Weak values, obtained from weak measurements, attempt to describe the properties of a quantum system as it evolves from an initial to a final state, without practically altering this evolution. Trajectories can be defined from weak measurements of the position, or inferred from weak values of the momentum operator. The former can be connected to the asymptotic form of the Feynman propagator and the latter to Bohmian trajectories. Employing a time-dependent oscillator as a model, this work analyzes to what extent weak measurements can shed light on the underlying dynamics of a quantum system expressed in terms of trajectories, in particular by comparing the two approaches.
We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum chaos related studies (an illustration involving quantum scars is given), and also in solving current weak-value related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit).
Standard quantum mechanics famously disallows to simultaneously measure particle position and momentum with arbitrary precision. In contrast, Bohmian mechanics, a hidden-variable interpretation, at the same time ascribes reality to the positions and momenta of quantum particles at the cost of a nonlocal ontology [1]. It thus allows to conceive of definite particle trajectories, while being fully compatible with the standard theory in all empirical predictions. To record such trajectories [2] we consider two optical settings using weak measurements in order to investigate different features of particle propagation in the Bohmian picture.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function
ψ. Given a Schrodinger wave function
ψ, we use its value
ψ0 at some fixed time (say,
t=0) to define the probability measure
∣ψ0∣2dx on the system configuration space
M (
=Rn). On the resulting probability space
M0, we introduce a stochastic process
ξ(t) corresponding to the Heisenberg position operator
XH(t) such that, in the Heisenberg state
∣ψh⟩ corresponding to
ψ0, the expectation value of
XH(t) equals that of
ξ(t) in
M0. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process
ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the
ψ0-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
Nondestructive weak measurements (WMs) made on a quantum particle are useful in order to extract information as the particle evolves from a prepared state to a finally detected state. The physical meaning of this information has been open to debate, particularly in view of the apparent discontinuous trajectories of the particle recorded by WMs. In this work we investigate the properties of vanishing weak values for projection operators as well as general observables. We then analyze the implications when inferring the past of a quantum particle. We provide a nonoptical example for which apparent discontinuous trajectories are obtained by WMs. Our approach is compared to previous results.
Abstract Pilot wave theory endows particles with de nite positions at all times governed by deterministic dynamics. However, individual particle trajectories are generically undetectable by experiment. This idea might seem to be contested in light of two proposals: (1) So-called ‘weak velocity measurements’, allegedly detecting Bohmian trajectories by weakly probing a quantum system without essentially disturbing it, and (2) the so-called ‘surrealistic’ trajectories experiment which supposedly establishes a conflict between the ‘actual’ position of a particle and its position derived from pilot wave theory. Although both attempts shed light on the nature of Bohmian particles, neither constitute empirical or theoretical evidence in favour or against pilot wave theory. Both instances admit a straightforward standard quantum mechanical interpretation compatible with the predictions of Bohmian theories. It is concluded that the puzzles arise from the absence of a coherent account of what quantum mechanical measurements signify.