Quantum Pha Oprator and Pha Stat Xin Ma CVS Halth Richardon Txa 75081 USA William Rhod Dpartmnt of Chmitry Florida Stat Univrity Tallaha Florida 3306 USA A impl olution i prntd to th long-tanding Dirac pha oprator problm for th quantum harmonic ocillator. A Hrmitian quantum pha oprator i formulatd that mirror th claical pha variabl with propr tim dpndnc and atifi trigonomtric idntiti. Th igntat of th pha oprator ar olvd in trm of Ggnbaur ultraphrical polynomial in th numbr tat rprntation. I. INTRODUCTION Lt Ĥ b th Hamiltonian of a quantum harmonic ocillator Hˆ qˆ pˆ Nˆ ( 1) (1.1) 1 whr ˆq and ˆp ar dimnionl poition and momntum oprator ˆN i th numbr oprator with igntat { n n 01...} ˆ N aˆ aˆ Nˆ n n n (1.) â and â ar annihilation and cration oprator with rlation ˆ ˆ ˆ ˆ ˆ ˆ a q ip a q ip (1.3) aˆ n n n 1 (1.4) aˆ n n 1 n 1 (1.5) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (1.6) [ a a ] 1 [ a N] a [ a N] a. 1
Th miing pha oprator for th quantum harmonic ocillator ha bn an nigma for ag. Dirac [1] in 197 attmptd to dfin a pha oprator ˆ by auming a polar dcompoition of th annihilation oprator with th potulatd commutation rlation ˆ ˆ 1/ aˆ i N (1.7) [ ˆ Nˆ ] i. (1.8) Unfortunatly Eq. (1.7) do not dfin a Hrmitian pha oprator ˆ bcau (1.8) lav n ˆ n undfind. Louill [] in 1963 potulatd th commutation rlation ˆ i i not unitary and Eq. [co ˆ Nˆ] iin ˆ [in ˆ Nˆ] i co ˆ (1.9) in plac of Eq. (1.8). Howvr Eq. (1.9) apparntly do not dfin a Hrmitian pha oprator ˆ ithr. Sukind and Glogowr (SG) [3] in 1964 dvlopd th coin and in oprator on th dcompoition of â Ĉ and Ŝ. Bad ˆ ˆ (1.10) 1/ aˆ ( N 1) E whr Ê i a on-idd unitary hift oprator ˆ 1 1 (1.11) Eˆ nn E n n n0 n0 with ˆ ˆ ˆ ˆ ˆ ˆ EE I E E I 0 0 (1.1) SG dfind oprator Ĉ and Ŝ a ˆ 1 ˆ ˆ 1 C E E Nˆ aˆ aˆ Nˆ 1/ 1/ ( ) [( 1) ( 1) ] (1.13) S ˆ 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ i E E i N a a N 1/ 1/ ( ) [( 1) ( 1) ] (1.14) with th commutation rlation quivalnt to Eq. (1.9) [ Cˆ Nˆ] isˆ [ Sˆ Nˆ] icˆ. (1.15)
Although Ĉ and Ŝ ar Hrmitian thy do not commut and do not atify th trigonomtric idntity ˆ ˆ 1 [ CS ] 0 0 i (1.16) Cˆ Sˆ Iˆ 0 0. (1.17) 1 Th ignvalu pctra of Ĉ and Ŝ wr found by Carruthr and Nito [4] to b continuou on [ 11] Cˆ co co co (0 ) (1.18) c c c c Sˆ in in in ( / / ). (1.19) Pgg and Barntt (PB) [5] in 1989 propod to dfin a pha oprator ˆ in an ( 1)-dimnional ubpac { n n 01... } by th following polar dcompoition aˆ Nˆ (1.0) i ˆ 1/ whr i ˆ i a cycling oprator unitary in i ˆ 01 1 1 0. (1.1) Th PB thory claim that on can u ˆ to calculat phyical rult uch a xpctation valu in and tak th limit aftr th calculation i don. Th lat trm 0 in Eq. (1.1) i artificial in ordr to connct th two nd of th finit pctrum of N ˆ to form a loop o that i ˆ i unitary i ˆ 0. (1.) Eq. (1.) not only lack jutification in phyic but alo crat a problm for th mathmatic of PB limiting proc [6]. It i quit obviou that calculation rult bad on th artificial circular pctrum of N ˆ in may not convrg a. For xampl lim 0 ( Nˆ ) 0 lim Nˆ lim. (1.3) iˆ ˆ i A a mattr of fact th PB formalim i quivalnt to what Louill and Gordon propod in 1961 [7-8]. In thi papr w dfin a Hrmitian pha oprator that proprly mirror th claical pha and i fr of th problm ncountrd by variou approach bad on polar dcompoition of â du to th 3
on-iddn of th numbr oprator pctrum [9]. Th dfinition of th pha oprator and th olution of it igntat ar givn in Sc. II. Th proprti of th pha oprator ar analyzd in Sc. III. II. FORMULATION OF PHASE OPERATOR A propr pha oprator ˆ for th quantum harmonic ocillator hould rmbl th claical pha variabl and i xpctd to hav th following dird proprti. whr 1) ˆ i Hrmitian in th infinit-dimnional Hilbrt pac H ˆ ˆ ˆ H H { n n 01...}. (.1) ) ˆ atifi th trigonomtric idntity ˆ ˆ ˆ (.) co in I. 3) ˆ ha propr tim dpndnc. Equation ˆ ( t) ˆ (0) t i not xpctd to hold tru at all tim in th Hinbrg pictur (othrwi ˆ( t) would ffctivly b a tim oprator). Howvr it hould hold tru at lat at th half priod tim t / to rflct th point of invrion whn qˆ( / ) qˆ(0) and pˆ( / ) pˆ(0) ˆ ( ) ˆ (0). (.3) In othr word an igntat of ˆ volv into th igntat at t /. W dfin our pha oprator ˆ uch that A. Dfinition of pha oprator ˆ ˆ 1 ˆ 1 ˆ (.4) 1/ 1/ co ( N ) q ( N ) ˆ ˆ 1 ˆ 1 ˆ (.5) 1/ 1/ in ( N ) p ( N ) ˆn ˆn1 co ˆ n ( 1) in ˆ n ( 1). ( n)! (n1)! (.6) n0 n0 4
Our dfinition of ˆ co and ˆ in ar compltly conitnt with co and in of th claical pha variabl rprnting th prcntag of th total nrgy ditributd to poition and momntum rpctivly at any tim. Clarly our pha oprator ˆ i Hrmitian and atifi th trigonomtric idntity (.). From Eq. (1.3) and (.4) - (.5) w find Applying Eq. (1.4) - (1.5) w hav 1 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (.7) 1/ 1/ co co in ( N ) ( a a )( N ). ˆ 1 ˆ 1 ˆ (.8) 1/ 1/ ( N ) a ( N ) n f ( n 1) n whr 1 1 ˆ ˆ ˆ (.9) 1/ 1/ ( N ) a ( N ) n f ( n 1) n nn ( 1) n 1... f( n) ( n1/ )( n3 / ) 0 n 0 (.10) which dcra aymptotically to 1 for n 1 8 f ( n) 1 lim f ( n) 1. (.11) 5 n From Eq. (1.6) it i traightforward to find th following commutation rlation 1 1 ˆ ˆ ˆ ˆ ˆ ˆ (.1) 1/ 1/ [co N] ( N ) ( a a )( N ) [co ˆ Nˆ] [co ˆ Nˆ] (.13) 1 [in ˆ Nˆ] [co ˆ Nˆ]. (.14) 1 Not that imply taking th invr coin of Eq. (.7) do not giv a complt dfinition of ˆ. W will giv xplicit dfinition of ˆ in th nxt ubction with it tim dpndnc takn into conidration. 5
Lt b an igntat of co ˆ B. Eigntat and ignvalu co ˆ. (.15) Expanding in th numbr tat { n } and applying Eq. (.7) - (.9) w hav ˆ 1 co f ( n 1) n f ( n 1) n n nn (.16) n 0 n 0 Comparing th cofficint of n in Eq. (.16) w find th following rcurrnc rlation for n f ( n 1) n n f ( n 1) n n 01... (.17) Now lt u dcompo th Hilbrt pac H into a dirct um of two orthogonal ubpac o H H H (.18) whr H and W can thn writ Sinc H and H o ar pannd by vn and odd numbr tat rpctivly H o ar clour of H { 0...} H { 1 3...}. (.19) o o H o H. (.0) ˆ co from Eq. (.7) - (.9) namly co ˆ H co ˆ o H (.1) both and o ar igntat of co ˆ with ignvalu. W thrfor hav o o ˆ 0 co o o. 0 (.) For vn and odd numbr tat rpctivly th rcurrnc rlation (.17) bcom f (n 1) ( n 1) n f (n 1) ( n 1) (.3) and f (n ) ( n 1) 1 o n 1 o f ( n) ( n 1) 1 o. (.4) 6
Th olution to th rcurrnc rlation (.3) - (.4) ar traightforward (.5) 1 1/8 (1/4) n N n (1/ 4)(1 ) Cn ( ) n 01... with (.6) 1 1/8 (3/4) n 1 o N n (3/ 4)(1 ) Cn ( ) n 01... ' o ' o ( ') ' o 0 (.7) whr ( C ) n ( ) ar Ggnbaur ultraphrical polynomial orthogonal on [ 11] with rpct to th wight function (1 ) 1/ for fixd 1/ 1 (1 ) 1/ ( ) ( ) ( ) ( ) C ( ) 1 m Cn d Nn mn (.8) and n 1 ( n ) ( n) n! ( ) N ( ). (.9) Sinc [co ˆ ˆ] 0 by dfinition (.6) co ˆ and ˆ hav a common t of igntat. Lt { ( 11)} b th common t of igntat of both co ˆ and ˆ dfind by th following unitary tranformation 1 1 1 ( 11) 1 1 o (.30) with ' ' ( ') ' 0. (.31) W dfin our pha oprator ˆ uch that ˆ 0 1 1 co ( 11). 0 (.3) Eigntat and will b rfrrd to a pha tat bcau thy volv into ach othr priodically and ar out of pha by a w will how in Sc. III. From Eq. (.3) any analytic function of ˆ can b aily xprd in th bai. For xampl tan ˆ corrponding to th claical xprion tan p/ q i givn by 7
ˆ tan 0 1 1 tan co ( 11). 0 tan (.33) For convninc w can u notation to collctivly dnot pha tat and 1/4 (1 ) (0 / ) co ( 11) 1/4 (1 ) ( 3 / ) (.34) with normalizd to a δ-function on th angular pctrum Th idntity oprator can b rolvd a ' ( '). (.35) 3 1 1 1 1 0 Iˆ d d d d. (.36) Th pctrum rprntation of ˆ thrfor i With th xplicit dfinition ˆ our 3 d 0 ˆ d. (.37) ˆ co and Ŝ oprator without th problm in Eq. (1.16) - (1.17). ˆ in can b viwd a a normalization of SG Ĉ and III. PROPERTIES OF PHASE OPERATOR A. Tim volution of pha oprator Th tim volution of pha tat and i dtrmind by th following quation (lt 1 for notational convninc) ithˆ 1 int it int n n n 1 n 1 o n0 n0 (3.1) ithˆ 1 int it int n n n 1 n 1 o. n0 n0 At tim t / 3 / w hav (3.) 8
i Hˆ 1 1i 1i 1i 1i i Hˆ 0 1 1 0 3 i Hˆ 1 1i 1i. 1i 1i (3.3) (3.4) (3.5) Eq. (3.4) how that and volv into ach othr at tim t. ˆ ˆ From Eq. (3.3) - (3.5) ˆ ith ( ) ˆ ith t (0) at tim t / 3 / can b xprd a i / ˆ( ) i / 0 ˆ( ) 0 3 i / ˆ( ). i / (3.6) (3.7) (3.8) Not that Eq. (3.7) i quivalnt to Eq. (.3) confirming ˆ( t) i a pha hift oprator by at tim t. Similarly for co ˆ ( t) co ˆ ( t) and co ˆ ( t) at tim t / 3 / w hav ˆ 0 iin co ( ) i in 0 (3.9) co ˆ ( ) co ˆ (0) (3.10) ˆ 3 0 i in co ( ) iin 0 (3.11) ˆ ˆ ˆ ˆ ˆ 3 ˆ (3.1) co ( ) in (0) co ( ) co (0) co ( ) in (0) ˆ ˆ ˆ ˆ ˆ 3 co ( ) co (0) co ( ) co (0) co ( ) co ˆ (0). (3.13) 9
Similar xprion for in ˆ ( t) in ˆ ( t) andin ˆ ( t) can b aily found by wapping "co" and "in" in Eq. (3.9) - (3.13). Although ˆ( t) co ˆ ( t) in ˆ ( t) ar not diagonal at t /and t 3 / co ˆ ( t) in ˆ ( t) and thu co ˆ ( t) in ˆ ( t) ar. Thrfor in pha tat th altrnating ignvalu of ˆ co btwn and vry quartr priod impli that pha information i at th granularity of quartr priod (a oppod to bing infinitly fin with a claical harmonic ocillator). whr B. Pha ditribution in numbr tat It i ay to how that in any numbr tat n co ˆ and in ˆ hav xactly th am momnt For xampl k1 ˆ k1 m ˆ k1( n) n co n n in n 0 k 01... (3.14) k / k (3.15) ˆ ˆ 1 m n n n n n C n ˆ n k k k m m k( ) co in co 1... k m0 m Ck ar binomial cofficint. From Eq. (.10) - (.11) w conclud that 1 m ( n) (3.16) 1 1 m4 ( n) f ( n 1) f ( n 1) (3.17) 4 16 1 3 m6 ( n) f ( n 1) f ( n 1). (3.18) 8 3 m ( n) u n 01 m ( n) u n 3... (3.19) k k k k whr th qual ign appli whn k 1 and u k ar momnt of uniform ditribution u k (k 1)!! ( k)!! k k co in k 1... (3.0) With u4 3 / 8 and u6 5 /16 w hav 7 9 5 m4 (0) u4 m4 (1) u4 m4 () u4 (3.1) 0 8 1 10
11 13 3 m6 (0) u6 m6 (1) u6 m6 () u6. (3.) 40 56 8 Th raon w hav mk uk for th two bottom numbr tat { 0 1 } i that f( n 1) in Eq. (.8) vanih giving no contribution to th um in Eq. (3.15) du to th on-idd bounddn of th numbr oprator ˆN. Thi can b aily n in Eq. (3.17) - (3.18). A Carruthr and Nito [4] howd in 1968 quantum pha in gnral i not of uniform ditribution in numbr tat n contrary to common blif. In th limit of n w hav k / m m k (3.3) m 1 1 ( 1)!! lim m ( n) C C u k 1... n k k k m k m0 ( k)!! confirming that numbr tat n approach uniform pha ditribution a n. C. Pha in cohrnt tat i In a cohrnt tat whr th xpctation valu of co ˆ i ( n1) ˆ co co n! ( n 5 / )( n 1/ ) (3.4) n0 From Eq. (.1) and Eq. (3.4) it immdiatly follow that ( n1) ˆ ˆ i[co H ] in. n! ( n 5 / )( n 1/ ) (3.5) n0 In highly xcitd cohrnt tat which i th claical limit of quantum tat w hav lim co ˆ co. (3.6) Th xpctation valu of th following commutator in th claical limit ar xactly th am a th corrponding claical Poion brackt (on th right hand id) lim i[co ˆ Hˆ ] in {co H} (3.7) ˆ ˆ (3.8) lim i[co H] in {co H} ˆ ˆ (3.9) lim i[in H] in {in H}. Eq. (3.6) - (3.9) confirm that our pha oprator ˆ ha corrct bhavior in th claical limit. 11
ACKNOWLEDGMENT Thi work wa upportd by Contract No. DE-FG05-86ER60473 btwn th Diviion of Biomdical and Environmntal Rarch of th Dpartmnt of Enrgy and Florida Stat Univrity. REFERENCES [1] P. A. M. Dirac Proc. R. Soc. London A 114 43 (197). [] W. H. Louill Phy. Ltt. 7 60 (1963). [3] L. Sukind and J. Glogowr Phyic 1 49 (1964). [4] P. Carruthr and M. M. Nito Rv. Mod. Phy. 40 411 (1968). [5] D. T. Pgg and S. M. Barntt Phy. Rv. A 39 1665 (1989). [6] X. Ma and W. Rhod Phy. Rv. A 43 576 (1991). [7] W. H. Louill and J. P. Gordon Bll Tlphon Laboratori Tch Mmo MM-61-14-34 (1961). [8] M. M. Nito Phy. Scr. 1993 5 (1993) or arxiv:hp-th/9304036 (1993). [9] X. Ma PhD dirtation Florida Stat Univ. (1989). 1