Set of square-integrable function 2 L : function space F

Transcription

1 Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E, o E E Ths appoach s moe unvesal than the smple example gven above. Fo example, n the case of nhomogeneous but lnea and sotopc medum such as photonc cystals the wave equaton govenng E can be wtten as H H c detemned (HW). H and c and E E, whee and ae opeatos to be In fndng solutons to these egenvalue equatons one often has to expand the wave functon (, H, E ) n tems of othe functons. Techncally, ths s called poectng o expandng the wave functon nto a gven functon space. You have aleady seen an example of ths method n ou dscusson of wave pacet and ts spectum decomposton. t, t g e d xd yd / g e d xd yd / wth g e d xd yd / In what follows we study the geneal popetes of the functon space, opeatos, and bases. If wave functon,t s squae ntegable then, t d If () s satsfed then we can always choose a multplcatve constant fo,t that, t d If,t satsfes () o ( ) we say,t s squae-ntegable. The set of squae-ntegable functons n mathematcs s called L set. () ( ) such

2 L set has the stuctue of Hlbet space. possess some egulaty. Fo example, be defned eveywhee, be contnuous eveywhee, and dffeentable. We also assume that ou wave pacet,t we eque that,t Assgnng the above physcal meanngs and constants to,t too wde fo ou puposes. We defne the functon space f as a subset of egula functons. L ; mples that L set s f L composed of suffcently Lneaty of functon space f whee and ae n Let f and f then f geneal complex; C. (HW), Inne Poduct of Two Functons The nne poduct of two functons and s defned by, d () The followng can be shown to be tue (HW),,,,,,,, (s a complex numbe), fo we Note that whle, d C have, d d (s a postve eal numbe), s called the nom of.,,, s the squae of the nom If,. If, we say and ae othogonal.

3 Schwat Inequalty,,,. The Equalty apples when nom of nom of Opeatos anothe Opeato A s a mathematcal entty whch assocates wth any f functon that may o may not belong to f A () Followngs ae examples of the opeatos Paty : x, y, x, y, Poston X : X x, y, x x, y, Dffeental x D x x, y, x, y, x D : Lnea opeato s such an opeato that satsfes the followng: A A A Whee C, Commutatos of two opeatos We defne: A B. In geneal A B B A A B The commutaton of A and B s defned as A B A B B A commutato opeato fo X and You can thn of Dx s.,. Show that as a vecto and opeato A as a matx. Bases n vecto functon space Dscete bass: Consde the countable set of functons s squae ntegable.,,,, n,, whee f.

4 The set * * s othonomal f, d Konece delta wth fo, fo If evey functon f, gven by c then we say the set, whee s the can be expanded n one and only one way n tem of, () When functons. Note that fom (), consttute a bass. consttute a bass, we sometme say that coeffcents of expanson,.e. c. s completely epesented n the s a complete set of bass by ts Coeffcents of ExpansonC The coeffcents of expanson c ae gven by (HW) c d, () In wods, the coeffcents of expanson ae the nne poduct of the wave functon and the bases functons. Clealy the coeffcents of expanson fo dffeent fom the coeffcents of expanson fo c s the same wave functon., even though c on the bass on anothe bass ae n geneal Whle a wave functon can be expessed on a bass n tems of sees of complex numbe (coeffcents of expanson, c ), the opeato A also can be expessed on the bass n tems of sees of numbes aanged n the fom of a matx. The nne poduct of two wave functons tems of the coeffcents of expanson Let f and f be expessed on the bass,.e. and accodng to n

5 c and b,, b b c b b c, b c b c c c, then d bc, d b c d Fo then, c c c One may daw analoges between bass functons Catesan coodnate ê vectos eˆ, x, y, eˆ, eˆ, eˆ x and unt vectos n. Fo example, n Catesan coodnates we have the bass y. Then note the followng analoges: Bass functons nt vectos n Catesan coodnates, eˆ eˆ ;, x, y, c M,m e ˆ x, y c, m eˆ M, b c M N mn eˆ eˆ mn mn Closue elaton We have aleady seen the othonomalty condton fo any two bass functons,.e. d. We now establsh a condton unde whch set condton s called closue elaton. Fo n the base,.e. consttutes a bass; ths to be a bass (a complete set) we must be able to expand any functon

6 c, whee, c, then () d,. () Intechangng and d we have. () Fo () to be tue must have the fom Theefoe, the closue elaton can be wtten as (Closue elaton) (4) Convesely f (4) s satsfed () s always tue, because d d. (5) Intechangng and obtan c. ode and followng the above pocedue n evese we wll Summay Othonomalty condton s gven by, d Closue elaton s gven by Note othonomalty nvolves bases wth dffeent ndces and, but the same agument ; howeve closue nvolves bases wth the same ndex, but dffeent aguments and. Bases that do not belong to the functon spacef : (bases that ae not squae-ntegable) So fa we have studed bases that wee squae-ntegable,.e. f, hee we elax ths condton and consde bases that L. We wll geneale ou esults fo expandng the wave functon on such bases and wll fnd the coeffcents of expanson, the othonomalty, and closue condtons.

7 We stat ou study wth the famla case of Foue tansfom and plane wave bass. Bases of Plane Waves and Foue Tansfom The One Dmensonal Case Consde the well nown Foue pa x x e d x xe dx () () Functons contnuous. Note that ntegal of Also wheeas fo x V x (plane waves) fom a set x e V x ove x dveges x x, was dscete fo x V V fo whch the ndex s V s not squae-ntegable., s contnuous. Expanson of wave functon x on V x bass We wte () and () as x x e x e d V xd wth V, and () x x e dx V xdx V, (4) Compang the above contnuous bass wth ou pevous dscete bass we see x V x d x c x and V xdx V, c dx x, s the coeffcent of expanson fo x n the x V bass. Ths coeffcent of expanson can be found fom the nne poduct of and the bass functons V. V

8 contnuous dscete Nom (squae of the nom) of the functon x Recall, x Fom Paseval s theoem (HW) t can be shown dx, x xdx x dx d Compae, d wth, Othonomalaton x xv xdx Fo the bass V, V V c d fo dscete bass. V the othonomalaton s gven by (HW), fo x Compae ths to x x dx Othonomalaton fo x bass t esults n Konece delta functon. Note that V, V V, V V V V, V V, V, bass. V esults n Dac-Delta functon whee as fo x Closue elaton fo V (x) Fo V t can be shown that closue elaton s gven by (HW) V x V x d x x. Snce x x x x x x V xv xd V xv xd x x x x then we can wte Compae the V xv x d x x x x x x fo dscete bass. (Closue elaton) fo contnuous bass wth

9 Extenson to Thee Dmensons In thee dmensons the plane wave bass s gven by V whee V / e Expanson of the wave functon s gven by e d d V / Coeffcents of expansons ae V, d V Inne poduct s gven by, d d Othonomalty condton s gven by V, V V, V Closue condton s gven by V V d V V d Delta functons as bass sng the popetes of Dac-Delta functon we have d () and d. () Whee stands fo x, y, and fo x, y, and x x y y. Let us defne bass L Note that Expanson of Functon whee on the Bass

10 d d, whee Compae these esults wth c (). Coeffcents of Expanson d,, whee d * * the dscete set c d, Note that coeffcents of expanson fo at each pont n space.. Compae these coeffcents of expanson wth on the bass s smla to c but whee s a contnuous ndex fo dscete ndex fo bass. ae the values of bass, s a Inne Poduct fo, d Compae the above to. Note ths s n fact the defnton of the nne poduct., b c. Othonomalty, d, d. Compae ths to Closue elaton d d Compae ths to.

11 Contnuous othonomal bases: genealaton We can geneale ou esults obtaned fo V and by ntoducng the contnuous othonomal bass W whch s a set of functon W labeled wth the contnuous ndex. The bass W wll satsfy the followng Expanson of wave d c W c functon Coeffcents of c W d W, c d, Expanson Scala poduct, d b c, b c Squae of the nom, c, c Othonomalaton elaton d W, W d W W, Closue elaton d W W