A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by ${\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}∕\ensuremath{\int}{\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}dt$ where ${\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)$ is the free particle wave function evaluated at the detector position $(x=0)$. Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.
A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by \ensuremath∣\ensuremathψf(0,t)\ensuremath∣2∕\ensuremath∫\ensuremath∣\ensuremathψf(0,t)\ensuremath∣2dt where \ensuremathψf(0,t) is the free particle wave function evaluated at the detector position (x=0). Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.
Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the \emph{time of arrival} (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider the setup where a detector keeps making instantaneous measurements at regular finite time intervals {\emph{till}} it detects the particle at some time t, which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.
Imagine an experiment where a quantum particle inside a box is released at some time in some initial state. A detector is placed at a fixed location inside the box and its clicking signifies arrival of the particle at the detector. What is the time of arrival (TOA) of the particle at the detector ? Within the paradigm of the measurement postulate of quantum mechanics, one can use the idea of projective measurements to define the TOA. We consider a setup where a detector keeps making instantaneous measurements at regular finite time intervals until it detects the particle at time t , which is defined as the TOA. This is a stochastic variable and, for a simple lattice model of a free particle in a one-dimensional box, we find interesting features such as power-law tails in its distribution and in the probability of survival (non-detection). We propose a perturbative calculational approach which yields results that compare very well with exact numerics.
We introduce a formalism for the calculation of the time of arrival t at a detector of particles traveling through interacting environments. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t. We interpret our results in terms of a Positive Operator Valued Measure. We then compute the probability distribution in times of arrival at a detector of those particles that, after their initial preparation, have undergone quantum tunneling or reflection due to the presence of potential barriers. We obtain the expected retardation or advancement for transmitted wave packets, and non-foreseen double bump structures for some cases of reflection.
In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times T(X) at position x=X for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at X, assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at t=0 with average wave number 〈k〉 and variance $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density J(x=X,t=T) only to terms of order $\ensuremath{\Delta}k/〈k〉.$ They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by ∣J(X,T)∣ (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT's theory, every particle in the ensemble reaches a point x=X, where $\ensuremath{\psi}(x,t)$ and J(x,t) are both zero for all t, while no particle ever reaches X according to the theory based on Bohmian mechanics. Some possible implications are discussed.
We are concerned with the justification of the statement, commonly (explicitly or implicitly) used in quantum scattering theory, that for a free non-relativistic quantum particle with initial wave function Ψ0(x), surrounded by detectors along a sphere of large radius R, the probability distribution of the detection time and place has asymptotic density (i.e., scattering cross section) σ(x,t)=m3ℏ−3Rt−4∣Ψ0(mx/ℏt)∣2 with Ψ0 the Fourier transform of Ψ0. We give two derivations of this formula, based on different macroscopic models of the detection process. The first one consists of a negative imaginary potential of strength λ>0 in the detector volume (i.e., outside the sphere of radius R) in the limit R→∞,λ→0,Rλ→∞. The second one consists of repeated nearly-projective measurements of (approximately) the observable 1∣x∣>R at times T,2T,3T,… in the limit R→∞,T→∞,T/R→0; this setup is similar to that of the quantum Zeno effect, except that there one considers T→0 instead of T→∞. We also provide a comparison to Bohmian mechanics: while in the absence of detectors, the arrival times and places of the Bohmian trajectories on the sphere of radius R have asymptotic distribution density given by the same formula as σ, their deviation from the detection times and places is not necessarily small, although it is small compared to R, so the effect of the presence of detectors on the particle can be neglected in the far-field regime. We also cover the generalization to surfaces with non-spherical shape, to the case of N non-interacting particles, to time-dependent surfaces, and to the Dirac equation.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.