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Canonical Pairs, Spatially Con?ned Motion and the Quantum Time of Arrival Problem

arXiv:quant-ph/0001062v1 18 Jan 2000

Eric A. Galapon§

Theoretical Physics Group, National Institute of Physics University of the Philippines, Diliman Quezon City 1101 Philippines Abstract. It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the con?nes of standard quantum mechanics. In this Letter we demonstrate the otherwise. We do so by pointing out that there is no a priori reason in demanding that canonical pairs form a system of imprimitivities. We then proceed to show that a class of self-adjoint and canonical time of arrival (TOA) operators can be constructed for a spatially con?ned particle. And then discuss the relationship between the non-self-adjointness of the TOA operator for the uncon?ned particle and the self-adjointness of the con?ned one.

PACS numbers: 03.65 Bz

The question of when a given particle prepared in some initial quantum state arrive at a given spatial point is a legitimate quantum mechanical problem requiring more than a parametric treatment of time. In standard quantum formulation, this raises the time of arrival at the level of quantum observable. And at this level the time of arrival (TOA) distribution is supposedly derivable from the spectral resolution of a certain self-adjoint TOA-operator canonically conjugate to the driving Hamiltonian. Thus the question of when translates to the question of what is the TOA-operator. But can one construct such an operator? The consensus is a resounding no. This consensus goes back to the well known Pauli’s theorem which asserts that the existence of a self-adjoint time operator (of any kind) implies that the spectrum of the Hamiltonian is the entire real line contrary to the generally discreet and semibounded Hamiltonian operator [1]. The embargo imposed by Pauli’s theorem has led to various treatments of the problem within and beyond the usual formulation of quantum mechanics [2, 4, 9, 8, 10]. However, we have recently shown—following Pauli’s own method of proof— the consistency of a bounded, self-adjoint time operator canonically conjugate to a Hamiltonian with a non-empty point spectrum, discreet or semibounded [3]. This denies the sweeping generalization of Pauli’s conclusion. Motivated by this development, we pose the question If Pauli’s well known argument can not be correct, then why there is a prevalent failure in constructing a self-adjoint and canonical TOA-operator? In this Letter we attempt to answer this question speci?cally for the free particle in one dimension within the con?nes of the standard single Hilbert space quantum mechanics. We approach the question by ?rst addressing the issue of quantum canonical pairs at the foundational level. Using the insight we gain in

§ email: egalapon@nip.upd.edu.ph

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clarifying canonical pairs, we proceed in investigating the TOA-problem under the assumption that the particle is con?ned. We show that self-adjoint and canonical TOA operators for the spatially con?ned particle can be constructed. We then discuss the relationship between the self-adjointness of the TOA operator in a bounded space and the non-self-adjointness of the same operator in unbounded space. We ?rst address the issue of quantum canonical pairs. A major impediment in constructing a self-adjoint time of arrival operator has been the conviction among workers that a given pair of self-adjoint operators, Q and P, satisfying the canonical commutation relation (QP ? PQ) ? i? I (where I is the identity operator of the Hilbert h space) yield a transitive system of imprimitivities over the entire real line [4]. That is if ? is a Borel subset of ?, and ? → EQ (?) and ? → EP (?) are the respective projection valued measures of Q and P, then for every α, β ∈ ?

?1 Uα EQ (?)Uα = EQ (?α ), ?1 Vβ EP (?)Vβ

(1) (2)

= EP (?β ),

where Uα = exp(iαP) (Vβ = exp(iβQ)), and ?α = {λ : λ ? α ∈ ?} (?β = {λ : λ ? β ∈ ?}). Equations (1) and (2) imply that the spectrum of Q and P is the entire real line. This automatically forbids the construction of a self-adjoint time operator canonically conjugate to a given semibounded or discrete Hamiltonian if equations (1) and (2) are imposed upon every physically acceptable canonical pair. However, the above conviction can be traced either from Pauli’s theorem or from analogy to the properties of the position,q, and momentum, p, operators in unbounded free space (for example ref.[5]). Having addressed Pauli’s objections [3], we point out that the analogy is false. It is well known that the pair (q, p) satisfy the canonical commutation relation and equations (1) and (2). However, it is not so well known that they satisfy (1) and (2) do not follow from them satisfying the canonical commutation relation. In fact it is the converse. That (q, p) satisfy (qp ? pq) ? i? I follows from h the fundamental axiom of quantum mechanics that the propositions for the location of an elementary particle in di?erent volume elements are compatible, and from the fundamental homogeneity of free space, i.e. points in ? are indistinguishable [6]. The former naturally leads to the self-adjoint position operator q in ?; while the later requires the existence of a unitary operator generated by the momentum operator such that the PV measure of q satisfy equation (1). Then symmetry dictates that the PV measure of p must satisfy equation (2). Now equation (1) and the fact that Uα and Vβ form a representation of the additive group of real numbers lead to the well known Weyl’s commutation relation, Uα Vβ = eiαβ Vβ Uα . This relation ?nally implies the canonical commutation relation (qp ? pq) ? i? I enjoyed by q and p. h For a free particle in a box, the points in the spatial space available to the particle are distinguishable, the walls being the distingushing factor, e.g. one point can be nearer to the left wall than another point. The bounded space for the particle then is not homogenous and equation (1) can not be imposed upon the position operator. Doing so is imposing homogeneity in an intrinsicaly inhomegenous space. However, it is known that the self-adjoint position and momentum operators for the trapped particle satisfy the canonical commutation relation without satisfying equations (1) and (2). It should then be clear that q and p (in ?) satisfy equations (1) and (2) not because they are canonically conjguate but because of an underlying quantum mechanical axiom and a fundamental property of free unboundned space. And that they are canonically conjugate because of these two. Therefore we are led to rede?ne and reinterpret quantum canonical pairs. We

Quantum Time of Arrival Problem

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expand the class of physically acceptable canonical pairs to include any given pair of densely de?ned, self-adjoint operators,(Q,P), in a separable, in?nite dimensional Hilbert space, H, satisfying the canonical commutation relation in some nontrivial, proper (dense or closed) subspace Dc ? H; that is, (QP ? PQ)? = i? ? for all ? ∈ Dc . h (The known fact that there are numerous non-unitarily equivalent solutions to the canonical commutation relation [3, 7] assures us of the richness of canonical pairs beyond those satisfying (1) and (2).) That a given pair is canonical in some sense— e.g. the pair satis?es equations (1) and (2), or one of the pair is bounded and thus do not satisfy (1) and (2)—is consequent to a set of underlying fundamental properties of the system under consideration or to the basic de?nitions of the operators involved or to some fundamental axioms of the theory. It is concievable to impose that a given pair be canonical as a priori requirement based, say, from its classical counterpart, but not the sense the pair is canonical without a deeper insight, say, into the underlying properties of the system. In other words, we don’t impose in what sense a pair is canonical if we don’t know much, we derive in what sense instead. Furthermore, we claim that if a given pair is known to be canonical in some sense, then we can learn more about the system or the pair by studying the structure of the sense the pair is canonical. Having cleared our way through equations (1) and (2), now we can take another look at the free time of arrival problem. Classically if the position of a given particle in one dimension is q and its momentum is p, its time of arrival at the origin is given by T = ??qp?1 where ? is the mass of the particle; T is canonically conjugate to the free Hamiltonian H = (2?)?1 p2 , i.e. {H, T } = 1. In the course of history of the quantum time of arrival problem (and related problems), T has served as the starting point in numerous attempts in constructing time of arrival operator. Various quantization schemes lead to the totally symmetric quantized form of T , qp?1 + p?1 q . (3) 2 in which T, q and p are the operator versions of T , q and p, respectively. Formally (3) is canonically conjugate to the free Hamiltonian, H = (2?)?1 p2 , i.e. [H, T] = i? . h Equation (3) has been the subject of numerous investigations in ? and known to have unequal de?ciency indices: T is not self-adjoint and lacks any self-adjoint extension in free unbounded space.[9]. The non-self-adjointness of T has always been ascribed to the semiboundedness of the Hamiltonian in accordance with Pauli’s theorem. But this is not necessarilly true. It is conceivable that even if T were self-adjoint and canonically conjugate to the Hamiltonian in some dense Dc , equations (1) and (2) remain unviolated as long as Dc is not invariant under either T or H. That is, T and H are canonically conjugate in some sense di?erent from the position and momentum operators in an unbounded space. Nevertheless equation (3) remains in the sorry state of non-self-adjointess in ?. Can we explain this and in the process construct a self-adjoint version of T? These we now attempt to do. In this letter we approach the problem with the additional assumption that the particle is known to be somewhere between two given points. That is the probability of ?nding the particle is zero outside and one in the entire length bounded by the two points. The particle is then essentially con?ned. Now let the length of the available spatial space be 2l and let the origin sit at the middle. If p = 0 and |q| < l, classically the time of arrivat at the origin is still given by T = ??qp?1 ; and T remains canonically conjugate to the free Hamiltonian. Then T = ??

Quantum Time of Arrival Problem

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equation (3) is still the totally symmetric quantized form of T even when the particle is con?ned. We are then led to investigate equation (3) when the particle is con?ned. At the level of equation (3), T is formal and its meaning is not precise until we de?ne the domain and actions of the operators involved. We attach the Hilbert space H = L2 [?l, l], the space of Lesbegue square integrable functions in the interval [?l, l], to our system. We de?ne the position, momentum, and Hamiltonian operators as follows D(q) = H (4) (qψ)(q) = q ψ(q) for all ψ ∈ D(q) D(pγ ) = φ ∈ H : φ a.c., φ′ ∈ H, φ(?l) = e?2i γ φ(l), 0 ≤ γ < 1 h ? dφ for all φ ∈ D(pγ ) (pγ φ) (q) = i dq D(Hγ ) = ? ∈ D(pγ ) : ?′′ ∈ H, ?′ (?l) = e?2i γ ?′ (l), 0 ≤ γ < 1 (Hγ ?)(q) = ? (5)

,(6) h ? 2 d2 ? for all ? ∈ D(Hγ ) 2 2? dq respectively. The di?erent values of γ correspond to di?erent physics, e.g. what happens to a given wavepacket when it reaches one of the boundaries. As de?ned q, pγ , and Hγ are densely de?ned, self-adjoint operators, with q and pγ canonically conjugate in a dense subspace of D(pγ ). The Hamiltonian is so de?ned such that it is consistent with the interpretation that the energy is purely kinetic. The momentum and the Hamiltonian then commute and have the common set of eigenvectors (γ) φn (q) = √1 exp i (γ + nπ) q , n = 0, ±1, ±2 · · · ; furthermore, both have pure l 2l point spectra. According to current thinking, the Hamiltonian, being discrete, it can not be canonically conjugate to any self-adjoint operator. We will show the otherwise though. Now we go back to equation (3), the formal operator T. First for non-periodic boundary conditions, i.e. for 0 < γ < 1. Since q appears in ?rst power in (3), T is an operator if the inverse of pγ exists. The inverse exists if the range of pγ is dense or its null space, N (pγ ), is the trivial subspace. For non-periodic boundary conditions the constants do not belong to D(pγ ), so that N (pγ ) is the trivial subspace. Thus the inverse p?1 exists. Because pγ is unbounded and self-adjoint, p?1 is bounded γ γ and likewise self-adjoint. Then it follows that for every γ ∈ (0, 1) T is a bounded, everywhere de?ned, self-adjoint operator! For a given γ we identify T with the operator Tγ = ?(qp?1 + p?1 q)2?1 derived from the formal operator (3) by replacing p with pγ . γ γ We shall refer to Tγ as the con?ned, non-periodic time of arrival operator for a given γ ∈ (0, 1). Now we seek an explicit coordinate representation of Tγ . We proceed by noting (γ) (γ) ∞ l that p?1 has the representation (p?1 ?)(q) = h n=?∞ (φn , ?)(γ + nπ)?1 φn (q) for γ γ ? all ?(q) ∈ H. This leads to the Fredholm integral operator representation of Tγ ,

l

(Tγ ?)(q) = where the kernel is given by Tγ (q, q ′ ) = ?

?l

Tγ (q, q ′ ) ?(q ′ ) dq ′ for all ?(q) ∈ H,

(7)

φn (q) φn (q ′ ) ? (q + q ′ ) 4? h γ + nπ n=?∞

∞

(γ)

?(γ)

Quantum Time of Arrival Problem = ? ? (q + q ′ ) ei γ H(q ? q ′ ) + e?i γ H(q ′ ? q) 4? sinγ h

5 (8)

? The kernel Tγ (q, q ′ ) is both symmetric and bounded, i.e. Tγ (q, q ′ ) = Tγ (q ′ , q) and ′ 2 ′ [?l×l] |Tγ (q, q )| dq dq < ∞, respectively. This rea?rms the self-adjointness of Tγ . Now it is straightforward to show that Hγ and Tγ for a given 0 < γ < 1 are canonically conjugate in the following sense: ((Hγ Tγ ? Tγ Hγ )?)(q) = i ? ?(q) for h

all ?(q) ∈ Dc

(γ)

where Dc

(γ) ∞ n=1

(γ)

= ?(q) ∈ D(Hγ ) :

∞ n=1

1

l ?l

?(q) dq = 0, ?(?) = 0, ?′ (?) = 0 .

∞ n=1

The subspace Dc

(γ) Dc =

is explicitly given by ?n φ(γ) (q), n n2 |?n |2 < ∞, (?1)n n ?n γ 2 + π 2 n2 =0 (9)

(γ)

?(q) =

(γ)

where φn = (lγ 2 + lπ 2 n2 )? 2 i γ sin nπq + nπ cos nπq exp( iγq ). Moreover Dc is l l l 1 (γ)⊥ (γ)⊥ = orthogonal to the subspace spanned by the states φ0 = (2l)? 2 exp( iγq ), φn l 1 (γ) nπq nπq iγq 2 2 2 ?2 i γ cos l + n π sin l exp l , n = 1, 2 · · ·. Thus Dc is closed. (lγ + lπ n ) For periodic boundary conditions, γ = 0, we face a di?erent problem. Now the range of the momentum operator p0 is no longer dense, because its null space, N (p0 ), (0) consists of the non-trivial one dimensional subspace spanned by the state φ0 (q), the state of vanishing momentum. The inverse of p0 then does not exist and equation (3) is meaningless. This is re?ected by the ill de?ned limit of the kernel (8) in the limit as γ → 0. Evidently the pathology arises from N (p0 ). But this subspace is not relevant to the problem because the question of when a given particle arrives makes sense only when the particle is in motion, otherwise it does not get anywhere (see also ref.[10]). We then expect that, with l ?xed, the non-periodic kernel (8) has a ?nite part corresponding to the non-vanishing momentum components in the limit as γ → 0. Indeed this is the case. The ?nite part can be extracted by removing the divergent contribution of the vanishing momentum eigenvalue to give ? ? (10) q 2 ? q ′2 T0 (q, q ′ ) = i (q + q ′ )sgn(q ? q ′ ) ? i 4? h 4? l h Equation (10) is also symmetric and bounded. This means that the ?nite periodic limit of (8) generates a self-adjoint integral operator, T0 , whose kernel is given by equation (10). T0 can be interpreted as the time of arrival operator for periodic boundary conditions for states with non-vanishing momentum. We shall refer to this as the con?ned, periodic time of arrival operator. It can be shown that H0 and T0 are also canonically conjugate; that is, ((H0 T0 ? T0 H0 ) ?)(q) = i? ?(q) for h l (k) k all ? ∈ Dc = ? ∈ D(H0 ) : ? (?) = 0, ?l q ?(q) dq = 0, k = 0, 1 . Moreover Dc = Dc (refer to equation (9)) so that φn spans Dc . Also Dc is orthogonal to (0)⊥ the subspace spanned by the states φn . Thus Dc is likewise closed. We point out that the pair (Tγ , Hγ ) for every γ ∈ [0, 1) satisfy the canonical (γ) commutation relation without violating equations (1) and (2) because Dc is closed: equations (1) and (2) imply that the generators of Uα and Vβ are canonically conjugate in a dense subspace Dc which is invariant under both generators. In our terminology, the pair (Hγ , Tγ ) are conjugate in a sense di?erent from the generators of Uα and Vβ . The sense that the pair (Tγ , Hγ ) is canonical can be further appreciated by noting that (γ) Dc carries information on the states with zero time of arrival expectation values. It (γ)⊥ (γ) (γ)⊥ (γ) can be shown that the states spanning Dc and Dc —the states {φn , φn }—have

(0) (0)

Quantum Time of Arrival Problem

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γ zero time of arrival expectation values. Morevover, Dc identi?es the states for which the the TOA-energy uncertainty relation holds, i.e. ?Tγ (?)?Eγ (?) ≥ ? 2?1 for all h (γ) γ ? ∈ Dc . Tγ being bounded, the uncertainty relation does not hold outside Dc , i.e. (γ) ?Tγ (?)?Eγ (?) ≥ 0 for all ? ∈ H \ Dc . Can we physically explain our above results and relate to the non-self-adjointness of equation (3) in unbounded space? When we ask ”when will the particle arrive?” we suppose that the particle was prepared in a state that will arrive; otherwise, we don’t ask when. Classically, for a free particle, the arrival is assured by imposing that the position is ?nite and it has non-vanishing momentum. When either of these is not satis?ed, we are dommed to wait inde?nitely. Now how do we translate these conditions in quantum mechanics? We may impose that the physical Hilbert space consists only of states with bounded supports in spatial space and allows equation (3) to be well de?ned in the neighborhood of vanishing momentum. For the con?ned particle for non-periodic boundary conditions, both of these are satis?ed. And equation (3) is bounded and self-adjoint. For periodic boundary conditions, the ?rst is satis?ed yet (3) is ill de?ned. And this is because the spectrum of p0 includes the zero eigenvalue, a non-arrival feature. A systematic removal of this non-arrival feature leads to a bounded and self-adjoint TOA. And the boundedness of these con?ned TOA operators assures that the particle will arrive within a ?nite interval of time. For the free particle in unbounded space, the topology induced by the L2 norm accomodates states whose tails extend to in?nity and yields states in the domain of the adjoint of (3) that equation (3) is otherwise unde?ned. It then appears that the non-selfadjointness or not being an operator of equation (3) at all is a consequence of the residual non-arrival features of T upon naive quantization. And removal of these non-arrival features has led us to a class of con?ned self-adjoint and canonical TOAoperators. In a seperate publication we shall deal with the Tγ ’s at a deeper level, e.g. their spectral properties and measurements, and explore the relationship between our results and the POVM program [8].

References

[1] Pauli W 1926 Hanbuch der Physik vol V/1 ed. S Flugge (Springer-Verlag, Berlin) p.60. [2] Muga J G, Sala R, Palao J P 1998 Superlattices and Microstructures 23 833 and references therein; Halliwell J J and Za?ris E 1997 quant-ph/97060; Blanchard Ph and Jadczyk A 1996 Helv. Phys. Acta 69 p 613; Holland PR 1993 The Quantum Theory of Motion (Press Syndicate, Cambridge University Press); Busch P et al 1994 Phys. Let. A 191 p 357; Perez A 1980 Am. J. Phys. 7 p 552 and references therein; Holevo A S 1978 Rep. Math. Phys. 13 p 379;Olhovsky V S and Recami E 1974 Nuovo Cimento 22 p 263. [3] Galapon E A quant-ph9908033, sumitted to JPhysA. [4] Toller M 1998 quant-ph/9805030 and referrences therein; Giannitrapani R 1997 Int. Jour. Theor. Phys. 36 p 1575; Eisenberg E and Horwitz L P 1997 Ad. Chem. Phys. XCIX p 245;Delgado V and Muga J G 1997 Phys. Rev. A 56 p 3425. [5] Gottfried K 1966 Quantum Mechanics (Benjamin/Cummings, Reading) vol. 1 p 248. [6] Mackey G W 1989 Unitary Group Representations in Physics, Probability, and Number Theory (Addison-Wesley Publishing Co., Inc) p 174-198; Jauch J M 1968 Foundations of Quantum Mechanics (Addison-Wesley Publishing Company) p195-206. [7] Nelson E 1959 Annals of Mathematics 70 572; Segal I E and Bongaarts P J M 1969 Applications of Mathematics to Problems in Theoretical Physics ed. Lurcat F (Gordon and Breach Science Publishers, Inc.) 107. [8] Egusquiza I, Muga J G 1999 Phys. rev. A accepted, quant-ph 9901055 and references therein. [9] Oppenheim J, Reznik B, Unruh W G 1999 Phys. Rev. A 59 1801; Kijowsky J 1971 Rep. Math. Phys. 6 362. [10] Grot N, Rovelli C and Tate R S 1996 Phys. Rev. A 54 p 4676.

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