Chapter 1: Review of Quantum Mechanics. Postulates of Quantum Mechanics: 1-3
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1 Chr : Rw of Qunum Mhns In hs lur you wll lrn..ll h you mgh h forgon: Posuls of qunum mhns Commuon rlons Shrongr n snrg urs Tm lomn Dnsy orors n nsy mrs Dohrn n qunum mhns C 47 Srng 9 Frhn Rn Cornll nrsy Posuls of Qunum Mhns: -3 Thr r srl osuls of qunum mhns. Ths r osuls sn hy nno r from som ohr r hory. Thy r nown only from rmns Frs hr osuls r: Th s of hysl sysm m s sr y or or, no y, h longs o lr s ry msurl quny l oson or momnum of rl s sr y n oror  h s n h lr s. 3Th only ossl ou om of msurmn of h quny s on of h gnlus of h oror C 47 Srng 9 Frhn Rn Cornll nrsy
2 C 47 Srng 9 Frhn Rn Cornll nrsy lr Ss lr s s us fny nm for or s wh h followng rors: Vors long o lr s f n only f: w u,, For som oron, no y +, f n n f 6 n 7 Innr rou: for n, h nnr rou hs h rors: 8 n oror n h lr s hs h rory h f hn u u u u w u w u u u u u u Ô Ô C 47 Srng 9 Frhn Rn Cornll nrsy Prors of lr Ss ml of lr S: Column ors u u h g f Ô gnors n gnlus: O gnlu gnor on Orors: Th on oror of s fn y: L: hn, Ô Ô u O u O O w u w w u Ths follows from h fnon of nnr rou Thrfor: O w rmn Orors: n oror s rmn or slf-on f, rmn orors h rl gnlus: Ô O O O O O u
3 Prors of lr Ss Bss Vors: s of ors h sn lr s s ll ss If,, 3... n sn lr s hn ny or w n h lr s n wrn s: w n Th mnmum numrs of ors n o sn lr s s ll h mnsonly of h lr s Orhogonl n Orhonorml Bss: ss s s orhogonl f f ss s s orhonorml f, For n orhonorml ss s, f r: w n hn h offns of nson w C 47 Srng 9 Frhn Rn Cornll nrsy Prors of lr Ss Coml Bss: n orhorml ss s onsr oml f: w w n n n n gnors of rmn Orors: ll gnors of rmn oror form oml s gnors n hosn o orhonorml O u O u O u u u w C 47 Srng 9 Frhn Rn Cornll nrsy w O u u w O u Comlnss rlon u u 3
4 Posul 4: Msurmn n Prols n Qunum Mhns Suos h qunum s of sysm s nown: Quson: f hysl osrl s msur, wh s h rsul? W now from osul 3 h h rsul hs o on of h gnlus of h oror  4 W nno now for rn h rsul of msurmn for s m, u h roly of gng s rsul s gn y: Dsusson: w n wr: n Th roly of gng s rsul s hn: C 47 Srng 9 Frhn Rn Cornll nrsy Th s s lnr suroson of h gnors of Th rols mus ll u o uny: n n n Posul 5: Colls of h Qunum S uon Msurmn Suos h qunum s of sysm s: Suos hysl osrl s msur n n h rsul s: Quson: Wh s h s of h sysm us fr h msurmn? Th qunum s rrsns nowlg ll h s nowl ou h sysm Th qunum s os msurmn mus rfl h rsul of h msurmn If son msurmn of s m mmly fr h frs msurmn, h rsul mus on wh roly nswr: Th s mmly fr h frs msurmn mus : n msurmn 5 Ths sun hng n h qunum s uon msurmn s ll h olls of h qunum s C 47 Srng 9 Frhn Rn Cornll nrsy 4
5 Posul 5: Colls of h Qunum S uon Msurmn Suos h qunum s of sysm s: Suos hysl osrl s msur n Bu now hs wo gnr gnlus: Suos h rsul of h msurmn s: Quson: Wh s h s of h sysm us fr h msurmn? Th msurmn nno sngush wn n Th s rgh fr h msurmn mus n h gnsus orrsonng o h gnlu : Msurmn ros h qunum s no h gnsus orrsonng o h msur gnlu C 47 Srng 9 Frhn Rn Cornll nrsy Som Common Osrls n Qunum Mhns Th Poson Oror: Orhogonly: Comlnss: Th Momnum Oror: Orhogonly: Comlnss: Poson Wfunon: Momnum Wfunon: C 47 Srng 9 Frhn Rn Cornll nrsy 5
6 6 C 47 Srng 9 Frhn Rn Cornll nrsy Momnum n Poson Commuon Rlon Suos my qunum s s oson gns: n I wn o n n momnum ss: Wh s hs? W n o now somhng mor ou h rlonsh wn oson n momnum n orr o fn Commuon Rlon:, Funmnl rory Wh os mn? Wh f quls? Wh f quls? Thr s n nm onnon wn oson n momnum msurmns n Ths onnon s mos lgnly rss y h oror ommuon rlon C 47 Srng 9 Frhn Rn Cornll nrsy Momnum n Poson Commuon Rlon Commuon Rlon:, Funmnl rory,, Cn hs g? X Ths s n ngrl quon for n h soluon s:
7 7 C 47 Srng 9 Frhn Rn Cornll nrsy Momnum n Poson Commuon Rlon I follows h: Thr s Fourr rnsform rlon wn h momnum n oson wfunons! : : C 47 Srng 9 Frhn Rn Cornll nrsy Momnum n Poson Commuon Rlon : 3 : 4
8 8 C 47 Srng 9 Frhn Rn Cornll nrsy Mn or on Vlus n Snr Dons of Orors Mn Vlu: Suos: Thn: Snr Don: Dfn: Thn: Oror for h unrny or fluuon n C 47 Srng 9 Frhn Rn Cornll nrsy Th Fourr Trnsform Rlon Bwn Momnum n Poson Wfunons 4 Cn w gnrlz hs osron o ll osrls n orors?
9 snrg nrny Rlons, If: B C Thn for ny qunum s: C B 4 ml: W now h:, Thrfor for ny qunum s: 4 Ths s no surrs sn h sm Fourr rnsform rlon, lry ol us h: 4 C 47 Srng 9 Frhn Rn Cornll nrsy snrg nrny Rlons n Msurmns, B C C B 4 Consr n rmn n whh on s l o msur h osrl wh n ury Msurmn of ~ C B 4 Th grr h ury n rmnng, h ggr h unrny n B us fr h msurmn ml: Poson n Momnum snrg Mroso Suos on rs o msur h oson of n lron wh ury On ns hoon of wlngh whr, ~ hoon of momnum: ~ Rsulng unrny n h rl momnum fr srng: ~ ~ ~ ~ C 47 Srng 9 Frhn Rn Cornll nrsy 9
10 C 47 Srng 9 Frhn Rn Cornll nrsy snrg nrny Rlons n Msurmns, 4 Ths ours sorl lomn C 47 Srng 9 Frhn Rn Cornll nrsy Th mlonn Oror Th oror orrsonng o nrgy hysl osrl s h mlonn oror V m For rl of mss m n onl V, h mlonn oror s:
11 C 47 Srng 9 Frhn Rn Cornll nrsy Posul 6: Tm Dlomn n Qunum Mhns Th m oluon of qunum s s gn y h quon: If h mlonn s m nnn hn h forml soluon su o h ounry onon h m = h s s s:...! Sonry ss: gnss of h mlonn r ll sonry ss Suos: Thn: Th roly of ng n rulr nrgy gns os no hng wh m: C 47 Srng 9 Frhn Rn Cornll nrsy Mr Rrsnon of Orors Suos w h oml ss s: Thn w n wr ny oror s: Mr rrsnon: Mr rrsnon s mng wn h lr s of hysl ss n h lr s of olumn ors
12 C 47 Srng 9 Frhn Rn Cornll nrsy Mr Rrsnon of Orors: Th mlonn Oror Suos: Thn: ry oror s gonl n s own gnss... C 47 Srng 9 Frhn Rn Cornll nrsy Th mlonn Oror wh Pruron Suos w h mlonn of rl n onl: V m o n w h foun ll s gnss n gnlus: o ow suos n onl onl s o h mlonn: V m o o o Ls wr h nw full mlonn n mr rrsnon usng h gnss of h orgnl mlonn:
13 3 C 47 Srng 9 Frhn Rn Cornll nrsy o Th orgnl mlonn ws: Th nw mlonn s: o Dynms of Two Ll Sysm C 47 Srng 9 Frhn Rn Cornll nrsy Inl Conon: Quson: Wh s? Wr h Soluon s: n lug no: Mully y rs n o g: nswr s: sn os Dynms of Two Ll Sysm
14 4 C 47 Srng 9 Frhn Rn Cornll nrsy Wh r h rols of fnng h rl n wll n wll m? sn os Th rl osll wn h wo wlls! Dynms of Two Ll Sysm sn os Soluon s: C 47 Srng 9 Frhn Rn Cornll nrsy o Soluon sng h gnss of h Coml mlonn: L: Ĥ Th gnss of h mlonn r: Dynms of Two Ll Sysm
15 5 C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm n h nl s n h gnss: Fn h s ny lr m: os Wh r h rols of fnng h rl n wll n wll m? sn C 47 Srng 9 Frhn Rn Cornll nrsy Frm s Goln Rul Consr sngl s oul o my ss onnuum of ss: D Suos: Thn fn: L: n hn us:
16 6 C 47 Srng 9 Frhn Rn Cornll nrsy Frm s Goln Rul D On gs h followng quons: Sol h son on:,,3, n susu n h frs on: C 47 Srng 9 Frhn Rn Cornll nrsy Frm s Goln Rul D Rl Rl W g: s h y r of h nl s Dy r or rnson r: D D
17 7 C 47 Srng 9 Frhn Rn Cornll nrsy Frm s Goln Rul: nohr Mh Rou D D D D D D D ssum onsn nrgy nnn nsy of ss n oulng C 47 Srng 9 Frhn Rn Cornll nrsy Shrongr Pur n snrg Purs Suos, gn n nl s, w n o fn h mn lu, som lr m Shrongr Pur: Fn usng: Thn lul: snrg Pur: o h, Dfn h m-nn oror s:
18 Shrongr Pur n snrg Purs Dffrn h o w.r.. o g:, snrg quon Fn usng, Thn lul:, Bounry onon: In h Shrongr ur h ss ol n m u h orors o no ol In h snrg ur h orors ol n m u h ss o no ol C 47 Srng 9 Frhn Rn Cornll nrsy snrg Pur Th mlonn s lwys m-nnn: Th ommuon rlon o no hng wh m: Suos:, B C Thn:, B B B C C Mor rsly, h qul-m ommuon rlons mnn hr form! C 47 Srng 9 Frhn Rn Cornll nrsy 8
19 9 C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm n h snrg Pur Dfn fw orors s follows: Th mlonn s hn: Gn w n o fn C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm n h snrg Pur You n rfy h followng ommuon rlons:,,,,, ml:, Fn usng:,, o h:
20 C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm n h snrg Pur Dfn h oulon ffrn oror s: quon for s: 4 Bounry onons: Soluon s: sn os Fnlly: sn sn os C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm n h snrg Pur W h: sn sn os sn os sn W n o fn: os Smlrly: sn Sm s on n h Shrongr ur!
21 Msurmns n Commuors n Qunum Mhns Commuon Rlons n Common gnors: If wo orors  n B ommu,..:, B hn hy n h h sm s of gnors Proof: Suos: n, B B B B B B B B B Thrfor, B s lso n gnor of  wh h sm gnlu s If  hs ll sn gnlus, hn B n hrfor s lso n gnor of B If hs som gnr gnlus, hn B ls ls n h sm gnsus. In hs s, h ors n hs gnsus n hosn so h hy r gnors of oh  n B C 47 Srng 9 Frhn Rn Cornll nrsy Msurmns n Commuors n Qunum Mhns Commuon Rlons n Smulnous Msurmns: Consr wo osrls n B: msur B u u Suos h osrl s msur for s S olls msur Suos h osrls n B r msur for s, B S rmns h sm msur S olls B msur u S olls u msur S olls u Msurmn of B surs n msurmn of surs B whn n B o no ommu. ur msurmns of n B nno on smulnously C 47 Srng 9 Frhn Rn Cornll nrsy
22 Msurmns n Commuors n Qunum Mhns Commuon Rlons n Smulnous Msurmns: Consr wo osrls n B: B u u, B Sn n B ommu, hy h ommon s of gnors B Suos h osrls n B r msur for s msur S olls B msur S rmns h sm msur S rmns h sm ur msurmns of n B n on smulnously n wll no ff h ohr C 47 Srng 9 Frhn Rn Cornll nrsy Suroson, Msurmns, n Dohrn Consr h followng lnr suroson s of rl m from gns of : n osrr ms msurmn o s f h rl ws n s or Dnng on h rsul h s ollss: Msurmn of LSSO: If n osrr loos qunum s, h sroys h lnr suroson sruur of h qunum s n ollss C 47 Srng 9 Frhn Rn Cornll nrsy
23 Suroson, Inron, n Dohrn Consr gn h followng lnr suroson s of rl m from gns of : Inron wh nronmn oms, ron, honons, LSSO: If nronmn grs of from r hng y h nron n wy h n l n osrr rmn h lu of y loong h nronmn, hn hs s quln o r msurmn of n h qunum s ollss C 47 Srng 9 Frhn Rn Cornll nrsy Dohrn ny nron wh h nronmn n sroy h lnr suroson n olls h qunum s Th rous, n rsn goo msur of h gr of suroson n qunum s Ths rous r gnr y h orors n : On n h s m gos y, nron wh h nronmn n m hs rous go o zro: Ths hnomnon whh rsuls n h sruon of qunum mhnl surosons s ll qunum mhnl ohrn C 47 Srng 9 Frhn Rn Cornll nrsy 3
24 Suos: Suroson n Dohrn nronmn W s nron wh h nronmn ns o sroy lnr surosons W mnu!! Dfn nw ss: nronmn LSSO: Whhr or no suroson s sroy ns on h nur of nron wh h nronmn.. on wh nformon s r y h nronmn urng h nron C 47 Srng 9 Frhn Rn Cornll nrsy Consr wo ry ffrn ss of ss: S lrg numr of nl os of: Lnr suroson ss ur ss Pur Ss n Ssl Murs O S B lrg numr of ss n n h ro of: : Ssl mur of ss Msurmn of O or h nr s Msurmn of O or h nr s Mn lu on: O Mn lu on: ow os on rrsn n/or sngush hs wo ss hn?? C 47 Srng 9 Frhn Rn Cornll nrsy 4
25 Dnsy Oror n Qunum Mhns Dnsy orors r usful wy o rrsn qunum ss Mos gnrlly, qunum s s no rrsn y s or u y nsy oror S Dnsy Oror for Pur Ss S : lrg numr of nl os of: For ur ss h nsy oror s: Lnr suroson ss ml: ur ss In mr rrsnon: Th gonl lmns n h ouon rols, n h off-gonl lmns rrsn ohrns C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Oror for Pur Ss For ur ss h nsy oror s: Th mn lu of n osrl s fn for ur s s: ow o w us h nsy oror o lul mn lus of osrls? Tr C 47 Srng 9 Frhn Rn Cornll nrsy Tr n n n n n n 5
26 Dnsy Oror for Ssl Murs Dnsy orors for h ssl mur s B s fn s: In mr rrsnon: Tr Off-gonl lmns r zro no ohrns! Tr S B lrg numr of ss n n h ro of: : Ssl mur of ss ow o w us h nsy oror o lul mn lus of osrls? C 47 Srng 9 Frhn Rn Cornll nrsy Dohrn n h Dnsy Mr Dohrn nronmn Dohrn nronmn Dohrn ms h off-gonl omonns of h nsy mr go o zro wh m! C 47 Srng 9 Frhn Rn Cornll nrsy 6
27 Dnsy Oror for Gnrl M Ss Suos w h ssl mur of wo ss: Th nsy oror s: In mr rrsnon: In mr rrsnon: h ho of ss! Whhr or no nsy oror hs off-gonl omonns ns on C 47 Srng 9 Frhn Rn Cornll nrsy Tm Dlomn: Shrongr n snrg Purs for Dnsy Orors h mn lu of n osrl m s: Gn s Shrongr Pur h s s m nn snrg Pur h oror s m nn Gn s Tr Tr h mn lu of n osrl m s: Shrongr Pur h s s m nn Tr snrg Pur h oror s m nn C 47 Srng 9 Frhn Rn Cornll nrsy 7
28 Tm oluon of Dnsy Orors n h Shrongr Pur snrg orors oy: Tr Tr Tr Tr Tr BC Tr C B Tr B C Whr: Dffrn w.r.. m o g:, quon for h m lomn of h nsy oror n h Shrongr ur C 47 Srng 9 Frhn Rn Cornll nrsy Tm oluon of Dnsy Orors n h Shrongr Pur Gn s sol:, Ths s no h sm s h snrg quon for ohr orors:, n hn lul: Tr Tr Shrongr ur Qunum Mhnl Corrlons of Osrls: n ng of h snrg ur s h on n lul orrlons: B Tr B C 47 Srng 9 Frhn Rn Cornll nrsy 8
29 9 C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm h Dnsy Mr n h Shrongr Pur Suos: Th gol s o fn: Tr Sr from: C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm h Dnsy Mr T h mr lmn of h o quon y mullyng from h lf y n from h rgh y o:
30 Dynms of Two Ll Sysm h Dnsy Mr On n on quons for ll h gonl n off-gonl lmns: C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm h Dnsy Mr I follows h: s s s n hn on ons: Bounry onons: s C 47 Srng 9 Frhn Rn Cornll nrsy 3
31 Dynms of Two Ll Sysm h Dnsy Mr Soluon s: n hrfor: os ρ sn os os Tr os Sm s on rlr! C 47 Srng 9 Frhn Rn Cornll nrsy Dynms h Dnsy Mr wh Dohrn Th gonl omonns r unff y ohrn: Th off-gonl omonns r ssum o y u o ohrn: Th quon w g now s: C 47 Srng 9 Frhn Rn Cornll nrsy 3
32 Dynms h Dnsy Mr wh Dohrn Bounry onons: Soluon s: os sn Thrfor: os sn os sn C 47 Srng 9 Frhn Rn Cornll nrsy Dynms h Dnsy Mr wh Dohrn s m : Thrfor s : C 47 Srng 9 Frhn Rn Cornll nrsy 3
33 33 C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm h Dnsy Mr n h snrg Pur Th mlonn s:,, Th snrg quons r: C 47 Srng 9 Frhn Rn Cornll nrsy Dynms of Two Ll Sysm h Dnsy Mr n h snrg Pur sn sn os Th gol s o fn: Tr Tr Whn: Soluon of h snrg quon s: os Tr Th rsul s: Sm s on rlr!
34 Dynms of Two Ll Sysm h Dnsy Mr n h snrg Pur wh Dohrn Consr h snrg oror quons: Thr rg w.r.. h nsy oror wll yl h fmlr quons: Tr Tr In h rsn of ohrn h o quons go mof: C 47 Srng 9 Frhn Rn Cornll nrsy Tr Tr Dynms of Two Ll Sysm h Dnsy Mr n h snrg Pur wh Dohrn I woul mng o nrou ohrn h followng wy: ohrn rms PROBLM: Commuon ruls r lwys ssf ro oror m oluon quons oy h snrg quon:, B, B, B C C C Commuon ruls wll g ol whn w ohrn rms rrrly s on o!! Lr w wll nrou qunum Lngn quons o g or hs rolm C 47 Srng 9 Frhn Rn Cornll nrsy 34
35 Jon or Pou lr Ss Th lr s of wo nnn qunum sysms s on y sng oghr h lr ss of h nul sysms: Innr Prou: Or smly: Orors: n oror n hs nlrg lr s s wrn s nsor rou: B h oror s n s own lr s: B C 47 Srng 9 Frhn Rn Cornll nrsy B Jon or Pou lr Ss Full lr S: Consr wo ffrn wo-ll sysms n Th full lr s onsss of h followng four ss whh form oml s: Comlnss: C 47 Srng 9 Frhn Rn Cornll nrsy 35
36 36 C 47 Srng 9 Frhn Rn Cornll nrsy Jon or Pou lr Ss Th mlonn for wo ffrn wo-ll sysms s: Whr: Or wh som us of noon: n gns of h omn sysm s, for ml: gnlu C 47 Srng 9 Frhn Rn Cornll nrsy nnngl Ss Ss longng o omn lr s of wo sysms, n, r of wo ys: nnngl ss nngl ss nnngl Ss: Ths ss n wrn s: sysm"" unqu s of sysm" " unqu s of mls:
37 nngl Ss: nngl Ss nngl ss nno forz or sr n h sm fshon, for ml: s n nngl s n nno wrn s: C 47 Srng 9 Frhn Rn Cornll nrsy nnngl Ss n Msurmns Consr h oml un-nngl s of wo ffrn wo-ll sysms: 3 3/4 LSSO: Msurmn of sysm os no ff Ms. of nrgy of h rsuls for sysm /4 Ms. of nrgy of / / 3 3 Ms. of nrgy of Ms. of nrgy of 3/4 /4 3/4 /4 C 47 Srng 9 Frhn Rn Cornll nrsy 37
38 nngl Ss n Msurmns Consr h nngl s of wo ffrn wo-ll sysms: 3 3/4 Ms. of nrgy of /4 LSSO: Msurmn of sysm ffs h rsuls for sysm PR Pro, 935 Ms. of nrgy of /4 3/4 Ms. of nrgy of Ms. of nrgy of C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Orors for Jon lr Ss nnngl Ss: If h qunum s of sysm onssng of wo susysms n s n unnngl s: hn h nsy oror s: Thrfor, h nsy oror n wrn s nsor rou of h nsy orors of h susysms ml: Whr: C 47 Srng 9 Frhn Rn Cornll nrsy 38
39 39 C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Orors for Jon lr Ss nngl Ss: Consr h nngl s: Th nsy oror s: Th nsy oror for nngl ss nno wrn s nsor rou of h nsy orors of h susysms: C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Orors for Jon lr Ss ml: Suos: Clulon of rg nrgy of h s: 4 qul o only h hrln o gsnonzro nswr Tr Tr 4 3
40 4 C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Orors of Susysms: Prl Trs Somms nsy oror for wo or mor sysms onns oo muh nformon If on s nrs n only sysm u hs h on nsy oror for sysm n sysm, hn on ns o r nsy oror for sysm : Tr nnngl Ss: For unnngl ss, w now h: Tr Tr n hs s ly wh w o g! C 47 Srng 9 Frhn Rn Cornll nrsy Dnsy Orors of Susysms: Prl Trs nngl Ss: Consr h nngl s: Dnsy oror of h susysm s on s follows: Tr Thrfor, s ssl mur of ss n
41 nnglmn n Dohrn Thr s n nm onnon wn nnglmn n ohrn Brf Rw: Dohrn nronmn Dohrn nronmn Dohrn ms h off-gonl omonns of h nsy mr go o zro wh m! C 47 Srng 9 Frhn Rn Cornll nrsy nnglmn n Dohrn Frs, w n o m mol of h nronmn Suos h muully orhogonl nronmn ss r: 3 Th nl qunum s of h sysm s: Th nl on s of h sysm + nronmn s: Th nronmn msurs h s of h sysm suh h som lr m h nronmn s rfls h sysm s s follows: Ths s n nngl s Inron wh h nronmn Thr s no s olls! C 47 Srng 9 Frhn Rn Cornll nrsy 4
42 4 C 47 Srng 9 Frhn Rn Cornll nrsy nnglmn n Dohrn Inron wh h nronmn Thr s no s olls! ow w fn h nsy oror for h sysm y ng h rl r of h full nsy oror : full full Tr nronmn Dohrn!! Inron wh h nronmn m h off-gonl omonns of h sysm nsy oror go o zro
Chapter 1: Review of Quantum Mechanics
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