Chapter 8 Theories of Systems

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1 ~~ 7 Char Thor of Sm - Lala Tranform Solon of Lnar Sm Lnar Sm F : Conr n a n- n- a n- n- a a f L n n- ' ' ' n n n a a a a f Eg - an b ranform no ' ' b an b Lala ranform Sol Lf ]F-f 7 C 7 C C C ] a L a 99 交大電信 ] Sol ] ]

2 ~~ 7 ' ' 交大電子所 ] Sol C C C C 7 9 an Sol Lf ] F-f-f ] ] ] ]

~~ 7 7 7 Eg orng o oal wavg hor h E-fl of wo nal wavg an flfll h ol mo qaon E E z E E E z E κ β κ β E E whr κ h olng offn Fn E z an E z Sol LE Σ an LE /zσ -E Σ - Smlarl LE Σ an LE /zσ -E Σ β κ κ β κ

3 ~~ Eg orng o oal wavg hor h E-fl of wo nal wavg an flfll h ol mo qaon E E z E E E z E κ β κ β E E whr κ h olng offn Fn E z an E z Sol LE Σ an LE /zσ -E Σ - Smlarl LE Σ an LE /zσ -E Σ β κ κ β κ β κ κ β β L - Fa] f -a ] o a L a an ] a a L a z z L z E β κ o ] an z z L z E β κ ] In h a h olng lngh L π/κ Whl h wavgng mo ravr a an of o mll of h olng lngh L L L h oal owr omll ranfrr no h ohr wavg bak afr a an of vn mll of h olng lngh L L L If h wavgng mo ravr a an of o mll of h half olng lngh L / L / L / h oal owr qall rb n h wo g

4 ~~ 7 - Mar Solon of Lnar Sm Sol - agonalzabl ] - ] Sol mv-rank-i Mll of - no agonalzabl Sol f- - - For - mv-rankimll of agonalzabl - or

5 ~~ 77 Sol f ± - ] o ] -- ] o ] ] o ] o o o o 成大電研 ] Sol an - h h ' ' Parlar olon: T g T T g T - T T - g Choo T h ha T T T T

6 ~~ 7 / / Sol f : agonalzabl an Homogno olon: h ~ ~ Parlar olon: T g T T g T - T T - g Choo T h ha T T T T / / ln T ln ln ln

7 ~~ 79 Eg Thr ar wo of anmal: wolf an fo nrang whn h am for om L w an f no h wolf an fo olaon rvl a m So frhr ha wolv mgh a fo a foo an fo mgh alo a wolv a foo b onl wolv ar hn b hman If hr ar no fo hn on mgh ha h wolv lakng an aqa foo l wol ln n nmbr a a ra of -w Whn fo ar rn hr a l of foo an o wolv ar a o h for a a ra of f Frhrmor h hang ra of h wolf olaon alo ovl n on a aonal hnng faor no a On h ohr han f hr no wolv hn h fo lakng an aqa foo l wol ln n nmbr a a ra of -f whn wolv ar rn h fo olaon nra b a ra of w Pla anwr h followng qon: a Formla h abov m b a of ffrnal qaon b U varaon of aramr o olv h m Wha ar h a-a olaon of h wolf an fo rvl? 台大電研 ] Sol a f w f w b f - - Homogno olon h : h h h f w ~ ~ Parlar olon: T g T T g T - T T - g L T flfll T T T / / / / T / / / / 9 o 9 7 o T f w ~ ~ 9 o 9 7 o o 7 f w o 7 : a-a olon

8 ~~ / / / / Sol / / / / / / / / I / / / / I ln ln C Φ / / / / Sol Homogno olon: ΦC No: Φ / / / / Φ Parlar olon: Φ ' Φ Φ / / / / Φ Φ / / / / Φ ' Φ / / / / ' ln ln ln ln ln ln

9 - Nonlnar Sm an Pha Plan F onomo m: whr F an G ar boh nnn G of Pha lan: Th -lan for h anal of h aonomo m Thorm F a b P P Q lm lm G Q W hav -aa-b : ral > hn a no Sabl no n a of < an < Unabl no f > > : ral < hn a al on : oml wh nonzro ral ar hn h on whh a ral aroah Sabl ral n a of R< Unabl ral f R> : r magnar hn a nr of a lo rv Eg lm lm - ± R> an nabl ral on ~~

10 ~~ Eg - - an -->: ngav an n a abl no Chk: Eg an -< a al on Chk: If n a of Eg ± a nr of a lo rv Chk: o o Cral on: flfll boh F an G Thorm C h lo raor of h aonomo m G F whr F an G C hn hr a la on ral on of h m nlo b C Thr of lm l: Sabl nabl an m-abl lm l

11 ~~

12 Eg Fn h raor of h followng m: Sol L roθ rnθ -rnθθ roθθ r rr - r -r r r r - rnθ-rnθθ - roθroθθ -r θ -r θ - r r θ r θ θ θ an r r r If r r ; l f r < ; l r > r r In h aml h n rl r a abl lm l - Som roma Solon of Nonlnar Ornar ffrnal Eqaon Eg For a ml nlm of ma m wh a hra of lngh l how ha θ g nθ Solv an oban ro 99 台大電研 9 交大電控所 ] θ Sol Toal nrg onrvav: mg l l oθ m l Conan θ g nθ θ θ g g θ L ' nθ n θ l g g nθθ oθ oθ whr θ h nal angl / θ g θ oθ oθ / g o o θ θ T θ θ Pro: T g oθ oθ If θ mall θ g nθ θ θ g g θ o n ~~

13 Van r Pol qaon ] Sol For φ ' o φ If mall n φ ] o φ ] n φ ] o φ ] φ ] n φ ] φ o φ ] o φ ] n φ ] φ ] o φ ] φ n φ ] n φ ] o φ ] w θ φ o θ oθ π π o q q q q φ / φ rnoll' qaon φ φ Eg Oban h arlar olon of ω Γ o Sol So ] ω ] ω ] ω ] / ] Γ o ω ] O Γ o ω Γ o ω Solv ω Eg Oban h aroma arlar olon of o Sol ; " o o o o o o o ~~

14 - Srm-Lovll Thor R Conr R QP]F mll b an hn l r R R R R qq P an ff Srm-Lovll form: r ] q]f Eg Fn h gnval an gnfnon of - wh bonar onon an ranform no h Srm-Lovll form 99 台大造船所 ] Sol r -r r ± ± If If If r rval olon < -> r±k k -k rval olon > -< r±k oknk n k nπ h orronng gnfnon nπ π ] - - ] Eg Fn h gnval an gnfnon of - wh bonar onon a <<a 99 台大機研 ] Sol r -r- r ± If n n If > k If < -k k k k a n n a k a k k k ln k ln h o k ln ] h n k ln ] nπ nπ h a k nπ ln l ] n ] lna ln a ln a Eg Fn h gnval an gnfnon of - 7 交大電子所 ] Sol r -r r If an If an - an - ~~

Eg Fn h h gnval an gnfnon of π 7 台聯大電研 ] Sol If - ab ab: rval olon If -k > aokbk a kn -k -n

orronng fnon φ n an φ m hn flfll φ φ f n m For h rglar Srm-Lovll roblm an wo gnfnon orronng

-k no olon / n nm f n m Eg Th gnfnon an hr orronng gnval of h aonar Hlmholz qaon -k ψ ar rn

15 Eg Fn h h gnval an gnfnon of π 7 台聯大電研 ] Sol If - ab ab: rval olon If -k > aokbk a kn -k -n an n If --k < a k b -k no olon Srm-Lovll horm For h wo n n an m of h Srm-Lovll roblm wh b orronng fnon φ n an φ m hn flfll φ φ f n m For h rglar Srm-Lovll roblm an wo gnfnon orronng o a gvn gnval ar lnarl nn Eg For π/ If ab ab: rval olon If k > aokbk a kn k n If -k < a k b -k no olon / n nm f n m Eg Th gnfnon an hr orronng gnval of h aonar Hlmholz qaon -k ψ ar rn a follow a n m Th fr gnfnon k 77 Th on gnfnon k 9 Th hr gnfnon k 97 Th nh gnfnon k 97 ~~ 7

oal wavg ar rn a follow If h gnmo n no an nfnl-long ragh wavg an roaga along h wavg

16 Eg Gvn a rfrav n rbon n h gnmoal fnon Φ φ φ of an oal wavg flfll k n Φβ Φ whr β h gnval an β rrn h ha onan of h lo wavg or h roagaon onan of h lol wavg Th gnmo of om oal wavg ar rn a follow If h gnmo n no an nfnl-long ragh wavg an roaga along h wavg who an formaon Howvr n a h n lgh no an gnmo om oal owr lo or an hn bom h gnmo graall ~~

Eg On-mnonal wav fnon Ψ n a qanm wll L wh nfnl har wall V flfll lwhr Ψ/

ha h m n π gnval E qanz a E n an h orronng gnfnon ml Ψ n / L nπ/l Eg

m E V Ψ I ΨII m Ψ II wh bonar ΨIII m E V Ψ III onon: Ψ I Ψ II Ψ II LΨ III

17 Eg On-mnonal wav fnon Ψ n a qanm wll L wh nfnl har wall V flfll lwhr Ψ/ meψ/ n L wh bonar onon ΨΨL I an b ranform no h gnval Ψ roblm a EΨ I rov ha h m n π gnval E qanz a E n an h orronng gnfnon ml Ψ n / L nπ/l Eg On-mnonal wav fnon Ψ n a qanm wll wh wo fn onal wall V L flfll V lwhr ΨI m E V Ψ I ΨII m Ψ II wh bonar ΨIII m E V Ψ III onon: Ψ I Ψ II Ψ II LΨ III L Ψ I Ψ II Ψ II LΨ III L n α ΨI C me me h gnfnon hav h form a ΨII o α ΨIII ~~ 9

18 Eg Tnnl Eff: On-mnonal wav fnon Ψ n a qanm barrr V > E L V flfll lwhr ΨI m EΨ I ΨII m E V Ψ II wh ΨIII m EΨ III bonar onon: Ψ I Ψ II Ψ II LΨ III L Ψ I Ψ II Ψ II LΨ III L n hn h gnfnon hav h form a k k ΨI k k me m V E me ΨII C whr k k k k k ΨIII F Th qanm mhan an rov ha h ranmon robabl T Ψ III / Ψ I F / kl kl ] K / K ~~ 9

Canonical Quantizing of Spinor Fields: Anti-Commutation Relations

Canonical Quantizing of Spinor Fields: Anti-Commutation Relations JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN - 857 Prn ISSN - 858 Onln h://www.rrh.org/jornl/j/j.hml Cnonl Qnzng of Snor Fl: An-Common Rlon D. Grn PhD Unvr of Brln* Ar Nw mg of hr nor ro on h