Matrix Representation In Quantum Mechanic, Schemes and Mind Maps of Quantum Mechanics

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Unit 4 | Mais Ke prrrevtation | © + Stratum Mechantus formutacten in 2. oli ffereut won “i — -_ Schroctinger's Wave Heiseuberg s Matri Mechawu Mechanics (Continuous Arasis System) ( Dicvete basi system) |Lineae Vechr Space| Tt cowists oh foo sett of | ements aucol op abyebric Auber: (1) a oet of vecto YF, 1 ---- andl a det Bt Bcablarsy a,b,c -----. . (2) Q cube for vecpy Aololitien Ano. a Aule fer Scaler prutti plication [Adaition Rute} This Rule har Some properties ee (2) Sh wd dG ate Vechs vf 4 Space, their Sum (b+ p )is ma vector of the Sewme Space. (by ped bey Ceommaterey (co) lpr P+ X -yr(¢th) (Ascciative ) ho (4) O+yp=P~to=- Y (met enist a 227 vector) (¢) Y +(-¥) = (-v)+Y = 0 [Symmetric or Tive se vec Mn (plication Rute 2 Py perties Like ! | (2) Product mh Geotar itt vecpr fv Another ; Gentnally. Vdd ant tim Veclr the ae any Latur Combinat he ar thd UL also a mn *4 Yi pies a8 bon) (+) alwtd)-ayr+ad ; (atb)y-ayt by | ernst) (c) a(by) = (ab)y —> [Awoutative (OO Te-yr=y & op =yord | IL-7 O vitor Scatar 0 > Rew Scalar [Sineation and Basis of a vechy Space | Vv A Sey Ron- 22m vers Pp dg... Pry dt sald fo be Linearly ivepensent if aud only if th | cpa pass f only tf the Selution > the 2 tits= 0 iy qug ._-e po 2 > Any =O ~ c=] Qut > there eubdls a set of BS alars | wolich qoe not at) Zen, Sothat it cau se euprettedl ay | . N i > Zadi t S HF, (=i panel §0, t 4; | set 15 seid to de Li neasly ale henclest Bincnsion?- 9 ib Jiven the Wwartimum numlser Liven independent vel - ace can have. “t d ber 4, Rinearly inole pendent Vectors At mantimum yum dao hos N, ie (d,,% ---4Fy) — “This space 7 said to be N- Amewionad’ , ai &, Ay a Re fe) D (6a,- 2a, \f + (-94, +3, )j > 0 | | Ay = ay [w--27 | | an I are Lintarly B2frsstect. | [he Hilbert Space (only Read) | St consists Pf a vects Y, bys Se of Seaton a,b,c wid satapy pour pospestia» (Berneted by H | OH Ba Linear Space, . GH has def [inne Pret ie( sly Foie | Scakar pydoluct anu elemeit yr with f us * dnner Product of tuo vec a cemplee Nurtber. | A Fentral comblen number oleneteol by | | (¥,4) woken (Cv. = Compl Noma) B Seater product rf p with f Lt equal A Comblex — conjugel of, Scatar product of df) voith a | (vid) =(P¥)" sue ©($, av, thta)= ald, vi) +b 4, Ys) tee [ (At. + hte) = a*(4v) + b*(do,0) [free |! ® (¥,y)- Ivo > o rs product af a we 4 with tel) qives + ve Veal Nimber aN (2) H AL Se nr abe : Thene exist a thy jae ti — HO SUch that I» even y oH aud € 70, there ebists at Le ast Or A Yy i | y — v, | < € (Oy Uy Complete ty I¥—Ye | = © Time &® The Physical state oh a system is yefruented in Pantin mechanics oe elomenl a Hilbert face, these cloments are” ca dled [Bat vec» | [Quantum state]: oot is a Linear vectr in a Linear “jector space which tects ott information stmt the deste. . Jf can be in the (orm ep ware fyrnction, mato anel ap inhan, Noawbos, bE reprucnted 4 (Cv) Kets 2, Clements oh a Vec4or Space [J ‘ olerioted /Ae presented Qs | >» @ | “> vi | Column vecfo ras'~ © elements * a clual Sfrace [ce Vector) d * Acpusented as < | > Ly] > [Row Ver} 61 2 [ Gra vechr] | ta-ket |s- Inner / Scalar Product <[7 > bra-kett — <4] ¥> Note: For every ket [y> there enislt a umigue , bra ! + Probebitity obiplitee | ix cL Leierl ha tatty of fay Dips ten in State [uy> Q [Un (simi Lave) ste, Be NP eajoy! Mi oe ile t Space oo ci [ines no ——— Learning we oh mati, = (Raasien) © Conjugate ef Q matin | | “A-firsi UW) _, ate fl-3 a] oO. -Y¥y (2) qj © ‘Tans pose of a matin — Duter Changi qieg: FOU L column ~ ~ — re I 4 "LE 3) carly 3) (2) Tran ese Conpugate ~e mabin ragr] MER 2] (yea Peat O°) “Ty” | * “| is i af gtat (>) snob, 2s Shes 0 tgs madi _ [ar - A] in --A | | Aji © ai) ay aij © flee mition | A Skew Hey mnifion (Anti her ual Hes) ian [an --a | | , we Mi tj | © Drtagat Her (D Unitary Mabie | AAT= ATA =I Aateata-z . AtT= 4! at = Act JAl= 2 (>) Stmilari ky Trav bor mation = PAP A 2 & 7 Similar Matrices © OL ALR matter ace Sinular tht Hees have : shame trace e pame het rel “Gees @) Diegonatizatton fe tiaiee election Hye a fe=rar | % ele ee be cing jin td A. Pr “J |e dj = oT oje Gera 4 ne ? Co fo Pea Cy] ~Pmatit (a ye a, x - . a,* ~) So now mu a comblen number 7epresenkd ay b, i bn | . = 8 a, by | n | Numeci cabs Ot Consider [pLtowing 2 Kets | lyy = ca) etl J¢7 = S ) ©) fina the bra <4], (8) Svaluate the Scalar proline <4] Vw? ora) BI-|4+& a 81+ [thy 1431 [ 4: Q. 2 Cousider tle stated | wz . ‘i | 4,7 > Th | b.4 7 ly? = “149 +2i/[4,> Where [di aul I¢.7 ane orthonormal, ® catetare [ys Ky aud

} not, noxmalize it (©) fre [v7 4 [47 Orthogonal ? we 'S er) sy] -G5 2 0) fou” t (6) The normalization 4b ) (-si)(si)+ 202) + ((-0) ° ~2517 44-1? ual to | tb 2, 254441 =] Zo | —_— aver mot normalize (a) [¥7 = [1] Other Melrod e Novmatize, Cel 4 ye | \4? “if |] wae cwlaarft ] cP f i] cee eer) 9.9 ape defi! ' ja)? fret)! (I+!)= 2 japr>= + CH? = 2 | are | Now, math ply vwoith = Ez] Ixy = Now , - | 1 gly? | Iv? set) } Ue elt ec) LUD? EO bates] it te vet (b) yr me ltir + pla Sep Chock normalise or vet ? (x)"* (zy = st% = zt x | (rot roret) 2 Normalize it lero Af di7e tly] Step g Noy mali zadion ; IAP, lal = | 2 4 a 4. J Nal ] JAI? = 4 4 A= oe | day Wr 2 Llyyt Bx [4 ig FR y7-B yr pur te [7 Je Now, Cheek normalize apain, (fs) * () +27 FOr wily 3} wwe wart probability, thou P _ ((5]°- Be lal 4