Analytical Study of a Special Case of Complex Canonical Transform

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1 lobl Jornl o Mhmcl Scncs: hory n Prccl Volm, Nmbr 3 00, pp Inrnonl Rsrch Pblcon Hos hp://wwwrphoscom Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm PR Dshmkh n AS h Pro Rm Mgh Ins o chnology & Rsrch, Bnr Rlwy, Amrv, In E-ml: pshmkh9@gmlcom Dp o Mhmcs, ov Vrbh Ins o Scnc & Hmns, Amrv, In E-ml: lkgh@gmlcom Absrc Cnoncl rnsormon whn ppl o qnm mchncs, n grop n o b xn rom rl omn o complx orr A [5] h nroc rconl plc rnsorm o orr s spcl cs o lnr cnoncl rnsorm wh complx ns n h rprsnv mrx In hs ppr, w hv s rconl plc rnsorm s mppng o R no h Brgmnn-Hlbr spc o nlyc ncons ovr h complx pln Som proprs o rconl plc rnsorm r scss Opron rnsorm orml r vlop n h nvrs o h rnsorm s lso obn whch wll b sl whl pplyng hs rnsorm o solv prl rnl qons sly m- npnn Schröngr qon or h qnm hrmonc oscllor s solv by sng rconl plc rnsorm Kywors: nrl ncon, Cnoncl rnsorm, rconl plc rnsorm AMS Sbjc co: 44A0 Inrocon nr cnoncl rnsorm s n ngrl rnsorm wh or prmrs, b, c,, nroc by Moshnsky [] n 97 Cnoncl rnsorms hv ply bsc rol n solvng mny problms o clsscl mchncs I provs cnoncl ormlsm or h rspons o vry lrg clss o physcl sysms P n Dng [] hv s gn ncons o lnr cnoncl rnsorm Srn A [4] prsn or ncrny

2 6 PR Dshmkh n AS h rlons bwn lnr cnoncl rnsorm prs n scss hr mplcons n som common opcl sysms Bss hs s lso r s sl ool n qnm mchncs b or hs pplcon h omn o lnr cnoncl rnsorm rqr xnson o h complx ns nr cnoncl rnsorm s gvn by, [ C ] b b b h lnr cnoncl rnsorm s chrcr by mrx A, bc c I hs or prmrs n on consrn wh hr grs o rom h rl cnoncl rnsormons r nry rnsorms rom R no R, whr s, complx cnoncl rnsorms r mppng rom R no h Brgmnn- Hlbr spc o nlyc ncons ovr h complx pln Forr rnsorm, rconl Forr rnsorm, Frsnl rnsorm, Dlon, ss-wrsrss rnsorm r spcl css o lnr cnoncl rnsorm whn,b,c, r rl orr A [5] h consr lnr cnoncl rnsorm hvng chrcrng mrx A, wh complx cocns In prclr, whn chrcrng mrx s, A B n 0 A, 0 h Brgmnn rnsorm n plc rnsorm r obn rspcvly H h lso nroc Frconl plc rnsorm s spcl cs o complx lnr cnoncl rnsorm whn chrcrsc mrx s sn sn Frconl plc rnsorm h Frconl plc rnsorm s gnrlon o plc rnsorm hs rnsorm s mor lxbl n pplcons I s powrl ool or h nlyss o m vryng sgnls I s clr h h proprs n pplcons o h convnonl plc rnsorm r spcl css o hos o rconl plc rnsorm Dnon: h Frconl plc rnsorm o sqr ngrbl ncon s n n rms o krnl, s K [ ] whr K, whr, K co co co, s no mlplo δ, s mlpl o

3 Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm 63 In hs ppr w shll sy hs rnsorm nlyclly h ppr s orgn s ollows Scon consss o som proprs o krnl o rconl plc rnsorm In Scon 3 w hv gvn h spporng proprs o h rnsorm so h cn b r s rconl opror Mny opron rnsorm orml r prov n Scon 4 whch r rhr s n pplcons Scon 5 s vo o obn nvrson orml In scon 6 w hv solv Schröngr qon by sng rconl plc rnsorm Scon 7 concls h ppr Proprs o krnl Som proprs o h krnl o rconl plc rnsorm r s blow K gvn by, s krnl o h rnsorm, kn s ncon o wh prmr K K, * K K K, K 3 K K β, K 4 * β ' K K δ ' 5 h proprs,, 3 r rvl h rvon o propry 4 s gvn n Appnx A Propry 5 cn b rv sng, n 4 Frconl plc rnsorm s h rconl opror Drn css o rconl plc rnsorm r scss n h ollowng scon, whn o, hn o n [ ] I[ ] whn, hn, [ ] [ ] I cn b shown h h rnsorm krnl s proc h opror hs h ollowng proprs: s lnr 3 0 I Iny 3 Invrs 33 / conssncy wh plc rnsorm 34

4 64 PR Dshmkh n AS h 35 s connos or ll vls o h orr prmr 36 h bov proprs sggs h s h rconl opror gnrng rconl plc rnsorm or ny gvn rcon n s h rconlon o ornry plc rnsorm Opronl rnsorm Forml In hs scon, w hv vlop opron rnsorm orml or rconl plc rnsorm W ls nmbr o mporn proprs o h Frconl plc rnsorm h proprs o rconl plc rnsorm r sl no only n rvng h rc n Invrs rnsorm o mny m-rqncy ncons b lso n obnng svrl rsls n sgnl procssng As n cs o h convnonl plc rnsorm n opronl clcls xss n rconl plc rnsorm Som o h opronl clcls or rconl plc rnsorm s ls n bl I SrNo Proprs Sclng Frs Shng bl-i Clcls [ ] [ ] co coϕ snϕ ϕ co sn sn ϕ snϕ sn 3 Scon shng v v sn v [ ] vsn 4 Mlplcon by [ ] sn 5 Mlplcon by [ ] [ ] sn [ ] sn '[ ] sn ' '[ ] 6 Drnon rl [ ] sn [ ] [ ] 7 Ingron rl n n [ g ] sc [ g ] 8 Dvson rl 0 0 co co [ ] [ ] 9 Smlry rl [ ] [ ] `

5 Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm 65 Frconl plc rnsorm o Smpl Fncons Frconl plc rnsorm o som smpl ncons r bl s ollows bl-ii SrNo Sgnl Frconl plc rnsorm wh δ n co n co 3 δ sn co co n n [ sc sn sc] n n sc n n sn sc Invrson hs scon s vo o obn h nvrson orml or h rconl plc rnsorm by sng h nvrs orml or convnonl plc rnsorm h mho s s s pr Sonkk,h[3] Frconl plc rnsorm o s gvn by [ ] co co co, hn by nvrson s possbl o rcovr h ncon by mns o h nvrson orml co co co [ ] Proo: h rconl plc rnsorm [ ] o s gvn by

6 66 PR Dshmkh n AS h [ ] co co co, co co [ co ] co U F, co { U, U kng nvrs plc rnsorm F co co Sbsng h vls o F co co co co co co U U [ ] co co co [ ], Solon o h m- Inpnn Schröngr Eqon or h Qnm Hrmonc Oscllor W now pply h rconl plc rnsorm o h solon o h mnpnn Schröngr qon or h qnm hrmonc oscllor h ψ - k ψ Eψ 6 m whr h s plnks consn, k s h sprng consn, m s h mss o oscllor n E s h nrgy 4 I 4mk h n E h k m, hn qon 6 cn b ψ rc o ψ 0 ψ ψ ψ 0 6 kng rconl plc rnsorm o qon 6,w wll g

7 Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm 67 0 ψ ψ ψ sn sn sn o sn sn sn 63 0 sn sn sn sn sn sn 0 sn ± co co n whr ±, w n h sss h rnl qon o rs orr [ ] 0 sn sn sn o 0 4 Mlplyng by [ ] 0 4 Solon o hs qon s [ ] 4 c W now obn ψ by clclng nvrs rconl plc rnsorm ψ K, whr K co, co n [ ] 4 c or hrmonc oscllor

8 68 PR Dshmkh n AS h Conclson W hv prov proprs o krnl o rconl plc rnsorm n sblsh opronl rnsorm orml W hv lso gvn nvrson orml or rconl plc rnsorm Fnlly n pplcon xmpl o solon o h m-npnn Schröngr qon ws gvn Appnx A K, K, β co snsn β co co csc β co co β β β sn sn β β co K, co β co β co β csc β co co β csc csc β sn β snβ β β β β sn sn β β csc sn sn sn sn co β co β csc β β Appnx B : Sclng propry co [ ] co co co co vcscϕ co, v n cscϕ co coϕ v co co ϕ v coϕ coϕ coϕv cscϕ

9 Anlycl Sy o Spcl Cs o Complx Cnoncl rnsorm 69 ϕ co co ϕ co co v ϕ ϕ sn sn, v v Appnx B : Frs shng propry ] [ co csc co co co csc co csc co co co co U U csc co co csc co whr { U U co csc co U U csc sn Appnx B 3: Mlplcon by co csc co co co csc co csc co co ] [ csc co ] [ ] [ sn ] [ ] [ co { co csc co co }

10 70 PR Dshmkh n AS h co [ ] co co [ ] sn {} [ ] sn [ ] sn [ ] Appnx C: [ ] sn [ ] [ ] sn sn [ ] sn sn sn sn sn Rrncs [] Moshnsky M, Qsn C : Ag97 nr cnoncl rnsormons n hr nry rprsnons, JMhPhys Vol, No8, [] P & Dng J : 00, Egn ncons o lnr cnoncl rnsorm, IEEE, rnssgnproc Vol50, -6 [3] Sonkk PKn h AS:008 Anlycy n opron rnsorm on gnrl rconl Hrly rnsorm, InJornl o Mh Anlyss, Vol, No0, [4] Srn A : Mr008 Uncrny prncpls n nr cnoncl rnsorm omns n som o hr mplcons n opcsjopsocam-a, Vol5, No3, [5] orr A : 003, nr n rl cnoncl rnsorms o rconl orr, Jornl o componl n Appl mhmcs, Ily 53,