The purpose of this article is to deeply study the evolution process and measurement behavior of wave function in quantum mechanics by computer simulation. Based on Schrodinger equation and quantum measurement postulate, a set of numerical simulation framework is constructed, and typical models such as one-dimensional infinite potential well, simple harmonic oscillator, double-slit interference and two-level system are simulated by Crank-Nicolson method. Probability sampling mechanism is introduced in the simulation process to reproduce the wave function collapse caused by measurement, and the key results in the evolution process are shown. It includes the change of maximum probability density at different time steps (0.78 → 0.63 → 0.78), the exchange of kinetic energy and potential energy under the condition of energy conservation, the distribution characteristics of measurement positions in the double-slit interference experiment and the statistical frequency of the measurement results of the two-level system (0.503:0.497). The research shows that the designed simulation method can effectively reproduce the dynamic evolution law of quantum system and the statistical characteristics of measurement, and provide a visual auxiliary tool for understanding quantum phenomena.
The purpose of this article is to deeply study the evolution process and measurement behavior of wave function in quantum mechanics by computer simulation. Based on Schrodinger equation and quantum measurement postulate, a set of numerical simulation framework is constructed, and typical models such as one-dimensional infinite potential well, simple harmonic oscillator, double-slit interference and two-level system are simulated by Crank-Nicolson method. Probability sampling mechanism is introduced in the simulation process to reproduce the wave function collapse caused by measurement, and the key results in the evolution process are shown. It includes the change of maximum probability density at different time steps (0.78 → 0.63 → 0.78), the exchange of kinetic energy and potential energy under the condition of energy conservation, the distribution characteristics of measurement positions in the double-slit interference experiment and the statistical frequency of the measurement results of the two-level system (0.503:0.497). The research shows that the designed simulation method can effectively reproduce the dynamic evolution law of quantum system and the statistical characteristics of measurement, and provide a visual auxiliary tool for understanding quantum phenomena.
This research paper presents a simulation of a particle in a quantum mechanical oscillator, modeled as a multidimensional wave function. The simulation was created by solving the time-dependent Schrodinger equation, which was discretized in both the spatial and time domains, and then approximated using a finite difference method. The objective of this research was to gain a deeper understanding of the behavior of a particle in a quantum mechanical oscillator and to provide visualizations and calculations of relevant physical observables. Initial conditions for the wave function, such as a Gaussian wave packet, were chosen and parameters for the simulation, such as the spatial step size and total simulation time, were set. The results of the simulation showed the evolution of the wave function over time, providing a clear picture of the particles behavior within the potential well of the oscillator. Additionally, physical observables, such as the average position, average kinetic energy, and average potential energy, were calculated and used to validate the simulation and the predictions of quantum mechanics. In conclusion, this research provides a comprehensive understanding of the behavior of a particle in a quantum mechanical oscillator and highlights the importance of simulations in advancing our knowledge of quantum mechanics. The results of this research can contribute to various fields, such as quantum computation and quantum control, and have the potential to drive further breakthroughs in these fields. KEYWORDS: Quantum mechanical oscillator, particle behavior, multidimensional wave function, Schrödinger equation, finite difference method, physical observables, simulation, quantum mechanics, quantum computation, quantum control.
This work extends the previous one-dimensional formulation of the double-slit problem to a two-dimensional framework within the Schrödinger equation, providing a more realistic representation of the physical configuration. The wavefunction is constructed through a spectral decomposition in terms of eigenstates, with a Gaussian distribution of initial energies. The temporal evolution of the system is explicitly analyzed, showing how interference patterns develop as a function of propagation time. In particular, the passage of electrons at different times is simulated, highlighting the progressive formation of fringes. A stochastic reconstruction of detection events is then introduced for the case of neutron beams, using physically realistic parameters, reproducing the emergence of the interference pattern from discrete particle-like impacts. The results show that both interference and particle-like behavior arise directly from the intrinsic mathematical structure of quantum mechanics, with no need for additional interpretative assumptions.
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.
Abstract Quantum mechanics admits two distinct evolutions: deterministic unitary dynamics governed by the Schrödinger equation and the probabilistic collapse of the wave function. We show that the continuous collapse of a quantum state under measurement can, on a trajectory-by-trajectory basis, be equivalently described as unitary evolution generated by a time- and state-dependent Hermitian Hamiltonian with stochastic parameters. While the ensemble dynamics remains non-unitary, each individual trajectory thus admits a unitary representation. We derive explicit forms of such Hamiltonians for projective measurements on arbitrary n -level systems and for continuous position measurements of a harmonic oscillator, and we propose experimental schemes to test these predictions. Our framework provides a new approach to modeling and controlling continuously monitored quantum systems using only state-dependent unitary resources.
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.