THERMAL STATES IN THE k-generalized HYPERGEOMETRIC COHERENT STATES REPRESENTATION

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1 THE PULISHING HOUSE PROCEEDINGS O THE ROMNIN CDEMY Sris O THE ROMNIN CDEMY Volum 9 Numbr 3/ THERML STTES IN THE -GENERLIED HYPERGEOMETRIC COHERENT STTES REPRESENTTION Duša POPOV Uivrsity Polithica Timisoara Dartmt of Physical oudatios of Egirig -dul Vasil Pârva No 33 Timisoara Romaia dusa_oov@yahoocou dusaoov@utro bstract I our rvious ar (Proc Romaia cad ) w hav built ad xamid som rortis of th -cohrt stats xrssd through th -hyrgomtric fuctios s a cotiuatio of that ar whr w hav focussd oly o th ur stats i th rst ar w dal with mixd (thrmal) stats for th systms with liar as wll as uadratic rgy sctra Ky words: cohrt stats -hyrgomtric fuctio thrmal stat dsity orator INTRODUCTION Th classical gralid hyrgomtric (GHG) fuctios hav a sris of alicatios i diffrt brachs of hysics [] mog thm thy ar usd for dfiig th most gral class of cohrt stats (CSs) gralid hyrgomtric cohrt stats (GHG-CSs) itroducd i [] ad alid to thrmal stats of th sudoharmoic oscillator i [3] Ev if th classical GHG fuctios turd out to b vry usful i trms of alicatios i th last dcads wr dfid th so calld gralid hyrgomtric (-GHG) fuctios { } { } ai b x as a formal owr sris [4 5] { } {( a ) } ( b ) i x ( a ) i ( b ) x! x ( ) () whr x C R ad N Particularly for th -GHG lad to th usual GHG fuctios I a rvious ar [6] w hav built ad xamid som rortis of th -GHG-CSs by usig th diagoal ordrig oratio tchiu (DOOT) W hav limitd oly to th -CSs of th arut-girardllo id (-GHG-G-CSs) although this rocdur ca b usd also i th cas of Klaudr-Prlomov or Gaau-Klaudr CSs I ordr to avoid rharsals w aal ad us hr th otatios ad formula of ar [6] W cosidr two Hrmitic cougat orators that act o th oc vctors > as ( ) ( ) > > > > () Thy ar coctd with th dimsiolss Hamilto orator H as follows: ( ) H > > > (3) ( ) Th dimsiolss rgy igvalus ( ) m m ad th structur costats ar

2 43 Duša POPOV ( ) m m ( b ( m ) ) ( a ( m ) ) ( ) ( ) m ( b ) ( a )! i which as w will s blow aar i th xrssio of -GHG-CSs Th followig uality holds > ( ) ( ) > Th vacuum stat roctor ca b obtaid through th stadard rocdur usig th DOOT ruls [6]: >< { } { } ai b Th -gralid hyrgomtric arut-girardllo cohrt stats (-GHG-G-CSs) > ar dfid as usual i as th igstats of th lowrig orator [7]: > > whr th comlx umbr x(iϕ) with ad ϕ π labls th -GHG-G-CSs Thir xasio ito th oc vctor basis > is > { } {( a ) } ( b ) i ( ) > I [6] w hav showd that th -GHG-G-CSs satisfy all Klaudr s ruirmts imosd to ay CS: a) cotiuity i comlx labl b) ormaliatio c) o orthogoality d) uity orator rsolutio with uiu ositiv wight fuctio of th itgratio masur ) tmoral stability ad f ) actio idtity [8] Grally th itgratio masur is xrssd usig th Mir s G-fuctio: (4) (5) (6) (7) ai Γ dϕ d µ d ( ) ( ) b π Γ ai / { ( ai) } {( b) } G i b / (8) whr MIXED STTES Lt us w xami th thrmal stats dscribd by th dsity orator ω H ( ) >< ω is a dimsiolss thrmal aramtr ad is th artitio fuctio: ()

3 3 Thrmal stats i th -gralid hyrgomtric cohrt stats rrstatio 433 Tr < > ( ) () y usig th DOOT E (5) ad thir cougat coutrart th dsity orator bcoms: ( ) ( ) (3) Th Husimi s -distributio fuctio dfid i th -GHG-G-CSs rrstatio is [9] ( ) ( ) ( ) < > ( ) { } { } a b i d µ Th dsity orator ca b xadd as a surositio of CSs roctors (s g [9]): ad it is ormalid to uity: ( ) (4) d µ P > < (5) To dtrmi th xrssio of th uasi-distributio fuctio ( ) P w follow th sam rocdur as thr for dtrmiig th itgratio masur with th fuctio chag ad th rsolutio of suitabl Stilts momt roblm (s [6]) This fuctio is also ormalid to uity: d() P ( ) Th xctatio valu of a hysical obsrvabl i a mixd (thrmal) stat is calculatd as < > Tr( ) d µ P < > (6) whr th otatio sigifis th diagoal ordrig (DOOT) of roduct orators ( ) [3] I aalogy with th Madl aramtr calculatd i th GHG-G-CSs w hav dfid th Madl aramtr for th thrmal stats calld th thrmal coutrart of th Madl aramtr (T ) []: ( ) ( T ) (7) Th sig (ositiv ro or gativ) of (T ) as fuctio of uilibrium tmratur T show whos id of statistics (sur-poissoia Poissoia or sub-poissoia) oby th thrmal stats s 3 SOME SIMPLE PRTICULR CSES W will illustrat th abov idas i two cass of dimsiolss rgy igvalus: a) th liar ( ) sctra ( ) ( ) ( ) ad b) th uadratic sctra with rsct to th ( ) rgy uatum umbr Hr is a ositiv costat ad is th dimsiolss ro rgy trm W will articulari without customiig it agai th mai rsults obtaid abov

4 434 Duša POPOV 4 a) Th liar sctra is charactristic for som uatum oscillators g th o-dimsioal uatum oscillator (HO-D) for which / [9] or th sudoharmoic oscillator (PHO) for which ( ) / α / mω r / 4 [] Th dimsiolss rgy igvalus m ar ( m ) ( m ) ( ) ( ) () m m m m (3) Cosutly ad so that th structur costats bcom ( ) ( ) m ( m) ( )! ( ) ad th GHG fuctio i this cas is th Kummr coflut -hyrgomtric fuctio: ( ) () ( ) ()! y articulariig ad w asily gt th vacuum roctor >< ( ) Th th GHG-G-CSs for th cas of liar sctra bcoms > ( ) ( ) (3) (33) (34) > () (35) whr a ad b whil th itgratio masur that assur th uity orator dcomositio is () d ϕ ( ) π d µ d Γ or th articular cas of HO-D ad (i if w glct th isigificat trm of ro dϕ d d d π π Th dsity orator for th cas of liar sctra is iddt o th costat () rgy) w rcovr th itgratio masur µ HO ( ) ω H >< whr ω is a dimsiolss thrmal aramtr ad ( ) is th os-eisti distributio fuctio or th avrag of th hoto umbr i th thrmal stat Cosutly by ormaliig th dsity orator to uity w obtai th artitio fuctio: ( ) (36) (37)

5 5 Thrmal stats i th -gralid hyrgomtric cohrt stats rrstatio 435 With th hl of th DOOT ruls ad th vacuum roctor th dsity orator bcoms: ( ) ( ) Th Husimi s - fuctio is obtaid by rlacig th orator roduct with th (38) [6] (39) d µ ad this ca b dmostratd by usig a itgral of id 764 (s [ 8]) P from th sctral dcomositio of th dsity orator () This fuctio is ormalid to uity i ( ) Th uasi-distributio fuctio () d µ ( P ) > < (3) is dtrmid by solvig a suitabl Stilts momt roblm ad th fial (ormaliabl to uity) rsult is P ( ) Th thrmal xctatio valu of itgr owrs s of umbr articl orator is (3) s s d X dx X (3) with X x( ) With th abov formula w obtai ad so that th thrmal coutrart of th Madl aramtr is ( ) s a fuctio of uilibrium absolut tmratur T ω / th thrmal Madl aramtr is a ositiv fuctio for all fiit tmraturs So th corrsodig thrmal stats oby sur Poissoia statistics ( ) b) Th uadratic sctra whos dimsiolss rgy igvalus ar is charactristic for som aharmoic oscillators g th Mors oscillator (MO) [] Hr th costat is strictly ositiv ad N Without affctig th grality of th roblm w ca cosidr so th ( ) () rgy igvalus ar Cosutly th structur costats bcoms s ( ) ( ) m ad th idxs of ormaliatio fuctio ar ad Th GHG fuctio associatd with this cas is m () ( m) ( )! max max ()! This fuctio ca b xrssd also i a uivalt mar ( ) (33) (34)

6 436 Duša POPOV 6 Γ J (35) whr w usd th xrssio of th hyrgomtric fuctio through ssl fuctio (i fact th ssl olyomial of th dgr th itgr art of [ ] max ) ( ) of th first id x J ν [3] Th vacuum roctor is Γ >< J (36) Th GHG- G-CSs for th cas of uadratic sctra is > > max () ) ( (37) whr b whil th HG-G-CSs roctor by rsctig th DOOT ruls is < > (38) Th itgratio masur that assurs th uity orator dcomositio is () d d d J K ϕ µ π (39) Usig th followig itgral of [] it ca b rovd that this itgratio masur assur th rsolutio of th uity orator I ordr to writ th ormalid caoical dsity orator w will rorgai th rgy xotial accordig to our rvious itroducd asat []: () x! (3) whr i th frot art aars th xotial orator actig o th variabl Th th dsity orator for th cas of uadratic sctra is max x >< (3) whr [ ] max ) ( (th itgr art) rrsts th umbr of boud stats of th xamid Mors systm (g th diatomic molcul) Th artitio fuctio is calculatd as

7 7 Thrmal stats i th -gralid hyrgomtric cohrt stats rrstatio 437 max () ( max ) x (3) With th DOOT ruls as wll as th xrssio of th vacuum roctor w ca writ th dsity orator i th followig mar: x (33) W obsrv that hr th GHG fuctio ( ) rlacig th roduct rducs to a olyomial of dgr max y with w obtai th Husimi s -distributio fuctio ( ) x (34) Th dsity orator ca b rrstd by th followig sctral dcomositio () d µ ( P ) ( ) > < (35) ad w will choos th uasi-distributio fuctio ( ) P P of th form ( ) x P ( ) (36) ftr a fuctio chag ad th rsolutio of a suitabl Stilts momt roblm th fial xrssio (ormaliabl to uity) bcoms P (37) K K x y itgratig this xrssio o th variabl lad to th dfiitio of th artitio fuctio which mas that ( ) ad usig th itgral of [] w P is ormalid to uity Th thrmal xctatios of th itgr owrs of umbr articl orator is as wll as th corrsodig thrmal Madl aramtr max () s s (38) max () max () ( ) N max () max () < > (39)

8 438 Duša POPOV 8 Th charactr of distributio that govrs ths thrmal stats must b xamid umrically 4 CONCLUDING REMRKS I this ar w hav xamid th statistical (thrmal) rortis of th gralid hyrgomtric arut-girardllo cohrt stats ( GHG-G-CSs) by usig th ruls of th diagoal ordrig oratio tchiu (DOOT) [3] W dalt with two ids of systms: liar (harmoic sudoharmoic oscillators) ad uadratic (Mors oscillator) rgy sctrum for which w calculatd both distributio fuctios: ( ) ad P ( ) Statistical charactristics of th GHG-G-CSs for thrmal stats wr xamid by calculatig th thrmal Madl aramtr Thir valus (gativ ro or ositiv) show what id of statistics is associatd to xamid stats (sub Poissoia Poissoia or sur Poissoia) as fuctio of th uilibrium tmratur T t th limit all formula ad xrssios for th GHG-G-CSs td to th corrsodig formula ad xrssios for th usual (classical) gralid hyrgomtric cohrt stats or th harmoic limit ad w rcovr formula ad xrssios for th caoical CSs of th o-dimsioal harmoic oscillator y cocludig th rst ar is a st forward rgardig th alicatios of th GHG fuctios cocrtly i dfiig th GHG-G-CSs ad xamiig thir thrmal rortis Through articulariatio of th idics ad aramtrs of GHG-G-CSs it ca b obtaid th all ow CSs with hysical sigificac I this cotxt th ar richs th litratur rfrrig to th CSs REERENCES M MTHI R K SXEN Gralid Hyrgomtric uctios with licatios i Statistics ad Physical Scics Lctur Nots i Mathmatics 348 Srigr-Vrlag rli 973 T PPL D H SCHILLER Gralid hyrgomtric cohrt stats J Phys : Math G D POPOV M POPOV Som oratorial rortis of th gralid hyrgomtric cohrt stats Phys Scr R DÍ E PRIGUN O hyrgomtric fuctios ad Pochhammr -symbol Divulgacios Matmáticas S MUEEN G M HIULLH Itgral Rrstatio of Som -Hyrgomtric uctios It Math orum D POPOV O som rortis of th -cohrt stats Proc Romaia cad O RUT L GIRRDELLO Nw cohrt stats associatd with o-comact grous Comm Math Phys J R KLUDER Cotiuous rrstatio thory I Postulats of cotiuous rrstatio thory J Math Phys D WLLS G J MILURN uatum Otics Srigr-Vrlag rli 995 D POPOV Photo-addd arut-girrdllo cohrt stats of th sudoharmoic oscillator J Phys : Math G I S GRDSHTEYN I M RYHIK Tabl of Itgrals Sris ad Products Svth Editio cadmic Prss mstrdam 7 D POPOV S H DONG N POP V SJERT S SIMON Costructio of th arut-girardllo uasi cohrt stats for th Mors ottial Phys htt://fuctioswolframcom/ssl-tyuctios/sslj/6/// 4 D POPOV Gaau-Klaudr uasi-cohrt stats for th Mors oscillator Phys Ltt Rcivd Dcmbr 7