Hermitian Operators. Eigenvectors of a Hermitian operator. Definition: an operator is said to be Hermitian if it satisfies: A =A


 Horatio Francis
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1 Heriti Opertors Defiitio: opertor is sid to be Heriti if it stisfies: A A Altertively clled self doit I QM we will see tht ll observble properties st be represeted by Heriti opertors Theore: ll eigevles of Heriti opertor re rel Proof: Strt fro Eigevle Eq.: A Eigevectors of Heriti opertor Note: ll eigevectors re defied oly p to ltiplictive cber costt A Ths we c choose the orliztio THEOREM: ll eigevectors correspodig to distict eigevles re orthogol Proof: A Strt fro eigevle eqtio: ( c ) ( c ) A Te the H.c. (of both sides): A Te H.c. with : A Use A A: Cobie to give: A A Cobie to give: This c be writte s: A ( ) 0 Sice # 0 it follows tht So either i which cse they re ot distict, or 0, which es the eigevectors re orthogol
2 Copleteess of Eigevectors of Heriti opertor THEOREM: If opertor i Mdiesiol Hilbert spce hs M distict eigevles (i.e. o degeercy), the its eigevectors for `coplete set of it vectors (i.e coplete bsis ) Proof: M orthoorl vectors st sp Mdiesiol spce. Ths we c se the to for represettio of the idetity opertor: Degeercy Defiitio: If there re t lest two lierly idepedet eigevectors ssocited with the se eigevle, the the eigevle is degeerte. The `degree of degeercy of eigevle is the ber of lierly idepedet eigevectors tht re ssocited with it Let d be the degeercy of the th eigevle The d is the diesio of the degeerte sbspce Eple: The d cse Let s refer to the two lierly idepedet eigevectors % d & There is soe opertor W sch tht for soe we hve: W % % % d W & & & Also we choose to orlize these sttes: % % d & & Lier idepedece es % & #. If they re ot orthogol (% & # 0), we c lwys se GrSchidt Orthogoliztio to get orthoorl set
3 GrSchidt Orthogoliztio Procedre: Let A secod orthogol vector is the, Proof:,, bt # # Therefore, C be cotied for higher degree of degeercy Alogy i d: r e r Reslt: Fro M lierly idepedet degeerte eigevectors we c lwys for M orthoorl it vectors which sp the Mdiesiol degeerte sbspce. If this is doe, the the eigevectors of Heriti opertor for coplete bsis eve with degeercy preset,, 0 r e r r + e r y r y + e r z r z r e r r ( e r ) + e r e r y ( r ) + r r e y z ( e z r ) r e # r e r e e r # e ( ) r Phy85/Lectre 4: Bsis sets d represettios A `bsis is set of orthogol it vectors i Hilbert spce logos to choosig coordite syste i D spce A bsis is coplete set of it vectors tht sps the stte spce Bsis sets coe i two flvors: discrete d cotios A discrete bsis is wht we hve bee cosiderig so fr. The it vectors c be lbeled by itegers, e.g. {,,, M}, where M c be either fiite or ifiite The ber of bsis vectors is either fiite or cotble ifiity. A cotios bsis is geerliztio whereby the it vectors re lbeled by rel bers, e.g. { }; i < <, where the pper d lower bods c be either fiite or ifiite The ber of bsis vectors is `cotble ifiity.
4 Properties of bsis vectors Eple property orthogolity orliztio stte epsio copoet/ wvefctio proector opertor epsio Mtri eleet discrete cotios # ( ) c d ( ) c ( ) d A A A dd A(, ) A A (, ) A A Cosider the reltio: ' A To ow ' _ or ' yo st ow its copoets i soe bsis Here we will go fro the bstrct for to the specific reltio betwee copoets Abstrct eqtio: Proect oto sigle it vector: Isert the proector: # A# # A# # A# # A# # # A # A # [ d ] d d # # # d d # ( % #) # d Trslte to vector ottio: Se procedre for cotios bsis: c c # A # A c A c # A # # d A # ( ) d # A(, )# ( )
5 Eple : Cobiig differet bsis sets i sigle epressio Let s sse we ow the copoets of ( i the bsis {,,, } c ) ( Let s sppose tht we oly ow the wvefctio of ' i the cotios bsis { } '() ) ' I dditio, we oly ow the tri eleets of A i the lterte cotios bsis { } A(,') ) A ' How wold we copte the tri eleet ( A '? Chge of Bsis Let the sets {,,, } d {,,, } be two differet orthoorl bsis sets Sppose we ow the copoets of ' i the bsis {,,, }, this es we ow the eleets {c }: How do we fid the copoets {C } of ' i the lterte bsis {,,, } A A # % # A d A dd d# A # # d # % dd c A(, ) ( ) We see tht i order to copte this ber, we eed the ierprodcts d ' This is esily hdled with Dirc ottio: A % %& %& %& d A dd d# dd d# c A (, # ) # ( ) The chge of bsis is ccoplished by ltiplyig the origil col vector by trsfortio tri U. A A # # These re the trsfortio coefficiets to go fro oe bsis to other
6 The Trsfortio tri The trsfortio tri loos lie this & U % M M M L# L L O Eple: D rottio Let s do filir proble sig the ew ottio Cosider clocwise rottio of diesiol Crtesi coordites: The cols of U re the copoets of the old it vectors i the ew bsis If we specify t lest oe bsis set i physicl ters, the we c defie other bsis sets by specifyig the eleets of the trsfortio tri
7 Cotied Sry Isert proector oto ow bsis Bsis sets c be cotios or discrete The iportt eqtios re: d # ( ) Chge of bsis is siple with Dirc ottio:. Write ow qtity. Isert proector oto ow bsis. Evlte the trsfortio tri eleets 4. Perfor the reqired stios C c
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