Canonical Pairs, Spatially Confined Motion and the Quantum Time of Arrival Problem

It has always been believed that no self-adjoint and canonical time of arrival operator can be constructed within the confines 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 confined particle. And then discuss the relationship between the non-self-adjointness of the TOA operator for the unconfined particle and the self-adjointness of the confined 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 specifically for the free particle in one dimension within the confines of the standard single Hilbert space quantum mechanics. We approach the question by first addressing the issue of quantum canonical pairs at the foundational level. Using the insight we gain in § email: [email protected] Quantum Time of Arrival Problem 2 clarifying canonical pairs, we proceed in investigating the TOA-problem under the assumption that the particle is confined. We show that self-adjoint and canonical TOA operators for the spatially confined 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 first 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) ⊂ ih̄ I (where I is the identity operator of the Hilbert space) yield a transitive system of imprimitivities over the entire real line [4]. That is if ∆ is a Borel subset of R, and ∆ → EQ(∆) and ∆ → EP(∆) are the respective projection valued measures of Q and P, then for every α, β ∈ R U−1 α EQ(∆)Uα = EQ(∆α), (1) V −1 β EP(∆)Vβ = EP(∆β), (2) 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) ⊂ ih̄ I follows from the fundamental axiom of quantum mechanics that the propositions for the location of an elementary particle in different volume elements are compatible, and from the fundamental homogeneity of free space, i.e. points in R are indistinguishable [6]. The former naturally leads to the self-adjoint position operator q in R; 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, UαVβ = e VβUα. This relation finally implies the canonical commutation relation (qp − pq) ⊂ ih̄ I enjoyed by q and p. 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 R) 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 redefine and reinterpret quantum canonical pairs. We Quantum Time of Arrival Problem 3 expand the class of physically acceptable canonical pairs to include any given pair of densely defined, self-adjoint operators,(Q,P), in a separable, infinite dimensional Hilbert space, H, satisfying the canonical commutation relation in some nontrivial, proper (dense or closed) subspace Dc ⊂ H; that is, (QP− PQ)φ = ih̄φ for all φ ∈ Dc. (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 satisfies 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 definitions 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 = −μqp−1 where μ is the mass of the particle; T is canonically conjugate to the free Hamiltonian H = (2μ)−1p2, 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 ,