Quantum Mechanics II. Chapter 6
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1 Quau Mecacs II Caer 6
2 Te race of uau ecacs Bor s eor o aoc srucure 93 oes o er o uersa ow vual aos erac w oe aoer A ore geeral a fuaeal aroac o aoc eoea s RQUIRD! Heseberg 95 a Scrger 96 eveloe uau ecacal forulaos. Scrger forula wc was sre b e e Brogle s aer wave s uc ore frel. aer was rove a wo eores are ecal. For e followg ears e eor was ale o a ffere sses a was rove o be cosse w eereal resuls.
3 Heseberg a Scrger Werer Heseberg Nobel rze 93 "for e creao of uau ecacs e alcao of wc as er ala le o e scover of e alloroc fors of roge" rw Scrger Nobel rze 933 "for e scover of ew roucve fors of aoc eor
4 Scrger e. A arcle se a oeal 3D Scrger euao z z f cf w l u u Q
5 al of e Scrger e. I was e b e e Brogle s aer wave coce a se u fro a free arcle case. No furer ueso was ase!!! Jus assue! Solve for a vare of scal sses Coare e resuls w eereal resuls v Wa abou oer scal laws or rcles? Newo s e. Mawell s es. ec. Te were o erve eer. Te were rove ol b e coarso w eereal resuls.
6 Raoale for Scrger e. A wave w wellefe f ω; a λ ; We we ser o e Scrger e. s cos A A Ae w w w Ae Ae w w w w \ w w f l w
7 Wave fuco Te soluo of e Scrger euao Carres forao abou e arcle s wavele beavor Ma Bor s erreao Ψ : e robabl es of fg a arcle a b Cole uber * a b Cojugae * º a b a b a b ³
8 Noralzao a robabl Te robabl P of a arcle beg bewee a s gve b Te robabl of e arcle beg bewee a s gve b Te wave fuco us be oralze so a e robabl of e arcle beg soewere o e as s.
9 Bouar coos: ψ soul beave well ψ us be fe everwere. To avo fe robables ψ us be sglevalue. To avo ulle values of e robabl For fe oeal ψ a ψ/ us be couous. Because ψ/ soul be sglevalue ψ as ± I orer o oralze ψ.
10 Teeee Scrger euao For ; eee of ú û ù ê ë é ß f f f f f f f f Searao of varables
11 Teeee Scrger euao l l l / ò ò f f f f f f f f f f Q e U e e w f \ /
12 Saoar or sea sae For a soluo ψ of e eeee Scrger euao Te robabl es becoes: Te robabl srbuos are cosa e. Ts s a sag wave eoea a s calle e saoar sae. cf Bor s saoar saes Mecacal sag wave s a aalog.
13 Bouar coos revse Te wave fuco a s ervave us be couous. [ U ] ò ò
14 Classcal vs uau ecacs Newoa ecacs Quau ecacs If e force acg o a arcle s ow a f al coos o oso a oeu are gve e e arcle s fuure s eere. DTRMINISTIC Deals w e robabl Ucera alwas ess. PROBABIISTIC Te sae for e Mawell s &M es.
15 Classcal a uau ecacs Newo s seco law a Scrger s wave euao are bo ffereal euaos. Newo s seco law ca be erve fro e Scrger wave euao so e laer s e ore fuaeal. o Te classcal ecacs eals w e average values ol. Classcal ecacs ol aears o be ore recse because eals w acroscoc eoea. Te uerlg uceraes acroscoc easurees are jus oo sall o be sgfca. o Te classcal ecacs s a aroao of e uau ecacs.
16 Quau ecacs: Te sgle scal rcle Quau ecacs Probablsc; croscoc Classcal ecacs Deersc; acroscoc Wave ocs Ra ocs
17 ear of Scrger e. If Ψ a Ψ are wo soluos e Ψ a Ψ aψ s also a soluo. Wave fucos ca suerose. Terefore erferece ca occur. Doublesl eere If sl ol s oe If sl ol s oe If bo are oe P P P * * Causg e erferece * * * *
18 oug s oublesl eere: revse?
19 ecao values Solvg e Scrger e. gves e wave fuco ψz wc coas all e forao abou e arcle. Probabl es: P Poso of e arcle? ò < > ò We ψ s roerl oralze ò
20 ecao values Te eecao value s e eece resul of e average of a easurees of a gve ua. A easurable ua for wc we ca calculae e eecao value s calle a scal observable. N N N N N N N44... N... 4 å å Te eecao values of scal observables for eale oso lear oeu agular oeu a eerg us be real because e eereal resuls of easurees are real. N N
21 Couous eecao values We ca cage fro scree o couous varables b usg e robabl es P of observg e arcle a a arcular. Usg e wave fuco e eecao value s: Te eecao value of a fuco G for a oralze wave fuco Ψ: G ò ò G * G
22 Poso oeraor A scal observable corresos o a scall easurable ua. For a scal observable ere ess a oeraor. For e oso of a arcle e oso a z self s s ow oeraor. ˆ ; ˆ ; zˆ z
23 Moeu oeraor ers of a. Coserg e ervave of e wave fuco of a free arcle w resec o : w Ae Ae w Ts suggess we efe e oeu oeraor as
24 erg oeraor Te e ervave of e freearcle wave fuco Ae wae w w w Te eerg oeraor
25 Oeraor euao Te oeraor for oal eerg Scrger euao P K
26 Oeraors a eecao values For a observable G G G G ò ò * * * ò ò ò ò * * ò ò * *
27 Oer was o calculae <>? ò * ò * ò ò * * ò * *!? Terefore e ol ossbl s ò ò * * ò *
28 gevalues a egefucos Te Scrger e. ca be sasfe ol for secal ψ s. gefucos Te corresog s are calle gevalues Wc are call uaze for a cofe sse.
29 Oeraors a egevalues H H gefuco Haloa oeraor Scrger e. Oeraor gevalue I geeral for a oeraor Ô O o A easuree of Ô ca ol el oe of e egevalues o.
30 Posulaes uau ecacs For a scall observable ua o ere ess a corresog oeraor Ô. For eac easuree ol oe of e egevalues of e corresog oeraor ca be easure. H O o For a sse ere ess a sae fuco ψ a coas all forao abou e sse a a ca be eresse as å c Te robabl a e egevalue o s easure for e sse s P c ò *
31 Scrger s ca åc
32 A arcle a bo Cofe < < Ife oeal well º Q î í ì > < < <
33 Ife suarewell oeal Þ Þ Ü A A A A B B A s s s s cos s Q \ \ A s Slar o a vbrag srg w fe es
34 Quaze eerg Noe a e eerg ees o e eger values of As Hece e eerg s uaze a ozero. Te secal case of s calle e grou sae eerg.
35 Noralzao 3... s s cos s \ ú û ù ê ë é ú û ù ê ë é ò ò ò ò A A A A A cos s Q
36 ecao values 8 cos 4 s 4 s ú û ù ê ë é ò ò s cos s * * * ú û ù ê ë é ò ò ò ò cf ± ± ± ò a a a a s cos s Q
37 A arcle a woesoal bo ß ß ß g g g g f f f f g g f f g f g f f g g f oerwse for < < < <
38 A arcle a woesoal bo s s s s g f \ \ 4
39 Degeerac Aalss of e Scrger wave euao wo ree esos rouces wo ree uau ubers a uaze e eerg. A uau sae s egeerae we ere s ore a oe wave fuco for a gve eerg. Degeerac resuls fro arcular roeres of e oeal eerg fuco a escrbes e sse. A erurbao of e oeal eerg ca reove e egeerac.
40 Fe suarewell oeal < <
41 Fe suarewell oeal Rego I & III Rego II III I Fe Fe e Ce De Ce º > Q Q B A II cos s º Q Te wave fuco s ozero ouse of e bo!!!
42 Fe suarewell oeal Bouar coos cos cos cos cos s cos s cos cos s Þ Fe B A A C Fe B A B C III II II I III II II I
43 Fe suarewell oeal soluo Te wave fucos eerae o e barrers o bo ses.
44 Peerao e Te eerao e s e sace ouse e oeal well were e robabl sgfcal ecreases. µ e ±» Te eerao sace a volaes classcal scs s rooroal o Plac s cosa.
45 Ife versus fe λ ges loger. Te waves eerae o e walls. ges saller. /λ becoes lower. /
46 Baga egeerg
47 Ga Al As
48 Secoucor uau wells
49 TM ages of QWs Aoc resoluo TM oogra of a AlGaAs/GaAs uau well
50 Sle aroc oscllaor SHO Uer a resorg force a objec oscllaes abou a eulbru. F Acos w f Qw º
51 For a oeal w a local u Near a oeal u o eulu ao o a crsal...!»»
52 Quau ecacal SHO º º 4 Q w a wa a Scrger e. for a arabolc oeal well
53 Soluos for e SHO Te eerg levels are evel sace. Te lowes eerg s /... e H µ a Q / 4!... e H \ \ Q w w w
54 Here oloals / µ H e... Oscllaor fucos
55 Soluos for e SHO I coras o e arcle a bo were e oscllaor wave fuco s a susoal curve s case e oscllaor beavor s ue o e oloal wc oaes a sall. Te eoeal al s rove b e Gaussa fuco wc oaes a large.
56 Coarso of classcal a uau robables I classcal ecacs P µ u w A Wc corresos o e uau ecacal case of a g. à e corresoece rcle Fro uau ecacs To classcal ecacs
57 Suare vs arabolc oeal wells
58 Tragular oeal well
59 Poeal se Rego I Rego II I Be Ae Q II De De Ce I II <
60 Poeal se II I II I II I e e e A D A B D B A D B A 4 ¹ II e A B R Toal refleco Ice reflece
61 Poeal barrer > Classcall a arcle jus asses e barrer w a lower see wle assg e barrer rego. Quau ecacall e arcle wave fuco ges arall reflece a arall rase. Ocal aalog: a glass lae
62 Refleco a rassso Te wave fuco cosss of a ce wave a reflece wave a a rase wave. Te Scrger wave euao Te corresog soluos As e wave oves fro lef o rg
63 Reflecace a rasace Te robabl of e arcles beg reflece R or rase T Because e arcles us be eer reflece or rase R T B alg e bouar coos ± a Tere s a suao wc e rassso robabl s T Resoace!!! T é ê ê ë é ê ë I I II II 4 s s II II ù ú û ù ú ú û
64 Poeal barrer < Classcall e arcle oes o ave eoug eerg o surou e oeal barrer. Quau ecacall owever ere s a sall bu fe robabl a e arcle ca eerae e barrer a eve eerge o e oer se. Rearable feaures of oer scs! Tere s ale eereal roof of s esece. Te rassso robabl a escrbes e eoeo of uelg s 4 s ú û ù ê ë é ú ú û ù ê ê ë é T
65 Tuelg 6 s 4 ¹ >>» ú û ù ê ë é e e T Q
66 Quau ecacs ocs??? If lg assg roug a glass reflecs fro a eral surface w a agle greaer a e crcal agle oal eral refleco TIR occurs. TIR ocs oeal se QM No rassso % refleco U I II
67 Ocal aalog o oeal se Uo TIR elecroagec fel sreg oes o go o zero abrul a e erface bu ecas eoeall. e e eerao of wave fuco o e oeal se
68 Ocal aalog o uelg across oeal barrer g s oall reflece b a rs va TIR. If we brg aoer rs ver close o e frs oe e lg aears e seco rs. Observe b Newo a ca be eosrae w wo rss a a laser. Te es of e seco lg bea ecreases eoeall as e sace bewee e wo rss creases. Te suao s aalogous o e uelg uau ecacs.
69 Quau uelg Joseso uelg Tuel oe Ala arcle eca Scag uelg croscoe ec.
70 Alaarcle eca Te eoeo of uelg elas e alaarcle eca of eav raoacve ucle. Ise e ucleus a ala arcle feels e srog sorrage aracve uclear force as well as e reulsve Coulob force. Te uclear force oaes se e uclear raus were e oeal s aroael a suare well. Te Coulob force oaes ouse e uclear raus.
71 Alaarcle eca Te oeal barrer a e uclear raus s several es greaer a e eerg of a ala arcle. Accorg o uau ecacs owever e ala arcle ca uel roug e barrer. Hece s s observe as raoacve eca.
72 STM S
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