Relation between wavefunctions and vectors: In the previous lecture we noted that:

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1 Rlatio tw wavuctios a vctos: I th pvious lctu w ot that: * Ψm ( x) Ψ ( x) x Ψ m Ψ m which claly mas that th commo ovlap itgal o th lt must a i pouct o two vctos. I what ss is ca w thi o th itgal as th scala pouct o two vctos wh w a us to thiig o th wavuctios i th itgal as commo uctios o x? Bo givig a omal asw i tms o th Diac lta uctio, o ca simply call th iitio o accptal quatum mchaical stat uctios, o which at vy poit i spac th xists a sigl complx um, th asolut squa o which is popotioal to poaility sity o iig th paticl. This is Postulat o quatum mchaics i most txt oos. Imagi, ow, iiitly log colum a ow vctos i which a sto all th ums which a th valus o Ψ(x) a Ψ*(x), spctivly at all possil valus o x. Thus, all itgals i this otatio a actually just scala poucts o two vctos. Thy a omally iiit caus i calculus th sult is oly xactly coct i th limit o x 0. I all actual calculatios o ums, o computs, x is a vy small ut iit x. This tuly tus th uctio ito a vcto o ums, as lag as w ca to ma y usig small valus o x a lag ag o x. This is a i, caus th huma mi caot gasp iiitsimal a iiity. This ia is tu o ay um o cooiats, o cous. A m

2 Wavuctios a pstatios o a a t vctos : Expaig aitay stat uctios as lia comiatios o th complt st o iguctios o a osval, yamical vaial opato A: A aitay stat vcto > may xpss as a suppositio (lia comiatio) o mms o a complt st: wh Giv that th c Thus, c A ˆ a, wh a a th igvalus o Aˆ, a c a ums c c Iˆ, th vy wily us "Itity"opato Th ums c a sai to : > i th A pstatio. O may o cous us ay oth complt st to pst >. Fo xampl, th c om th complt st usig th iguctios o th Hmitia opato, /x, i.., si(x) a cos(x), is th Foui pstatio, tt ow as th Foui Tasom. Th st o ums < B > is similaly sai to th opato B i th A pstatio. Th Itity opato Nxt w itouc th spcial complt st ow as th positio iguctios.

3 Positio Eiguctios Th positio opato x op is simpl multiplicatio y th positio x. Th iguctios a th igious a wily us Diac lta uctios, with th symol δ(x-a), wh a is a al um, a th poptis: δ ( x a) 0 δ ( x a) δ ( x a) x xˆ δ ( x a) i i x x a a aδ ( x a) To atioaliz th omalizatio, o may thi o th uctio a xa to vy lag ut iit i a vy aow, ut iit, gio such that aa u th cuv. Thus xpaig > i th iguctios o x, i.., th Diac lta uctio, chags > ito th wavuctio Ψ (x) > i th positio pstatio, i aalogy with th pvious pag.

4 Th Vaiatio Picipl a Vaiatio Mthos O caot ovmphasiz th imms impotac o th vaiatio picipl to quatum chmisty, ut it is qually impotat i umous oth ils, may o thm i o-quatum giig applicatios. It tus out that igvalus a igvctos always mg wh a itial quatio is sujct to ouay coitios. Schöig himsl ivo a vaiatio picipl to suppot his omulatio o th Schöig Equatio. Quotig om A vaiatioal picipl is a scitiic picipl us withi th calculus o vaiatios, which vlops gal mthos o iig uctios which xtmiz th valu o quatitis that p upo thos uctios. Fo xampl...th shap o a chai susp at oth s... is ou y miimizig th gavitatioal pottial gy Exampls Lo Rayligh's vaiatioal picipl Ela's vaiatioal picipl Fmat's picipl i gomtical optics Th picipl o last actio i mchaics, lctomagtic thoy, a quatum mchaics Mauptuis' picipl i classical mchaics Th Eisti quatio also ivolvs a vaiatioal picipl, th Eisti Hilt actio Gauss's picipl o last costait Htz's picipl o last cuvatu Palatii vaiatio Th vaiatioal mtho i quatum mchaics Th iit lmt mtho

5 Th Vaiatio Picipl o Quatum Systms Th xpctatio valu o gy is th xpctatio valu o H. I quatum mchaics th vaiatio picipl is usually xpss as: * tial * tial H tial tial τ τ Th pow o th vaiatio picipl is that ay appoximat wavuctio o th gou stat o a quatum systm is guaat to giv a gy xpctatio valu, <E appox >, that will high tha th tu gou stat gy, E 0.. This allows o to vay th shap o th tial wavuctio y ay mas availal util th ivativ o <E appox > with spct to all vaials ig vai 0, a satisi that this will th most accuat wav uctio o th typ ig us as ϕ tial. Th is o ag o iig a gy that is too low. I aitio to th aov wavuctio schm, Koh a Sham sha th Nol Piz ctly o povig a hlpig implmt th so call sity uctioal thoy (DFT). With DFT, o ictly vais th lcto sity ϕ* tial ϕ tial to ach th gy miimum, ista o vayig ϕ tial, which must always squa to gt th gy ayway. Th a it vaiatio mthos which ca classii ito two oa classs: ()lia vaiatio mtho, a () o-lia vaiatio mthos. W will gi with a simpl o-lia vaiatio mtho xampl, which pvas most o th computatioal mthos w will cout i chmisty. E E 0

6 A quic xampl is usig th xact om o th gou stat wavuctio o a hyog-li atom o io. W guss that: uits i atomic a tial i atomic uits w ow that ˆ ˆ ), ( i atomic uits o 0 Z a Z agula aial T T L m Z Z m H θ H E E H tial tial tial tial * * 4 4 si π π τ τ ϕ θ τ τ τ

7 lgth uits! : Fo lgth uits! : Fo!!

8 Hyog-li io with chag Z Z Z Z E( ) appox 0 Z Z

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