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American Journal of Physics

Time development in quantum mechanics using a reduced Hilbert space approach

Mario Belloni, Wolfgang Christian

Published Mar 13, 2008• 8 citations
Publisher

Abstract

We have created a suite of open source programs that numerically calculate and visualize the evolution of arbitrary initial quantum-mechanical bound states. The calculations are based on the expansion of an arbitrary wave function in terms of basis vectors in a reduced Hilbert space. The approach is stable, fast, and accurate at depicting the long-time dependence of complicated bound states. Several real-time visualizations, such as the position and momentum expectation values and the Wigner quasiprobability distribution for the position and momentum, can be shown. We use these computational tools to study the time-dependent properties of quantum-mechanical systems and discuss the effectiveness of the algorithm.

DOI
10.1119/1.2837810

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We have created a suite of open source programs that numerically calculate and visualize the evolution of arbitrary initial quantum-mechanical bound states. The calculations are based on the expansion of an arbitrary wave function in terms of basis vectors in a reduced Hilbert space. The approach is stable, fast, and accurate at depicting the long-time dependence of complicated bound states. Several real-time visualizations, such as the position and momentum expectation values and the Wigner quasiprobability distribution for the position and momentum, can be shown. We use these computational tools to study the time-dependent properties of quantum-mechanical systems and discuss the effectiveness of the algorithm.
We develop methodology for performing time dependent quantum mechanical calculations by representing the wave function as a sum of Gaussian wave packets (GWP) each characterized by a set of parameters such as width, position, momentum, and phase. The problem of computing the time evolution of the wave function is thus reduced to that of finding the time evolution of the parameters in the Gaussians. This parameter motion is determined by minimizing the error made by replacing the exact wave function in the time dependent Schrödinger equation with its Gaussian representation approximant. This leads to first order differential equations for the time dependence of the parameters, and those describing the packet position and the momentum of each packet have some resemblance with the classical equations of motion. The paper studies numerically the strategy needed to achieve the best GWP representation of time dependent processes. The issues discussed are the representation of the initial wave function, the numerical stability and the solution of the differential equations giving the evolution of the parameters, and the analysis of the final wave function. Extensive comparisons are made with an approximate method which assumes that the Gaussians are independent and their width is smaller than the length scale over which the potential changes. This approximation greatly simplifies the calculations and has the advantage of a greater resemblance to classical mechanics, thus being more intuitive. We find, however, that its range of applications is limited to problems involving localized degrees of freedom that participate in the dynamic process for a very short time. Finally we give particular attention to the notion that the GWP representation of the wave function reduces the dynamics of one quantum degree of freedom to that of a set of pseudoparticles (each represented by one packet) moving according to a pseudoclassical (i.e., classical-like) mechanics whose ‘‘phase space’’ is described by a position and momentum as well as a complex phase and width.
In classical mechanics the state of a system is defined by the numerical values of a set of dynamical variables such as the position coordinates and the velocities. In quantum mechanics the state of a system at an instant is represented by a unit vector in a Hilbert space, where a set of axes are the eigenvectors of a complete set of observables. The description of quantum dynamics where the wave function is time-dependent and the operators describing the observables are time-independent is known as the Schrodinger picture. The use of total time derivative of operator that depends on the time evolution of the wave function as well as on any intrinsic time-dependence in the operators simplifies the development of quantum mechanics and allows the development to more closely follow the corresponding development of classical mechanics. The difference between the Schrodinger and Heisenberg pictures is best understood as the difference in the basic set of observables.
We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.
The implementation of Monte Carlo methods acting in the quantum phase space is hindered by the fact that quantum phase-space information is available only through quasiprobability densities. In this work, we present a first-principles Monte Carlo method employing a hidden variables representation. This allows the full quantum time evolution of an arbitrary initial quantum state to be calculated by a classical Monte Carlo algorithm, even for systems subjected to time-dependent potentials. Guidelines for implementing a practical algorithm are presented.
This study offers a comprehensive exploration of quantum mechanics by integrating theoretical analysis with simulation-based experimentation to investigate fundamental principles such as energy quantization, operator expectations, and wavefunction behavior. Utilizing a mixed-methods approach, the research employed numerical solutions to the time-independent Schrödinger equation and qualitative analysis of foundational quantum postulates to model and interpret the behavior of discrete quantum states. Nine detailed datasets were generated, each containing over 20 quantum states, capturing energy levels, expectation values of position and momentum, and probability density maxima. The analysis revealed consistent non-linear spacing in energy levels, symmetrical behavior in position and momentum distributions, and the emergence of wavefunction localization in high-probability states. Twelve complex visualizations—including line, bar, pie, scatter, and hybrid plots—further illustrated the dynamic relationships among observables. These results confirm core theoretical constructs such as the Heisenberg uncertainty principle and the correspondence between potential confinement and probability density peaks. Violin plots and scatter matrices identified subtle structural symmetries and degeneracies in state distributions, while hybrid line-bar visualizations highlighted state-dependent transitions in expectation values. Additionally, the study demonstrated the scalability and accuracy of simulation frameworks in modeling quantum systems and emphasized their pedagogical value in visualizing abstract quantum behaviors. Collectively, the results substantiate quantum mechanical theory while offering data-driven insights into operator correlations and system evolution. This research not only reinforces classical interpretations but also paves the way for further exploration in quantum computing, photonic systems, and educational simulations.
Abstract Nowadays quantum physics is crucial for several scientific applications, where it is no longer possible to neglect the environmental interaction, like dissipation and decoherence. In these cases, the quantum systems are usually treated as open systems and their time-evolution is described by a density matrix in frames of the master equation, instead of the Hilbert-space vector and the Schrodinger equation. The visualization of such quantum systems allows users to calculate and study the sensitivity of the parameters, like excitation photon numbers or photonnumber distribution functions or Wigner functions. In this paper, a cloud service for numerical calculations and visualization of photonic dissipative systems is presented, which enables numerical simulations and visualizations of a wide variety of Hamiltonians, including those with arbitrary time-dependences widely used in many physics applications. The service allows creating graphics and charts for interacting complex systems and simulating their time evolution with many available timeevolution drivers.