Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

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1 Proprtis of Phas Spac Wavfunctions and Eignvalu Equation of Momntum Disprsion Oprator Ravo Tokiniaina Ranaivoson 1, Raolina Andriambololona 2, Hanitriarivo Rakotoson 3 raolinasp@yahoo.fr 1 ;jacqulinraolina@hotmail.com 1 ;raolina.andriambololona@gmail. com 1 ; tokhiniaina@gmail.com 2,infotsara@gmail.com 3 Thortical Physics Dpartmnt Institut National ds Scincs t Tchniqus Nucléairs (INSTN- Madagascar) BP 4279 Antananarivo 101, Madagascar, instn@moov.mg Abstract: This papr is a continuation of our prvious works about coordinat, momntum, disprsion oprators and phas spac rprsntation of quantum mchanics. It concrns a study on th proprtis of wavfunctions in th phas spac rprsntation and th momntum disprsion oprator, its rprsntations and ignvalu quation. Aftr th rcall of som rsults from our prvious paprs, w giv most of th main proprtis of th phas spac wavfunctions and considr som xampls of thm. Thn w stablish th ignvalu quation for th diffrntial oprator corrsponding to th momntum disprsion oprator in th phas spac rprsntation. It is shown in particular that any phas spac wavfunction is solution of this quation. Kywords:Quantum mchanics, Eignvalu quation, Disprsion oprator, Phas spac rprsntation, Wavfunction 1-Introduction Th prsnt work can b considrd as part of a sris of studis rlatd to a phas spac rprsntation of quantum thory introducd and dvlopd in [1], [2] and [3]. Bcaus of th uncrtainty rlation [4], th problm of considring phas spac, which mix momntum with coordinat, in quantum thory is an intrsting challng. Many works rlatd to this subjct hav bn alrady prformd. W may quot for instanc [5-12]. Many of ths works ar basd on th approach introducd by Wignr in [5]. In th rfrnc [1], w hav considrd an approach using th currnt formulation of quantum mchanics basd on linar oprators thory and Hilbrt spac (as in [4] for instanc). Our prvious work [1] may b considrd as xtnsion in th framwork of quantum thory of th rsults obtaind in [14]. Th phas spac rprsntation that w hav dfind is basd on th introduction of quantum stats, dnotd,,,. Ths stats ar dfind by th mans valus, and statistical disprsions (varianc) ( ) and ( ) of coordinat and momntum ( bing a positiv intgr numbr). W hav th rlations 1

2 ( ) =(2+1)() =(2+1) ( ) =(2+1)() =(2+1)B =() B =() (1) = ħ 2,,,,,,=,,,,,,=!,,,"( ) $ ",,,%=(2+1)() (2)!,,,"( ) $ ",,,%=(2+1)() and bing rspctivly th momntum and coordinat oprators. Th wavfunctions corrsponding to a stat,,, rspctivly in coordinat and momntum rprsntations ar th Harmonic Hrmit-Gaussian functions & and thir Fourir transforms &' dfind in [1] and [14]. ( ( )*+ & (,,,)=,,,= ) 0 * ħ (3) -2! 2/ ( ;) ( ( <*= &' (,,,)=,,,= -2! 2/ ) &' (,,,)= 1 (,,,)0 *8 >2 ħa 0 *1> *8+ (>39) ħ (4) ( is a Hrmit polynomial of ordr. It has bn stablishd that a stat,,, is an ignstat of th momntum disprsion oprator B =ℶ 7 and th coordinat disprsion oprator B = B ℶ7. Th xplicit xprssions of ths oprators ar and th corrsponding ignvalus quations ar B =ℶ 7 = 1 D 2 E( ) +4B( ) F B = B ℶ7 = 1 (5) 2 E4( ) +( ) F H B,,, =(2+1)B,,, B,,, =(2+1),,, (6) As th ignvalus of th momntum and coordinat disprsion oprators ar proportional and as thy hav th sam ignstats, it is sufficint to considr only th momntum disprsion oprator ℶ 7. In th rfrnc [2], w hav tackld th problm of finding th rprsntations of coordinat, momntum and disprsion oprators in th framwork of th phas spac rprsntation. W hav stablishd that thy can b at th sam tim rprsntd both with matrix and diffrntial oprators in th basis {,,, } dfining this phas spac rprsntation. 2

3 In this papr, our goal is to prform a study on th proprtis of wavfunctions in th phas spac rprsntation and thir rlation with th ignvalu quation of th momntum disprsion oprator ℶ 7. W show and vrify xplicitly, in particular, that ths wavfunctions ar as xpctd th ignfuctions of th diffrntial oprator which rprsnts ℶ 7. 2-Phas spac rprsntations of momntum disprsion oprator Lt us considr th momntum disprsion oprator ℶ 7. It can b put in th form in which ℶ 7 = 1 2 E ( ) +4 B ħ ( ) F=4Bℶ 7 =B( + ) (7) ℶ 7 = 1 4 ( + ) = ( ) 2 ) =( 2B = 2 ħ ( )= 2 ħ ( ) = ( ) 2 =( ) 2 = 2 ħ ( )= 2B ħ ( ) (8) In th papr [2], w hav stablishd that in th phas spac rprsntation, w hav for th oprators and on on hand th matrix rprsntations N =,,, O,,,= 1 2 ( OP N*Q+ O+1P N7Q ) N =,,, O,,,= ; (9) 2 ( OP N*Q O+1P N7Q ) which satisfy th commutation rlation and on th othr hand th diffrntial oprators rprsntations: S N S S N S =;P N (10) U= 1 2 (;ħ V V )= 2(; V V ħ ) U= ; 2 V (11) V = ; ħ V 2V From th rlation (8) and (9),w obtain for th matrix rprsntation of th oprator ℶ 7 ℶ 7 N =,,, ℶ 7 O,,,= 1 4 ( S S N + S S N )= 1 4 (2+1)P N (12) and from th rlation (8) and (11), w obtain for its diffrntial oprator rprsntation ℶU 7 = 1 4 ( U$ +U $ )= 1 2 V V E V BV 2;B ħ V V + B ħ $ F (13) 3

4 Using th rlation (7), (12), (13) and (1), w obtain rspctivly for th matrix and diffrntial oprator rprsntations of th momntum disprsion oprator ℶ 7 ℶ 7 N =,,, ℶ 7 O,,,=4Bℶ 7 N =(2+1 )BPN (14) V ℶU 7 =4BℶU 7 =E ħ$ V 2BV 2BV 4;B ħ V V +2B ħ $ F (15) W may rmark that th xprssion of ℶ 7 N corrsponds to th fact that th lmnts of th basis {,,, } dfining th phas spac rprsntation ar th ignvctors of ℶ 7. 3-Phas spac wavfunctions A phas spac wavfunction Ψ (,,) of a particl is a phas spac rprsntation of its quantum vctor stat X. It is qual to th innr product of th vctor X with an lmnt of th basis {,,, } dfining th phas spac rprsntation. As stablishd in our paprs [1], w hav xplicitly th rlations Ψ (,,)=,,, X=@ & (,,,)X()A =@ &' (,,,)X[()A (16) in which, th functions X()= X and X[()= X ar th wavfunctions corrsponding to th stat X rspctivly in coordinat and momntum rprsntations. Th functions & (,,,) and &' (,,,) ar th complx conjugats of th wavfunctions & (,,,) and &' (,,,) corrsponding to th stat,,, rspctivly in coordinat and momntum rprsntations. Th xprssions of & and &' ar givn in (3) and (4). Thr ar two possibilitis to xpand a stat X in th basis {,,, } of th phas spac rprsntation. Th first on corrsponds to any fixd valu of, and and th xpansion is obtaind by varying th positiv intgr : X =\Ψ (,,),,, (17) Th scond on corrsponds to any fixd valu of and and th xpansion is obtaind by varying th ral numbr and X Ψ (,,),,, AA 2/ħ (18) W can also hav two anothrs kind of xpansion of a th stat X : th first on is obtaind by fixing th valus of and and varying th valus of th ral numbr and th positiv ral numbr X Ψ (,,) E( ),,, F AA (19) /ħ ^ 4

5 th scond on is obtaind by fixing th valus of and and varying th valus of th ral numbr and th positiv ral numbr = ħ X Ψ (,,) ^ E( ),,, F AA /ħ (20) To th four abov rlations (17), (18), (19) and (20) corrspond th following rlations btwn X(),X[(),& (,,,) and &' (,,,) : X()= \Ψ (,,)& (,,,) X[()= \Ψ (,,)&' (,,,) Ψ (,,)& (,,,) Ψ (,,)&' (,,,) Ψ (,,) ^ Ψ (,,) It can b stablishd that w hav also th rlations ^ AA 2/ħ AA 2/ħ E( )& (,,,)F AA /ħ E( )&' (,,,)F AA /ħ \ Ψ (,,) Ψ (,,) AA=1 (22) 2/ħ In th framwork of th probabilistic intrprtation of quantum mchanics, ths rlations prmits us to giv th following physical intrprtation of th phas spac wavfunctions: Th rlations (17) and (21) allow to intrprt th function Ψ (,,) as th probability to find a particl in a stat,,,, for a fixd valu of, and knowing that th stat of this particl is X. Th rlations (18) and (22) allow to intrprt th function _`(+,=,) 5 as th dnsity of probability to find a particl in a stat,,,, for a fixd valu of and, knowing that th stat of this particl is X. aħ 5

6 W may list thr particular xampls of phas spac wav functions by choosing rspctivly a particular valu of th stat vctor X For X = For X = Ψ (,,)=,,, =& (,,,) Ψ (,,)=,,, =&' (,,,) For X = b, b, b, Ψ (,,)=,,, b, b, b, =d (,,, b, b, ) w may stablish an xplicit xprssion of th function d (,,; b, b, ) by prforming, for instanc, a calculation in th coordinat rprsntation d (,,; b, b, )=@& (,,,)& (,,,)A (23) using th xplicit xprssion of th function & (,,,) as givn in th rlation (3) and proprtis of Hrmit polynomial, w can prform th calculation of th intgral and find th xplicit xprssion (s appndix) d (,,; b, b, b )=g (,,, b, b, b )0 * (434 ) 5 h( 5 i 5 ) * (939 ) 5 h( 5 i 5 ) *j ( 5 4i 5 4 ) ħ 5 i 5 (=*= ) (24) in which g (,,, b, b, b ) is th polynomial b g (,,, b, b, b )=\\E 2( 1) *N (;) S7N!!( b ) *S7N7k 5() *N7S7 k 5 SN l!( l)!o!( O)!E2( + )F`i`ik b 5 ( 7 *S*N( w may considr thr intrsting particular cass: For = b (and = ), w hav d (,,; b, b,)=g (,, b, b,)0 *p434 q 5 b -2( + b ) )( ( b ) S7N( )F (25) -2( + ) r 5 * p939 5 q r 5 *8p434 qp939 q 5ħ (26) with (,,; b, b, b ( 1) *N (;) S7N!! )=\\E 2 7 l!( l)!o!( O)! g b S N ( 7 *S*N( b 2 )( S7N( b )F (27) 2 6

7 For = b (and = ), = b and = b w hav d (,,;,,)=,,, b,,,=p 0 ;s =H 1 ;s = Th introduction of th function d (,,; b, b, ) prmits to dduc intrsting proprtis of th stats,,, and th phas spac wavfunctions. In fact, as w hav for any stat X X =\Ψ (,,),,, (28) It follows in particular that, for X =,,,, w hav th rlation,,, =\d (,,; b, b, b ),,, (29) Lt us now considr any stat X.It rsults from (28) that in th basis {,,, } w hav and in th basis,,, X =\Ψ (,,),,, (30) X=\Ψ (,, ),,, (31) Insrting th rlation (29) in (31) and idntifying with (30), w may dduc th intrsting proprty which holds tru for any phas spac wavfunction Ψ (,,) Ψ (,,)=\d (,,; b, b, b )Ψ (,, ) (32) 4-Diffrntial quation satisfid by phas spac wavfunctions On on hand, from th rlations (14) and (17), it can b dducd that for any phas spac wavfunctions Ψ (,,)=,,, X w hav th rlation,,, ℶ 7 X=\(2+1 )BP N Ψ N (,,)=(2+1)BΨ (,,) (33) N and on th othr hand, from th dfinition of diffrntial oprator rprsntation, as givn in our work [2], w hav,,, ℶ 7 X= ℶU 7 Ψ (,,) (34) in which ℶU 7 is th diffrntial oprator rprsntation of th momntum disprsion oprator ℶ 7 givn in th rlation (15). 7

8 V ℶU 7 = ħ$ V 2BV 2BV 4;B ħ V V +2B ħ $ (35) It follows from th rlations (33), (34) and (35) that any phas spac wavfunction Ψ (,,) satisfis th diffrntial quation E ħ$ V V 2BV 2BV 4;B ħ V V +2B ħ $ FΨ =(2+1)BΨ (36) This quation is also th ignvalu quation for th diffrntial oprator rprsntation ℶU 7 of th momntum disprsion oprator ℶ 7. And thn, according to it, any phas spac wavfunction Ψ is an ignfunction of ℶU 7 with th ignvalu qual to (2+1)B. Th quation (36) can b also chckd xplicitly using th proprtis (32) of th phas spac wavfunctions. In fact, according to this rlation w hav Ψ (,,)=\d (,,; b, b, b )Ψ (,, ) so, taking into account th xprssion (35) of ℶU 7, w hav ℶU 7 Ψ (,,)=\EℶU 7 d (,,; b, b, b )FΨ (,, ) But according to th rlation (23) d (,,; b, b, )=,,, b, b, b, =@& (,,,)& (,,,)A so, taking again into account th xprssion (35) of ℶU 7, w hav ℶU 7 d (,,; b, b, )=@EℶU 7 & (,,,)F& (, b, b, b )A (37) Using th xplicit xprssion of & givn in (3) and th xprssion of ℶU 7 givn in (35), w can stablish aftr a long but straightforward calculation th rlation ℶU 7 & (,,,)=(2+1)B& (,,,) (38) This rlation mans that th function & (,,,) is an ignfunction of ℶU 7 with th ignvalu (2+1)B. Using th rlation (38), w can dduc from th rlation (37) that ℶU 7 d (,,; b, b, b )=(2+1)Bd (,,; b, b, b ) (39) Us of th rlations (32), (35) and (39) allows to hav, as xpctd, an xplicit chcking of (36). 8

9 5- Conclusion In th phas spac rprsntation, th momntum disprsion oprator ℶ 7 can b at th sam tim rprsntd ithr with th diagonal matrix ℶ N 7 givn in th rlation (14) or with th diffrntial oprator ℶU 7 givn in th rlation (15).This fact can b xploitd to stablish th quation (36) which is th ignvalu quation of th oprator ℶU 7. According to this quation, any phas spac wavfunction Ψ (,,)=,,,B X is an ignfunction of ℶU 7 with ignvalu qual to (2+1)B. This rsult can b considrd as bing logically xpctd sinc th lmnts of th basis {,,, } dfining th phas spac rprsntation thmslvs ar th ignstat of ℶ 7. In th sction 3, w hav don som rcall concrning th phas spac wavfunctions and giv most of thir main proprtis. Among ths proprtis, w may notic a particular on which is givn in th rlation (32). This rlation shows that th valus of a phas spac wavfunction Ψ corrsponding to diffrnt valus of th paramtrs and variabls,, and can b linkd using th function d (,,; b, b, b ) dfind in th rlation (23). As discussd in th last part of th sction 4, this particular proprty of phas spac wavfunction givn in th rlation (32) prmits also to prform an xplicit chcking of th ignvalu quation (36) of ℶU 7. Appndix Establishmnt of th xprssion on th function u v v (w,x,y;w b,x b,y b ) in th coordinat rprsntation, w hav d (,,; b, b, )=,,, b, b, b, =@& (,,,)& (,,,)A Using th xplicit xprssion of th function & givn in th rlation (3). W obtain d (,,; b, b, b )=E i h ( ( 2 )( ( b 2 ) b 0 *{( 5 i 5 h 5 5))5 *E 5 4i *8(939 ) ħ F)} AF To prform th calculation of th intgral, w introduc th gnratric function z() of th Hrmit Polynomials. Using this function, w obtain th 0 (234 5 ) 7(234 5 )}* 5 *} 5 A z(,{)=0 ) * 5 = \ { (! () = \ \E {! ( ( 2 )( ( b 2 ) b 0 *{( 5 i 5 h 5 5))5 *E 5 4i *8(939 ) ħ F)} AF 9

10 Th xprssion of th intgral in th first mmbr of this rlation can b obtaind asily using th rlation this obtaind xprssion can b put in th 0 (234 5 ) 7(234 in which is th function 5 )}* 5 *} 5 A (, b,, b,, b,ˆ)= 0 0 *()5 7 )) A= / 0ƒ5 h / =ħ ( + b ) 0* 4 ( 34 9 ) 7( 39 )Š* 5 *Š 5 5( 5 i 5 ) 5( 5 i 5 ) =\\ Œ! =\\ Œ! ˆ Ž! ˆ Ž! (434 ) 5 h( 5 i 5 ) * (939 ) 5 h( 5 i 5 ) *j ( 5 4i 5 4 ) ħ b ( ( -2( + b ) )( ( b -2( + b ) ) V 7 (V ) (Vˆ),Š = + { + ~ ; ; b ˆ= + {+ + ~ b 5 i 5 (=*= ) Using th rlations btwn (,ˆ) and ({,~), w can stablish th following rlation V 7 V{ V~,} =\\E ( 1) *N (;) S7N! b!( b ) *S7N () *S7N SN V 7 ( + )`i` 5 (V ) 7 *S*N (Vˆ) S7N,Š =\ \E ( 1)* (;) S7N! b!( b ) *S7N () *S7N SN l!( l)!o!( b O)!p + b q `i` 5 F ( 7 *S*N( so w hav for th xprssion of (as function of { and ~) =\ \ {! ~! V 7 = \ \ { (V{) (V~)!,} ( b ) *S7N () *S7N `i` p + b 5 q ~! ( 7 *S*N( b -2( + b ) )( b S7N( -2( + b ) ) {\\E ( 1)* (;) S7N! b! l!( l)!o!( b O)! SN b -2( + b ) )( b S7N( -2( + b ) )} 10

11 Introducing this xprssion (of ) in th following 0 (234 5 ) 7(234 =\ \E {! ~! 5 )}* 5 *} 5 A / =ħ ( + b ) ( ( 2 )( ( b 2 ) b (434 ) 5 h( 5 i 5 ) * (939 ) 5 h( 5 i 5 ) *j ( 5 4i 5 4 ) ħ 0 *{( 5 i 5 h 5 5))5 *E 5 4i i 5 (=*= ) *8(939 ) ħ F)} AF w can obtain th xprssion of th intgral containing product of Hrmit polynomials and this rsult prmits us to dduc that d (,,; b, b, b ) i h 5 5 ( ( -2 77Q!/ 2 )( ( b 2 ) b =g (,,, b, b, b )0 * (434 ) 5 h( 5 i 5 ) * (939 ) 5 h( 5 i 5 ) *j ( 5 4i 5 4 ) ħ 5 i 5 (=*= ) 0 *{( 5 i 5 h 5 5))5 *E 5 4i *8(939 ) ħ F)} AF in which g (,,, b, b, b ) is th polynomial b g (,,, b, b, b )=\\E 2( 1) *N (;) S7N!!( b ) *S7N7k 5() *N7S7 k 5 SN l!( l)!o!( O)!E2( + )F`i`ik b 5 ( 7 *S*N( b -2( + b ) )( ( b ) S7N( -2( + ) )F 11

12 Rfrncs [1] Ravo Tokiniaina Ranaivoson, Raolina Andriambololona, Rakotoson Hanitriarivo, Roland Raboanary, "Study on a Phas Spac Rprsntation of Quantum Thory", arxiv: v3 [quant-ph],intrnational Journal of Latst Rsarch in Scinc and Tchnology Volum 2, Issu 2: pp26-35, [2] Hanitriarivo Rakotoson, Raolina Andriambololona, Ravo Tokiniaina Ranaivoson, Raboanary Roland, "Coordinat, momntum and disprsion oprators in phas spac rprsntation",arxiv: [quant-ph],intrnational Journal of Latst Rsarch in Scinc and Tchnology ISSN (Onlin): Volum 6, Issu 4: Pag No. 8-13, July-August [3] Raolina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson, Damo Emil Randriamisy, "Disprsion Oprator Algbra and Linar Canonical Transformation",arXiv: v2 [quant-ph], Intrnational Journal of Thortical Physics,Volum 56, Issu 4, pp , Springr, April 2017 [4] Raolina Andriambololona, "Mécaniqu quantiqu", Collction LIRA, Institut National ds Scincs t Tchniqus Nucléairs (INSTN- Madagascar),1990 [5] E.P. Wignr, "On th quantum corrction for thrmodynamic quilibrium", Phys. Rv 40, , 1932 [6] H.J. Gronwold, "On th Principls of lmntary quantummchanics",physica12, 1946 [7] J.E. Moyal, "Quantum mchanics as a statistical thory", Procdings of th Cambridg Philosophical Socity 45, , 1949 [8] G Torrs-Vga, J.H. Frdrick, "A quantum mchanical rprsntation in phas spac", J. Chrn. Phys. 98 (4), 1993 [9] H.-W. L, "Thory and application of th quantum phas-spac distribution functions",phys.rp 259, Issu 3, , 1995 [10] K. B Mollr T. G Jorgnsn,G. Torrs-Vga,"On cohrnt-stat rprsntations of quantum mchanics: Wav mchanics in phas spac". Journal of Chmical Physics, 106(17), DOI: / , 1997 [11] A. Nassimi, "Quantum Mchanics in Phas Spac",arXiv: [quant-ph], 2008 [12] T.L Curtright,C.K. Zachos, Quantum Mchanics in Phas Spac,arXiv: v2 [physics.hist-ph], [13] D. K. Frry, Phas-spac functions: can thy giv a diffrnt viw of quantum Mchanics, Journal of Computational Elctronics, Volum 14, Issu 4,pp ,Dcmbr [14] Ravo Tokiniaina Ranaivoson, Raolina Andriambololona, Rakotoson Hanitriarivo. "Tim- Frquncy analysis and harmonic Gaussian functions",pur and Applid Mathmatics Journal.Vol. 2, No. 2,2013, pp doi: /j.pamj