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Aspects of particle localisation in quantum mechanics

G.F. Melloy

Published Jan 1, 2001• 0 citations
Publisher

Abstract

The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n

DOI
10.14264/uql.2019.665

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The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n
A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac’s position operator x. It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the 4-vector density (ρ(x,t),j(x,t)/c) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalized at the same time. 1
A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac's position operator x . It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the four-vector density ( ( x , t ), j ( x , t )/ c ) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalized at the same time.
This thesis investigates the description of localised states and observables in quantum mechanics.Motivated by the fact that actual experimental apparatus is of only finite size, a method of localising any observable ("selfadjoint operator on Hilbert space) is introduced, which is applicable to both bounded and unbounded operators.This method makes use of objects called localising isometries, which also prove useful in modifying the evolution of non-relativistic particles, so that they remain well localised at all times: this modification does not jeopardise the well tested predictions of standard quantum mechanics.It is mathematically more convenient when describing systems of well separated identical particles.Pairs of correlated particles, such as those considered in EPR-type thought experiments, undergo chronological disordering at large times, and the analysis of this phenomenon is used to argue physically in favour of asymptotic algebras, which do not contain two-particle observables at infinite times.The second part of the thesis investigates the description of the position of a spinless relativistic particle.Existing Lorentz covariant currents are non-local, i.e. they describe particles which are not strictly localised in a bounded region of space, and these currents are modified so as to describe strictly localised particles, which evolve causally.vi
Highly localized positive-energy states of the free Dirac electron are constructed and shown to evolve in a simple way under the action of Dirac's equation. When the initial uncertainty in position is small on the scale of the Compton wavelength, there is an associated uncertainty in the mean energy that is large compared with the rest mass of the electron. However, this does not lead to any breakdown of the one-particle description, associated with the possibility of pair-production, but rather leads to a rapid expansion of the probability density outwards from the point of localization, at speeds close to the speed of light.
Non-classical light sources are now being developed in more and more laboratories. Their noise reduction properties are potentially important for technology. Examples are squeezed and antibunched light sources. It is tempting to picture in some of these processes single photons flying through space. In non-relativistic quantum mechanics it is easy to discuss the properties of particle localisation in a general context without recourse to a particular experimental configuration. A position operator exists and the localised states are just the eigenstates of this operator. The situation in relativistic quantum mechanics (or field theory) is not as straightforward. The photon is relativistic and it has the addi-tional property of being massless. It has no position operator associated with it. Yet, nevertheless, it is possible to pose the question of finding a generic construction for a wave packet for the photon which is localised as far as possible in some (imaginary) box in space. The difficulty of the construction follows from the fact that in an electromagnetic wave the polarisation is perpendicular to the direction of propagation. Thus we need to construct a vector field which is localised and has this transverse property related to the masslessness of the photon.
The concept of position and its measurement are fundamental to modern navigation technology as well as to theoretical physics. Nevertheless, a consistent notion of position has not been found in modern physics because of apparent inconsistencies between quantum mechanics and relativity. In quantum mechanics the probability distribution for a particle's position is determined by its wave function. A wave function strictly localized in space describes a particle within that locality, and nowhere else. But the quantum equations of motion predict the function will disperse, after any length of time at all, to occupy all space. Therefore the theoretical prediction is that, shortly after detection in a laboratory, a particle may be found anywhere in the universe, including regions requiring faster-than-light travel to reach. This contradicts the theory of relativity, which precludes a body traveling faster than the speed of light. The conflict arises from unexamined assumptions regarding the time of the measurement event and the time of the system under measurement. The prevailing view in physics is that time is only a parameter, having a definite value, describing dynamic changes in physical systems. This view endures even though quantum mechanics recognizes position as a quantum variable with inherent uncertainty, and relativity indicates that time should be placed on the same footing as position. But if time is recognized as a quantum variable with inherent uncertainty, the conceptual problems with position are resolved. The particle has finite probability of being in the past (or future) of the observer, while the quantum mechanical operator describing the position measurement contains a term which extrapolates from the intrinsic time of the particle to the time of the observer. Being potentially well in the past of the measurement event, the particle has time to travel to remote regions without exceeding the speed of light, and the conflict with relativity is resolved. A relativistic quantum theory can therefore be constructed incorporating a well-defined concept of position.