lecture notes on qm1 - C h ap ter Q u an tumM ech an ics...

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Unformatted text preview: C h ap ter Q u an tumM ech an ics inH ilb ert Sp ace s .T h eA b strac tH ilb ertSp ace T h eessen t ia lresu lts inquan tumm echan icsa reg iv en th roughpu re lya lg eb ra icre la t ion s . Sp ec icresu ltscanb ed er iv ed ,e .g .,fo rv ecto rs ` andm a tr icesb e ing lin ea rm ap s ; how ev er ,tho seresu ltsa reessen t ia lly ind ep end en to fth esp ec icrep resen ta t iono fth e op era to rs .F o rth esp ec icresu ltson lya lg eb ra icre la t ion sb etw eenop era to rsandab stra ct p rop ert ieso fth eH ilb ertspa ceen ter .T h ispo in to fv iew a llow stocon s id erp rob lem s in fu ll g en era lityand th en con s id era sp ec icrep resen ta t iono fth eba s isv ecto rso fth eH ilb ert spa ceand th eop era to rs(e .g .,m a tr ices ,d ieren t ia lop era to rs) . I .T h eab stra ctH ilb ertspa ce ` isg iv enbya seto fe lem en ts H = ( j i ; j ' i ; ji ; ) , fo rw h ichadd it ionandm u lt ip lica t ionw ith com p lexnum b ers isd en ed j i + j ' i = j + ' iH ( .) a j i = j a iH ( .) tog eth erw itha sca la rp rodu ct h ' j iC : ( .) W ith resp ectto ( .)and ( .) , H isa lin ea rv ecto rspa ce ,i.e ., j i + j ' i = j ' i + j i ( j i + j ' i )+ j x i = j i + ( j ' i + j x i ) j i + ji = j i j i + j? i = ( .) T h e la sttw o re la t ion ssta teth eex isten ceo fa-v ecto rand th eex isten ceo fan ega t iv e v ecto rw ith resp ectto j i . j i = j i a ( b j i )= ( a b ) j i ( a + b ) j i = a j i + b j i a ( j i + j ' i )= a j i + a j ' i ( .) II .W ith resp ectto th esca la rp rodu ct , H isaun ita ryv ecto rspa ce h j i ( .) and h j i = ) j i = ( .) h ' j i = h j ' i h ' j a i = a h ' j i h ' j + i = h ' j i + h ' j i ( .) B ecau seo f( .)aN o rm canb ed en ed k k = q h j i ; ( .) w h ereth esp ec iccha ra cter ist icso fth eno rm d ep endon th ev ecto rspa ce . O n eha s jh ' j i j k ' kk k k ' + kk ' k + k k ( .) and h a' j i = a h ' j i : ( .) F u rth erpo stu la tesa re : III . H iscom p lete . IV . H issepa rab le . AH ilb ertspa ceb e ing sepa rab lem ean stha tth ereex istsa seto fv ecto rsd en se in H and coun tab le .L et f j i ;; j k ig b ea sequ en ceo fv ecto rs in H .I fw etak eou tev ery v ecto r j k i from th issequ en ce ,w h ichcanb erep resen teda s lin ea rcom b ina t iono fth e p rev iou sv ecto r j i ; j k ? i ,th enw eob ta inaseto flin ea r ind ep end en tv ecto rs f j ' i ; j ' n ig inw h ich th eo r ig ina lsequ en ce iscon ta in ed .T h eseto f f j ' n ig isv ia con stru ct iond en se in H .O n ecana ssum etha t f j ' n ig isa seto fo rthogona lv ecto rs( if no t ,u see .g .,G ram-S chm id to rthogona liza t ion ) . h ' m j ' n i = m n : ( .) I f f j ' n ig isd en se in H ,w ecan expand ev erya rb itra ryv ecto ra cco rd ing to th isba s is j i = X n = j ' n i a n ( .) and thu s h ' m j i = X n = h ' m j ' n i a n = X n = m n a n = a m ( .) fromw h ich fo llow tha tea chv ecto rcanb erep resen teda s j i = X n = j ' n ih ' n j i : ( .) T h ere la t ion ( .)ison lyva lid if f j ' n ig isacom p leteset ,andw ehav eth ecom p...
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This note was uploaded on 12/12/2009 for the course PHYS 270 taught by Professor Wilfredlee during the Spring '09 term at École Normale Supérieure.

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lecture notes on qm1 - C h ap ter Q u an tumM ech an ics...

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