The non-negatvty of probabltes and the collapse of state Slobodan Prvanovć Insttute of Physcs, P.O. Box 57, 11080 Belgrade, Serba Abstract The dynamcal equaton, beng the combnaton of Schrödnger and Louvlle equatons, produces noncausal evoluton when the ntal state of nteractng quantum and classcal mechancal systems s as t s demanded n dscussons regardng the problem of measurement. It s found that state of quantum mechancal system nstantaneously collapses due to the non-negatvty of probabltes. 1 Introducton Quantum and classcal mechancs are causal theores. By ths we mean that durng evolutons, that are governed by dynamcal equatons of these theores, states cannot change ther purtes. (Of course, ths holds only n the cases wth no stochastc terms n the Hamltonan.) However, there are stuatons n whch purty of state can be changed. Ths noncontnuous change happens when quantum system nteracts wth some classcal system. An example of ths s a process of measurement wth a well known reducton - collapse, of state. Theory that unfes quantum and classcal mechancs by descrbng nteracton of classcal and quantum systems has to be based on such dynamcal equaton whch can produce noncausal evolutons. The dynamcal equaton of hybrd systems - quantum and classcal systems n nteracton, whch was frstly ntroduced by Aleksandrov [1], produces noncausal evoluton n a case 1
addressng the problem of measurement. More precsely, If the state of quantum system before the measurement s the superposton of the egenstates of measured observable,.e., ψ, and f the apparatus before the measurement s n the state wth sharp values of poston and momentum, then the pure ntal noncorrealted state has to evolve nto some mxed correlated state. The equaton of moton governng ths process s just the combnaton of Schrödnger - von Neumann, and Louvlle equatons. Interestng s that for ths transton only the regular type Hamltonan s needed (that s Hamltonan wth no stochastc terms). On the other hand, mportant role s played by the non-negatvty of states, whch resembles the non-negatvty of probabltes, and ths s what we shall dscuss n ths artcle. In order to nvestgate mentoned non-negatvty of states, we shall ntroduce operator form of classcal mechancs. Our approach to ths problem s very smlar to the one proposed by Sudarshan et al. n [2-4]. 2 Operator form of classcal mechancs The classcal mechancs, n dfference to quantum mechancs, s characterzed by the possblty of smultaneous measurement of both poston and momentum wth vanshng devatons. Due to ths, the algebra representng observables of classcal mechancs has to be the commutatve one. In the drect product of two rgged Hlbert spaces H q H p one can defne commutatve algebra of classcal observables as the algebra (over R) of polynomals of the operators ˆq cm =ˆq Î and ˆp cm = Î ˆp. These operators represent coordnate and momentum of classcal system. States can be defned, lke n standard phase space formulaton, as functons of poston and momentum, whch are now operators. That s, pure states are defned by: δ(ˆq q(t)) δ(ˆp p(t)) = δ(q q(t))δ(p p(t)) q q p p dqdp = = q(t) q(t) p(t) p(t), (1) whle (noncoherent) mxtures are ρ(ˆq cm, ˆp cm,t). These states are postve and Hermtan operators normalzed to δ 2 (0) f ρ(q, p, t) R, ρ(q, p, t) 0and ρ(q, p, t) dq dp = 1. If one calculates the mean values by the Ansatz: f = Trf(ˆq cm, ˆp cm )ρ(ˆq cm, ˆp cm,t), (2) Trρ(ˆq cm, ˆp cm ) 2
then f wll be equal to standardly calculated: f = f(q, p)ρ(q, p, t)dqdp. (3) The dynamcal equaton n operator formulaton s defned as: ρ(ˆq cm, ˆp cm,t) = t H(ˆq cm, ˆp cm ) ρ(ˆq cm, ˆp cm,t) ρ(ˆq cm, ˆp cm,t) H(ˆq cm, ˆp cm ). (4) ˆq cm ˆp cm ˆq cm ˆp cm It s obvous that ths form s equvalent to the standard classcal mechancs snce the latter appears through the kernels of the operator formulaton (expressed wth respect to the bass q p ). Let us further remark that ths formulaton of classcal mechancs employs formalsm of standard quantum mechancs. More precsely, the drect product of two rgged Hlbert spacesh q H p used here s the carbon copy of the one used n quantum mechancs when the coordnates of system wth two degrees of freedom are under consderaton. The only dfference comes from the fact that we have neglected non-commutng operators here snce they have no physcal meanng. All other aspects of the formalsm are the same or smlar. Wthout gong nto detals snce t s beyond the scope of ths artcle, t should be stressed that ths holds for all formal problems and respectve solutons as well. 3 Hybrd systems One can use operator form of classcal mechancs n order to analyze the nteracton between classcal and quantum systems. Mathematcal framework s based on drect product of the Hlbert space and two rgged Hlbert spaces (n case when consdered classcal and quantum systems have only one degree of freedom). The frst Hlbert space H qm s as n the standard quantum mechancs, whle the other two are rgged Hlbert spaces that were dscussed n prevous secton. So, for descrpton of so called hybrd systems one uses H qm H q cm Hp cm. The state of the composte system s the statstcal operator ˆρ qm (t) ˆρ cm (t), where the frst one acts n H qm representng the state of quantum 3
system and second one acts n Hcm q Hp cm representng the classcal system. The propertes of these operators are as n standard quantum mechancs and as gven n prevous secton. The evoluton of hybrd systems state s governed by the Hamltonan Ĥ = ˆV qm ˆV cm,where: ˆV qm = V qm (ˆq qm, ˆp qm )=V qm (ˆq Î Î, ˆp Î Î), and: ˆV cm = V cm (ˆq cm, ˆp cm )=V cm (Î ˆq Î,Î Î ˆp). Snce t s Hermtan, operator ˆV qm can be dagonalzed n form: v ψ ψ Î Î. Obvously, the operator ˆV cm s dagonal wth respect to the bass q p. The dynamcal equaton for hybrd systems s the generalzaton of Schrödnger and Louvlle equatons or, more precsely, ther combnaton gven by: ˆρ qm (t) ˆρ cm (t) t = 1 h [ ˆV qm, ˆρ qm (t)] ˆρ cm (t) ˆV cm + + ˆρ qm(t) ˆV qm + ˆV qmˆρ qm (t) {ˆV cm, ˆρ cm (t)}, (5) 2 where operator form of the Posson bracket {, } s defned by (4). Smlar equaton appeared n [1,5-7]. There one can fnd detaled dscussons regardng the propertes of there gven dynamcal equatons of hybrd systems. 4 Process of measurement In lterature, an deal quantum measurement s consdered as nteracton between the quantum system, descrbed by the state ψ(t) n a Hlbert space H qm, and the measurng apparatus - classcal system, ntally n the state φ(t o ). The measurement process s such that: a.) the quantum system, before the measurement beng n one of the egenstates of the measured observable, say ψ (t o ), does not change the state durng the measurement (repeated measurement has to gve the same results ) and b.) the classcal system undergoes transton from ntal state φ(t o ) to φ (t). Ths 4
transton enables one to fnd out what s the state of measured quantum mechancal system. The problem of the measurement s the followng: f the ntal state of the quantum system was superposton of the egenstates of measured observable, that s f ψ(t o ) = c ψ (t o ), then, due to assumed lnearty of the evoluton, the state of the composte system would be ψ(t) = c ψ (t) φ (t), whch s n contradcton wth the obvous fact that classcal system cannot be n superposed states. Many other processes can be related to ths one n more or less straghtforward manner. Wthn the operator formulaton of the classcal mechancs and hybrd systems, the process of measurement can be descrbed as follows. The ntal state: ˆρ qm (t o ) ˆρ cm (t o )= c c j ψ (t o ) ψ j (t o ) q(t o ) q(t o ) p(t o ) p(t o ), j (6) evolves accordng to the dynamcal equaton (5) where ˆV qm s the measured observable. The last term on the RHS of (5), due to whch ˆρ cm (t) depends on ˆρ qm (t), n ths case becomes: j v + v j c c j 2 ψ (t) ψ j (t) {ˆV cm, ˆρ cm (t)}. Ths term suggests that correlated state can be assumed n the form of: c c j ψ (t) ψ j (t) q j (t) q j (t) p j (t) p j (t). (7) j But, such operator, despte of beng the soluton of (5), s not non-negatve one,. e., some events would have negatve probabltes f ths operator s taken as the state of composte system. The only meanngful soluton of dynamcal equaton s: c 2 ψ ψ q (t) q (t) p (t) p (t). (8) Ths operator s Hermtan and postve one. 5
5 Dscusson The ntal state of hybrd systems (6) s dempotent (up to the norm) whle the evolved state n consdered case (8) s not. Thus, n the absence of some ad hoc ntroduced stochastc terms n the Hamltonan and/or nonlnear terms n the equaton of moton, ths equaton produces noncausal evoluton: the ntal noncorrelated pure state evolves n mxed correlated state. From the evolved state t follows that to each state of the measured quantum system ψ (whch s the egenstate of the measured observable), there corresponds one state of the measurng apparatus (wth sharp values of poston and momentum) q (t) p (t) and each of these states happens wth the probablty c (t o ) 2. Consequently, soluton (8) s n agreement wth the projecton postulate of orthodox quantum mechancs. The formal descrpton of the collapse of quantum mechancal state could be the followng. Intal state of the hybrd system should be seen as c j (t) ψ ψ j q (t) q j (t) p (t) p j (t) j for t = t o snce ths correlated state s desgned to be as pure, Hermtan and non-negatve for t t o as s the ntal one. The partal dervatons wthn the Posson bracket on the rght hand sde of the dynamcal equaton, whch s the generator of tme transformaton, for t>t o annhlate nondagonal classcal mechancal terms of the state accordng to ˆq q (t) q j (t) = ˆq δ(ˆq q (t)) δ,j, ˆp p (t) p j (t) = ˆp δ(ˆp p (t)) δ,j, snce the classcal mechancal j terms of desgned state for t>t o do not commute wth coordnate and momentum of the classcal system, the meanng of whch s that they are not functons of the only avalable observables. For t = t o these dervatves do not vansh snce q (t o )=q o and p (t o )=p o for all. Ths means that dynamcal equaton nstantaneously changes j terms of classcal mechancal state at t o and then forbds further tme evoluton of these terms,. e., these terms become constant. Snce there s no other possblty for the state of hybrd system to be non-negatve operator, 6
j terms of classcal mechancal state has to vansh n order to be tme ndependent and, n ths way, they annhlate j terms of quantum mechancal state. Ths s seen as the collapse of state of quantum mechancal system. Smlar reasonng holds n some other cases of the nteracton between classcal and quantum systems. The pure ntal states can evolve n noncoherent mxtures, whle noncoherent mxtures cannot evolve nto coherent mxtures (pure states),. e. the process s rreversble. Therefore, the entropy ncreases or stays constant as the consequence of the superposton of two lnear dynamcal equatons and the non-negatvty of probablty. References [1] I. V. Aleksandrov, Zet. Nat., 36a, (1981), 902 [2] T. N. Shery and E. C. G. Sudarshan, Phys. Rev., 13D, (1978) 4580 [3] T. N. Shery and E. C. G. Sudarshan, Phys. Rev., 20D, (1979) 857 [4] S.R.Gautamet al., Phys. Rev., 20D, (1979) 3081 [5] W. Boucher and J. Trashen, Phys. Rev., 37D, (1988) 3522 [6] O.V.PrezhdoandV.V.Ksl,Phys.Rev.,56A, (1997) 162 [7] A. Anderson, Phys. Rev. Lett., 74, (1995) 621 7