We propose to modify the collapse axiom of quantum measurement theory by replacing the instantaneous with a continuous collapse of the wave function in finite time $τ$ . We apply it to coordinate measurement of a free quantum particle that is initially confined to a domain $D\subset\rR^d$ and is observed continuously by illuminating $\rR^d-D$. The continuous collapse axiom (CCA) defines the post-measurement wave function (PMWF)in $D$ after a negative measurement as the solution of Schrödinger's equation at time $τ$ with instantaneously collapsed initial condition and homogeneous Dirichlet condition on the boundary of $D$ . The CCA applies to all cases that exhibit the Zeno effect. It rids quantum mechanics of the unphysical artifacts caused by instantaneous collapse and introduces no new artifacts.We propose to modify the collapse axiom of quantum measurement theory by replacing the instantaneous with a continuous collapse of the wave function in finite time $τ$. We apply it to coordinate measurement of a free quantum particle that is initially confined to a domain $D\subset\rR^d$ and is observed continuously by illuminating $\rR^d-D$. The continuous collapse axiom (CCA) defines the post-measurement wave function (PMWF)in $D$ after a negative measurement as the solution of Schrödinger's equation at time $τ$ with instantaneously collapsed initial condition and homogeneous Dirichlet condition on the boundary of $D$. The CCA applies to all cases that exhibit the Zeno effect. It rids quantum mechanics of the unphysical artifacts caused by instantaneous collapse and introduces no new artifacts.

arXiv:1103.2526Mar 13, 20110 citations

Dynamics of a particle in the double-slit experiment with measurement

Alexey A. Kryukov

Abstract Spontaneous collapse models use non-linear stochastic modifications of the Schrödinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schrödinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schrödinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement. Our analysis shows that, in this basic case, the evolution of the particle’s state can be effectively captured through a random walk on a two-dimensional submanifold of the state space. This random walk reproduces the Born rule for the probability of finding the particle near the slits, conditioned on its arrival at one of them. To ensure that this condition is satisfied, we introduce a drift term representing a change in the variance of the position observable for the state. It is argued that the drift term accounts for the energy transfer and trapping incurred during the particle’s interaction with the detector. A drift-free model, based on equivalence classes of states indistinguishable by the detector, is also considered. The resulting random walk, with or without drift, serves as a suitable model for describing the transition from the initial state to an eigenstate of the measured observable in the experiment, offering new insights into its potential underlying mechanisms.Abstract Spontaneous collapse models use non-linear stochastic modifications of the Schrödinger equation to suppress superpositions of eigenstates of the measured observable and drive the state to an eigenstate. It was recently demonstrated that the Born rule for transition probabilities can be modeled using the linear Schrödinger equation with a Hamiltonian represented by a random matrix from the Gaussian unitary ensemble. The matrices representing the Hamiltonian at different time points throughout the observation period are assumed to be independent. Instead of suppressing superpositions, such Schrödinger evolution makes the state perform an isotropic random walk on the projective space of states. The relative frequency of reaching different eigenstates of an arbitrary observable in the random walk is shown to satisfy the Born rule. Here, we apply this methodology to investigate the behavior of a particle in the context of the double-slit experiment with measurement. Our analysis shows that, in this basic case, the evolution of the particle’s state can be effectively captured through a random walk on a two-dimensional submanifold of the state space. This random walk reproduces the Born rule for the probability of finding the particle near the slits, conditioned on its arrival at one of them. To ensure that this condition is satisfied, we introduce a drift term representing a change in the variance of the position observable for the state. It is argued that the drift term accounts for the energy transfer and trapping incurred during the particle’s interaction with the detector. A drift-free model, based on equivalence classes of states indistinguishable by the detector, is also considered. The resulting random walk, with or without drift, serves as a suitable model for describing the transition from the initial state to an eigenstate of the measured observable in the experiment, offering new insights into its potential underlying mechanisms.

arXiv:2412.00584Journal of Physics A Mathematical and TheoreticalMay 21, 20251 citation

Quantum dynamics under continuous projective measurements: Non-Hermitian description and the continuum-space limit

Varun Dubey, Cédric Bernardin, Abhishek Dhar

The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave-function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schr\"odinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave-function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schr\"odinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.

arXiv:2012.01196Physical Review AMar 29, 202138 citations

Relativistic Quantum Mechanics and Quantum Field Theory

Hrvoje Nikolić

This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.

arXiv:1205.1992May 24, 201911 citations

Measurement as absorption of Feynman trajectories: Collapse of the wave function can be avoided

Avi Marchewka, Z. Schuss

We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such a detector vanishes on the detector. The trace the absorbed particles leave on the detector is identified as the absorption current density on the detector. This density is calculated from the solution of Schr\"odinger's equation with a reflecting boundary at the detector. This current density is not the usual Schr\"odinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of four postulates. In the resulting theory the quantum-mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two measurement experiments are proposed to measure time of arrival and the probability density function of a freely propagating two-dimensional Gaussian packet from the measurement of the absorption current on two planes.We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such a detector vanishes on the detector. The trace the absorbed particles leave on the detector is identified as the absorption current density on the detector. This density is calculated from the solution of Schr\"odinger's equation with a reflecting boundary at the detector. This current density is not the usual Schr\"odinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of four postulates. In the resulting theory the quantum-mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two measurement experiments are proposed to measure time of arrival and the probability density function of a freely propagating two-dimensional Gaussian packet from the measurement of the absorption current on two planes.

arXiv:quant-ph/9906078Physical Review AApr 10, 20028 citations

ДИНАМИКА ИЗМЕРЕНИЙ КООРДИНАТ КВАНТОВЫХ НЕРЕЛЯТИВИСТСКИХ ЧАСТИЦ

Цой Валерий Иванович

Background and Objectives: There are a few different ways to change the quantum mechanical wave function.One is the continuous evolution according to the Schrodinger equation; the others are the collapse provided by strong measurements and the weak continuous measurement.The question arises: Can the strong measurement be considered as a dynamical process.Methods: To answer the question, a dynamic model is proposed to simulate numerically the measurement of the coordinates of quantum particles.By measuring the coordinate, we mean the capture of a particle in a narrow box and the excitation of the stationary wave function in it.We consider the amplitude of the probability of transition to a stationary state as a wave function, and the non-stationary Schrodinger equation is solved using this initial function.The transition to a stationary state was modeled as a gated process.Results and Conclusion: It is shown that when measuring coordinates, along with the instant collapse, there are cases when the measurement takes a short but finite time.In particular, the projective measurement of the coordinate of one of the entangled particles takes some time and is accompanied by the correlation in the probability density of the particles coordinates.Moreover, the correlation is lost over time.Background and Objectives: There are a few different ways to change the quantum mechanical wave function.One is the continuous evolution according to the Schrodinger equation; the others are the collapse provided by strong measurements and the weak continuous measurement.The question arises: Can the strong measurement be considered as a dynamical process.Methods: To answer the question, a dynamic model is proposed to simulate numerically the measurement of the coordinates of quantum particles.By measuring the coordinate, we mean the capture of a particle in a narrow box and the excitation of the stationary wave function in it.We consider the amplitude of the probability of transition to a stationary state as a wave function, and the non-stationary Schrodinger equation is solved using this initial function.The transition to a stationary state was modeled as a gated process.Results and Conclusion: It is shown that when measuring coordinates, along with the instant collapse, there are cases when the measurement takes a short but finite time.In particular, the projective measurement of the coordinate of one of the entangled particles takes some time and is accompanied by the correlation in the probability density of the particles coordinates.Moreover, the correlation is lost over time.

Известия Саратовского университета. Новая серия. Серия ФизикаJan 1, 20200 citations

We propose to modify the collapse axiom of quantum measurement theory by replacing the instantaneous with a continuous collapse of the wave function in finite time

τ

. We apply it to coordinate measurement of a free quantum particle that is initially confined to a domain $D\subset\rR^d$ and is observed continuously by illuminating $\rR^d-D$. The continuous collapse axiom (CCA) defines the post-measurement wave function (PMWF)in

D

after a negative measurement as the solution of Schrödinger's equation at time

τ

with instantaneously collapsed initial condition and homogeneous Dirichlet condition on the boundary of

D

. The CCA applies to all cases that exhibit the Zeno effect. It rids quantum mechanics of the unphysical artifacts caused by instantaneous collapse and introduces no new artifacts.