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arXiv:quant-ph/0404012 · Journal of Physics A Mathematical and General

Time as an observable in nonrelativistic quantum mechanics

G. E. Hahne

Published Jun 11, 2003• 6 citations
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Abstract

The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional spacetime entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e. is a time = constant hyperplane in the Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the spacetime coordinates are ( t , x , y , z ), the paper analyses in detail the case that the hyperplane is of the type z = constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z = constant hyperplane, and average dwell times and delay times between a pair of z = constant hyperplanes, is presented.

DOI
10.1088/0305-4470/36/25/316

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The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation.Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate.In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial.A nontrivial result is obtained if the plane is not of this type.When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant.Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface.Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory.A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.

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The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional spacetime entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e. is a time = constant hyperplane in the Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the spacetime coordinates are ( t , x , y , z ), the paper analyses in detail the case that the hyperplane is of the type z = constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z = constant hyperplane, and average dwell times and delay times between a pair of z = constant hyperplanes, is presented.
The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation.Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate.In conventional formulations the hyperplane is always spacelike, i.e., is a time=constant hyperplane in Galilean relativity, and the result is then trivial.A nontrivial result is obtained if the plane is not of this type.When the space-time coordinates are (t, x, y, z), the paper analyzes in detail the case that the hyperplane is of the type z=constant.Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface.Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory.A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction contained between a pair of z=constant hyperplanes, is presented.
In a previous paper [G. E. Hahne, J. Phys. A 36, 7149 (2003)] the author studied a nontraditional boundary value problem associated with Schr\"odinger's partial differential equation for the wave function of a structureless particle moving in four-dimensional spacetime: in this boundary value problem, instead of the conventional specification of initial wave-function values on a time=constant\text{time}=\text{constant}time=constant surface, suitable time-dependent boundary and normal-derivative values are given on a three-dimensional space-time surface surrounding a slablike region of interaction in four-dimensional spacetime. The particle's time coordinate plays a natural role as an operator and observable in the modified formalism. In the present paper, the formalism is extended to describe a system of two nonrelativistic particles---each with its own time coordinate---scattering from background potentials and from one another in four-dimensional spacetime. The two-body interaction is taken as a generic noninstantaneous action-at-a-distance, which depends independently on the space-time positions of the two particles. The dynamics is expressed in terms of an integral equation for the wave function, that is, a nonrelativistic version of the Bethe-Salpeter equation. An optical theorem is derived for the transition operator associated with scattering processes; when the theorem holds, the pointwise probability current density derivable from the wave function is conserved globally, that is, in a region covering the space-time domain of significant interparticle interaction. A general formula for the expected dwell-correlation time for the two particles in the space-time region in terms of the scattering matrices is worked out.
Muñoz, Seidel, and Muga [Phys. Rev. A 79, 012108 (2009)10.1103/PhysRevA.79.012108], following an earlier proposal by Pollak and Miller [Phys. Rev. Lett. 53, 115 (1984)10.1103/PhysRevLett.53.115] in the context of a theory of a collinear chemical reaction, showed that suitable moments of a two-flux correlation function could be manipulated to yield expressions for the mean quantum dwell time and mean square quantum dwell time for a structureless particle scattering from a time-independent potential energy field between two parallel lines in a two-dimensional spacetime. The present work proposes a generalization to a charged, nonrelativistic particle scattering from a transient, spatially confined electromagnetic vector potential in four-dimensional spacetime. The geometry of the spacetime domain is that of the slab between a pair of parallel planes, in particular, those defined by constant values of the third (z) spatial coordinate. The mean Nth power, N = 1, 2, 3, …, of the quantum dwell time in the slab is given by an expression involving an N-flux-correlation function. All these means are shown to be nonnegative. The N = 1 formula reduces to an S-matrix result published previously [G. E. Hahne, J. Phys. A 36, 7149 (2003)10.1088/0305-4470/36/25/316]; an explicit formula for N = 2, and of the variance of the dwell time in terms of the S-matrix, is worked out. A formula representing an incommensurability principle between variances of the output-minus-input flux of a pair of dynamical variables (such as the particle's time flux and others) is derived.
By taking into account the quantum aspects intrinsic to any operational definition of spatio-temporal relationships, a stochastic concept of spacetime emerges. In relation to its classical counterpart is realized as a stochastic mean around which quantum fluctuations become negligible only in the limit of macroscopic spacetime intervals. The test-particle propagators used in the proposed quantum concept of spacetime are derived by solving in a consistent manner the localizability problem for relativistic particles. This is achieved in the framework of the stochastic phase space formulation of quantum mechanics, which in the nonrelativistic context is shown to result from systems of imprimitivity related to phase space conserved probability currents derivable from bona fide convariant probability densities in stochastic phase spaces of one particle systems, which can be interpreted as due to measurements performed with extended rather than pointlike test particles. The associated particle propagators can be therefore consistently related to coordinate probability densities measurable by the exchange of photons in between test particles from a chosen standard. Quantum spacetime is defined as the family of propagators corresponding to all conceivable coherent flows of test particles. This family of free-fall propagators has to satisfy certain self-consistency conditions as well as consistent laws ofmore » motion which inplicitly determine the stochastic geometro-dynamics of quantum space-time. Field theory on quantum spacetime retains many of the formal features of conventional quantum field theory. On a fundamental epistemological level stochastic geometries emerge as essential prerequisites in the construction of spacetime models that would be operationally based and yet consistent with the relativity principle as well as with the uncertinty principle.« less
Some suggestions on the probabilistic structure of quantization are first obtained by studying local properties of random variables such as continuity and differentiability with respect to a topology in the sample space. This analysis is extended to analytic functions on the complex plane and conditions are formulated under which probability densities are expressed as sums of absolute squares of complex numbers. Physical restrictions are introduced through the Schrödinger equation for a single bound-particle system. As a result, observations on a physical system become identified with the random selection of points in a topological measure space and the physical observables such as energy, position, and momentum, as well as time, are identified with measurable functions on appropriate spaces. Time, considered as a function, appears to be multiple-valued with spacings between multiple values and a detailed functional structure that characterizes and is characterized by the physical system under observation and the physical observables being measured. Its detailed functional structure is related to the physically measurable probabilities of quantum theory and it is seen to serve in the capacity of a conditioning random variable in the computation of quantum mechanical expectations.
In quantum theory we refer to the probability of finding a particle between positions xxx and x+dxx+dxx+dx at the instant ttt, although we have no capacity of predicting exactly when the detection occurs. In this work, we first present an extended nonrelativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between xxx and x+dxx+dxx+dx during [t,t+dtt,\phantom{\rule{0.16em}{0ex}}t+dtt,t+dt]. Then, we find a Schr\"odinger-like equation for a ``mirror'' wave function $\ensuremath{\phi}(t,x)$ associated with the probability of measuring the system between ttt and t+dtt+dtt+dt, given that detection occurs at xxx. In this framework, it is shown that energy measurements of a stationary state display a nonzero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here.