 mf v KJ 34 ms (c) d 5 m Using t w gt: t s v m s 34 No T minimum tim to pass troug t door is ovr 5 tims t ag of t Univrs P Js 3 p 663 kg m s p m (a) lctrons: K 34 3 p m J kv T rlativistic answr is mor prcisly corrct: 4 K p c + m c m c 4 9 kv 3 8 kv potons: Eγ pc Sction 86 T Quantum Particl *P86 E K mu f v pas and mu mu u f v mu pas Tis is diffrnt from t spd u at wic t particl transports mass, nrgy, and momntum P87 As a bonus, w bgin by proving tat t pas spd vp ω is not t spd of t particl k p c m c vp ω k + γ mv 4 4 γ m v c γ m v + m c c c v c c + c + c v v c KJ γ + v n fact, t pas spd is largr tan t spd of ligt A point of constant pas in t wav function carris no mass, no nrgy, and no information Now for t group spd: continud on nt pag c v

11 v dω dω de d dk d k dp dp m c + p c 4 vg m c + p c + pc v g g c γ m v γ m v + m c c 4 p c p c v v c m c v v c + c c v v c v v + c v v c Captr 8 78 t is tis spd at wic mass, nrgy, and momntum ar transportd Sction 87 T Doubl-Slit Eprimnt Rvisitd P88 Considr t first brigt band away from t cntr: N M O QP a f 8 4 dsinθ m 6 m sin tan m KJ so mv mv m v and K mv V m m V m Js V C 9 kg m 5 V P89 (a) mv Js kg 4 m s b g or dstructiv intrfrnc in a multipl-slit primnt, dsinθ m+ t first minimum 7 m K J, wit m for Tn, θ sin 84 d H G K J a fb g so y tanθ y tan θ m tan mm (c) W cannot say t nutron passd troug on slit W can only say it passd troug t slits

12 78 Quantum Pysics *P83 W find t spd of ac lctron from nrgy consrvation in t firing procss: K f + Uf mv V V 6 v m a f C 45 V kg ms 8 m T tim of fligt is t 74 8 s T currnt wn lctrons ar 8 cm 6 v 398 m s 9 q apart is t t 6 C 7 A 8 74 s Sction 88 T Uncrtainty Principl P83 or t lctron, p m v 9 kg 5 m s 4 56 kg m s or t bullt, p m v b g p 6 66 Js 4π 3 4π 456 kg m s b gb g 6 mm 4 3 kg 5 m s kg m s p 58 3 m 4π P83 (a) p m v so v m π Js 4π 4π kg m b ga f 5 ms T duck migt mov by 5 ms 5s 5 m b ga f Wit original position uncrtainty of m, w can tink of growing to m + 5 m 5 m P833 y py and d py p 4π Eliminat p y and solv for 4π p y d 3 b g : 4π kg b m s g m 3 34 m 6 66 Js T answr, m, is 9 tims gratr tan t diamtr of t obsrvabl Univrs!

13 Captr P834 rom t uncrtainty principl E t or mc t Trfor, m m 4π c t m 4π te R a f a f 34 π a f m m 6 66 Js MV KJ s 35 MV 6 J 8 P835 (a) At t top of t laddr, t woman olds a pllt insid a small rgion i Tus, t uncrtainty principl rquirs r to rlas it wit typical oriontal momntum p m v t falls to t floor in a travl tim givn by H + gt H as t, so i g t total widt of t impact points is + + i H G b g K J + v t f i i m i H g A i wr To minimi f, w rquir A m d d d i H g f b ig or i A so i A T minimum widt of t impact points is A d fi i + min d f i min i A KJ m H G A i H g Js O a f NM m QP O 4 NM 5 kg 98 ms QP KJ m Sction 89 An ntrprtation of Quantum Mcanics P836 Probability P ψ a a af a a π a + a a d H G K J H G ak J tan H G a K J π π π P π π 4 4 tan tan a f NM K JO QP a a

14 784 Quantum Pysics i a f 5 a f a f P837 (a) ψ A Acos 5 + Aisin 5 Acos k + Aisin k gos troug a full cycl wn cangs by and wn k cangs by π Tn k π wr π k 5 m Tn π m 6 m Js 4 p 57 kg m s 6 m (c) m 9 3 kg 4 K m v p m m 57 kg m s 9 3 kg J J JV 95 5 V Sction 8 A Particl in a Bo P838 or an lctron wav to fit into an infinitly dp potntial wll, an intgral numbr of alf-wavlngts must qual t widt of t wll n 9 9 m so n p n (a) Sinc K m m m or K 6V n 4 p n V G P838 Wit n 4, K 63 V

15 P839 (a) W can draw a diagram tat parallls our tratmnt of standing mcanical wavs n ac stat, w masur t distanc d from on nod to anotr (N to N), and bas our solution upon tat: Captr Sinc d N to N and p p d p 6 66 Js Nt, K m 8md d M 3 89 kg M NM 34 O QP Evaluating, 6 38 J m V m K K d d n stat, d m K 37 7 V n stat, d 5 m K 5 V n stat 3, d 333 m K V n stat 4, d 5 m K 4 63 V G P839 Wn t lctron falls from stat to stat, it puts out nrgy c E 5 V 37 7 V 3 V f into mitting a poton of wavlngt 34 8 c 6 66 J s3 m s nm E 9 3 V 6 J V a f T wavlngts of t otr spctral lins w find similarly: Transition E V f a f nm

16 786 Quantum Pysics P84 T confind proton can b dscribd in t sam way as a standing wav on a string At lvl, t nod-to-nod distanc of t standing wav is 4 m, so t wavlngt is twic tis distanc: p 4 m T proton s kintic nrgy is p 6 66 Js K mv m m kg m 3 39 J 5 MV 9 6 J V 34 G P84 n t first citd stat, lvl, t nod-to-nod distanc is alf as long as in stat T momntum is two tims largr and t nrgy is four tims largr: K 8 MV T proton as mass, as carg, movs slowly compard to ligt in a standing wav stat, and stays insid t nuclus Wn it falls from lvl to lvl, its nrgy cang is 5 MV 8 MV 66 MV Trfor, w know tat a poton (a travling wav wit no mass and no carg) is mittd at t spd of ligt, and tat it as an nrgy of +66 MV ts frquncy is f E 6 9 H 6 6 V 6 J V Js 49 c 3 And its wavlngt is f 49 8 ms s 3 m Tis is a gamma ray, according to t lctromagntic spctrum cart in Captr 4 *P84 (a) T nrgis of t confind lctron ar En m n ts nrgy gain in t quantum 8 ump from stat to stat 4 is 4 and tis is t poton nrgy: 8m 5 8m c f Tn 8mc 5 and 5 8mc H G K J t rprsnt t wavlngt of t poton mittd: c m m m Tn c 5 m 8 5 and 5 c 8m 4

17 Sction 8 T Quantum Particl Undr Boundary Conditions Captr Sction 8 T Scrödingr Equation P84 ψaf Acos k+ Bsin k ψ ka sin k + kbcos k a f a f ψ m me k A cos k k Bsin k + E U ψ A cos k B sin k Trfor t Scrödingr quation is satisfid if ψ m ψ E U K Ja f me or kaacosk+ Bsinkf K J a A cos k + B sin k f k Tis is tru as an idntity (functional quality) for all if E m P843 W av ψ A Scrödingr s quation: Sinc k b g and ikωt a f b g π π p p ψ ψ ik and E U p m ψ k ψ ψ m k ψ a ψ E U f Tus tis quation balancs P844 (a) Wit ψaf A ik d d A ik Aik ik and d ψ Ak ik d d ψ k 4 Tn + m d m A ik π p m v ψ ψ ψ mv ψ Kψ 4π m m m Wit ψ nπ A k dψ sin sin, Ak k d ψ cos and Ak k sin d d af H G K J d ψ 4 Tn + m d m Ak k π p sin ψ ψ Kψ 4π m m P845 (a) d H G π K J 4π sin cos d NM 4π 4π 4π sin + cos 6π K J O QP continud on nt pag

18 788 Quantum Pysics 4π sin 4π 5 π Probability H G K J sin d 49 NM a π πf (c) (d) Probability sin 4 sin π NM 6 Probability 4π sin 399 4π QP 4 O O QP 5 49 n t n grap in igur 83, it is mor probabl to find t particl itr nar or 3 tan at t cntr, wr t probability dnsity is ro 4 4 Nvrtlss, t symmtry of t distribution mans tat t avrag position is P846 Normaliation rquirs ψ d nπ or A sin d all spac nπ A sin d A or A K J H G K J P847 T dsird probability is P ψ d wr 4 4 H G K J cos θ sin θ 4 K J + K J K J π sin d 4π Tus, P sin 5 4π 4 a f P848 (a) ψ A KJ Scrödingr s quation dψ A d d ψ m E U ψ d a f m EA d ψ A d A bcoms KJ + m A m Tis will b tru for all if bot and me me + me me 4 bot ts conditions ar satisfid for a particl of nrgy E m continud on nt pag 4

19 or normaliation, A d A + KJ N M A O Q NM P A A (c) P d d ψ KJ P 8 58 N M O 6 QP H G 5 K J A 5 O Q 3 N Captr KJ d 5 6 P 3 M O Q P Sction 83 Tunnling Troug a Potntial Enrgy Barrir C P849 T mu a Ef wr C C (a) T 3, a % canc of transmission R T 99, a 99% canc of rflction G P849 P85 C 3 9 a f kg m s Js C 9 36 a f T p 3 6 m 95 m p 6 88 T m G P85 *P85 T original tunnling probability is T wr C c a f a f 3 9 mue π 9 kg 6 J Js C 448 c 4 V nm T poton nrgy is f 7 V, to mak t lctron s nw kintic nrgy 546 nm V and its dcay cofficint insid t barrir 3 9 m π 9 kg 47 6 J C Js a Now t factor of incras in transmission probability is C C a f m m 9 C C f m

20 79 Quantum Pysics Sction 84 Contt Connction T Cosmic Tmpratur P85 T radiation wavlngt of 5 nm tat is obsrvd by obsrvrs on Eart is not t tru wavlngt,, mittd by t star bcaus of t Dopplr ffct T tru wavlngt is rlatd to t obsrvd wavlngt using: c c vc + vc b g : b g vc + vc T tmpratur of t star is givn by ma T mk: a f a f 5 nm nm mk 898 mk T : T ma 3 K P853 (a) Win s law: ma T m K mk 898 mk 3 Tus, ma 6 m 6 mm T 73 K Tis is a microwav P854 W suppos tat t firball of t Big Bang is a black body a f σ T ( ) 5 67 Wm K 73 K 3 5 Wm As a bonus, w can find t currnt powr of dirct radiation from t Big Bang in t sction of t univrs obsrvabl to us f it is fiftn billion yars old, t firball is a prfct spr of radius fiftn billion ligt yars, cntrd at t point alfway btwn your ys: 8 7 a f syr KJ ms P A ( 4πr ) 3 5 Wm 4π 5 ly 3 56 ly yr P W Additional Problms a f a f a f V V A V π r *P855 T condition on lctric powr dlivrd to t filamnt is P V R ρ l ρ l P ρ l so r πa Vf Hr P 75 W, ρ 73 7 Ω m, and V V As t filamnt radiats in 4 4 stady stat, it must mit all of tis powr troug its latral surfac ara P σ AT σ πrl T W combin t conditions by substitution: H G K J continud on nt pag

21 and r a f P σ π P856 V S f H G K J φ H G P ρ l π V V P σ π ρ l T 3 4 l a f K J lt H G K J H a VfP V a75 W f m K 4 σ π ρ T G W 45 π 7 3 Ω m 9 K m m l H G P ρ l V K J H G πa f a f 4 Captr a f a f KJ 7 75 W 7 3 Ω m 333 m 5 J r 98 m π V rom two points on t grap 4 4 H K H G K J and 3 3 V 4 H φ H G K J φ 3 Combining ts two prssions w find: G P856 (a) φ 7 V 4 5 V s (c) At t cutoff wavlngt c φ H G K J c c c 5 9 c 4 V s 6 C 8 3 ms 9 a7 V f6 J V 73 nm

22 79 Quantum Pysics P857 W want an Einstin plot of K ma vrsus f, nm f, 4 H Kma, V V (a) slop ± 8% 4 H c VS f φ (c) K ma 9 Js a f KJ Js ± 8% 4 f (TH) G P857 at f 344 H 9 φ f 3 J 4 V P858 rom t pat t lctrons follow in t magntic fild, t maimum kintic nrgy is sn to b: BR Kma c rom t potolctric quation, Kma f φ φ m Tus, t work function is φ c c K B R ma m P859 (a) mgyi mv v f f gy 98 ms 5 m 33 ms mv i a f Js 8 75 kg 3 3 m s b gb g 37 m a not obsrvabl f E t Js so E s π 3 J (c) 3 E E 6 J % 75 kg 98 m s 5 m b g a f

23 Captr P86 (a) m Js 4 p 33 kg m s m (c) p E m 7 V P86 (a) S t figur S t figur G P86(a) G P86 (c) (d) ψ is continuous and ψ as ± T function can b normalid t dscribs a particl bound nar, by a vry dp, vry narrow wll of potntial nrgy Sinc ψ is symmtric, ψ d ψ d α A or A d α KJ Tis givs A α α b g b g K J α α α α () P a d 63 α α α P86 (a) Us Scrödingr s quation wit solutions ψ m ψ E U a f ik ik ψ A + B [rgion ] ψ C ik [rgion ] G P86(a) continud on nt pag

24 794 Quantum Pysics Wr and k k me a meu f Tn, matcing functions and drivativs at bψ g bψ g givs A + B C and dψ dψ d K J H G d K J givs kaa Bf kc Tn k k B + k k A and C + k k A ncidnt wav A ik rflcts B ik, wit probability R B A b b k + k k k g g b b k k k + k g g Wit E 7 V and U 5 V T rflction probability is R k k a a EU E T probability of transmission is T R f f 9 P863 ψ d or a on-dimnsional bo of widt, ψ n nπ sin H G K J Tus, nπ sin d H G K J 3 n π (from intgral tabls)

25 *P864 (a) T rquirmnt tat n so p n nc E pc + mc En mc H G bg K J + K E mc n n H G K J + mc nc mc is still valid Captr Taking m, m kg, and n, w find K 469 J Nonrlativistic, E Js 8m 3 89 kg m 6 Comparing tis to K, w s tat tis valu is too larg by 8 6% 4 J P865 or a particl wit wav function ψ a f a a for > and for < (a) ψaf af, < and ψ a a f a f a f Prob < ψ d d a f a, > (c) Normaliation ψ d ψ d+ ψ d d + H G K J a f a a d a a G P865 a d 865 a a d a a < < H G Prob ψ K J a Tis wav function would rquir t potntial nrgy to b + for < and for Tis potntial nrgy function cannot b ralistic in dtail

26 796 Quantum Pysics ANSWERS TO EVEN PROBEMS P8 (a) K ; 5 nm P84 (a) 57 V; 8 µ V; (c) 9 nv; (d) 484 nm visibl, 968 cm and 65 m radio wavs P potons P88 (a) 63 kg; 8 W; (c) 5 3 Cs 99 Cmin; (d) 989 µ m ; () J; (f) poton s P8 (a) 38 V; 334 TH P8 48 days, absurdly larg P84 78 V, kg m s P86 kv, 478 V c P88 (a) 88 pm; P8 397 fm P8 8 nm P84 (a) 34 m s ; s; (c) no, t tim is ovr 5 tims t ag of t univrs u P86 vpas P88 5 V P83 7 pa P83 (a) 5 m s ; 5 m P P836 P838 (a) n 4; 63 V P84 66 MV, fm, a gamma ray k P84 s t solution, E m P844 s t solution P846 s t solution P848 (a) E m ; rquiring A d givs KJ A 5 K J 6 ; (c) P P K P µ W m P856 (a) 7 V; 4 fv s; (c) 73 nm P858 c B R m P86 (a) m; 33 4 kg m s ; (c) 7 V P86 (a) s t solution; R 9, T 98 nc 4 P864 (a) m c mc K J + ; 469 fj, 86%