Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Visualizing Quantum Mechanical Phenomena

Johannes Schmidt-Ehrenberg, Hans‐Christian Hege

Published Dec 2, 1999• 0 citations
OpenAlex

Abstract

In this paper we discuss several ways to visualize stationary and non-stationary quantum mechanical systems. We demonstrate an approach for the quantitative interpretation of probability density isovalues which yields a reasonable correlation between isosurfaces for different timesteps. As an intuitive quantity for visualizing the momentum of a quantum system we pro-pose the probability flow density, which can be treated by vector field visu-alization techniques. Finally, we discuss the visualization of non-stationary systems by a sequence of single timestep images. 1

Related papers

Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials

Richard W. Robinett

After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.

arXiv:quant-ph/0307056•European Journal of Physics•Jan 30, 2002•11 citations

Quantum wavepackets: Proofs (almost) without words

Yu-xi Liu

We present a geometrical way of understanding the dynamics of wavefunctions in a free space, using the phase-space formulation of quantum mechanics. By visualizing the Wigner function, the spreading, shearing, the so-called "negative probability flow" of wavefunctions, and the long-time asymptotic dispersion, are intuited visually. These results are not new, but previous derivations were analytical, whereas this paper presents elementary geometric arguments that are almost "proofs without words", and suitable for a first course in quantum mechanics.

arXiv:2409.02962•Sep 3, 2024•0 citations

Visualization of dynamics in coupled multi-spin systems

Jingyan Xu, Dmitry Budker, Danila A. Barskiy

Abstract. Since the dawn of quantum mechanics ways to visualize spins and their interactions attracted attention of researchers and philosophers of science. Angular momentum probability (AMP) surfaces are known for visualizing density matrices; a plotted surface represents probability of finding maximum projection of angular momentum along any direction in space. While AMP surfaces visually reveal symmetries of density matrices which directly translate to measured properties, they focus solely on one total-angular-momentum manifold, neglecting interaction between different manifolds often encountered in the NMR of multispin systems. In this work, we extend applicability of this visualization method and introduce its extension, angular momentum coherence (AMC) surface approach. Three examples of nuclear spin dynamics in two-spin systems are presented and visualized using the AMP/AMC surfaces: (I) spin-lock induced crossing (SLIC) sequence with initial state being the singlet state between two spins; (II) interplay between chemical exchange and spin dynamics during the high-field signal amplification by reversible exchange (SABRE) experiment; (III) zero- to ultralow-field (ZULF) NMR experiment in the presence of magnetic field applied perpendicular to the sensitive axis of the detector. The presented visualization technique extends applicability of AMP surfaces to coupled multispin systems and will facilitate intuitive understanding of spin dynamics during complex NMR experiments as exemplified here by the considered cases. The temporal sequences (“the movies”) of such surfaces show phenomena like interconversion of polarization moments (Auzinsh et al., 2010) and, as particularly relevant in NMR and demonstrated here, polarization transfer between different spins. Such effects are difficult to grasp by looking at (time-dependent) numerical values of density-matrix elements.

Magnetic Resonance•Aug 9, 2022•4 citations

Visualizing Quantum Dynamics

Tyler N. Thompson

My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.

•Jan 1, 2018•0 citations

Visualizing quantum mechanics in phase space

Heiko Bauke, Noya Ruth Itzhak

We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave functions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features.

arXiv:1101.2683•Jan 11, 2011•6 citations

Comparing classical and quantum probability distributions for an asymmetric infinite well

M. A. Doncheski, Richard W. Robinett

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

arXiv:quant-ph/0307014•European Journal of Physics•Apr 5, 2000•27 citations
In this paper we discuss several ways to visualize stationary and non-stationary quantum mechanical systems. We demonstrate an approach for the quantitative interpretation of probability density isovalues which yields a reasonable correlation between isosurfaces for different timesteps. As an intuitive quantity for visualizing the momentum of a quantum system we pro-pose the probability flow density, which can be treated by vector field visu-alization techniques. Finally, we discuss the visualization of non-stationary systems by a sequence of single timestep images. 1
After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.
We present a geometrical way of understanding the dynamics of wavefunctions in a free space, using the phase-space formulation of quantum mechanics. By visualizing the Wigner function, the spreading, shearing, the so-called "negative probability flow" of wavefunctions, and the long-time asymptotic dispersion, are intuited visually. These results are not new, but previous derivations were analytical, whereas this paper presents elementary geometric arguments that are almost "proofs without words", and suitable for a first course in quantum mechanics.
Abstract. Since the dawn of quantum mechanics ways to visualize spins and their interactions attracted attention of researchers and philosophers of science. Angular momentum probability (AMP) surfaces are known for visualizing density matrices; a plotted surface represents probability of finding maximum projection of angular momentum along any direction in space. While AMP surfaces visually reveal symmetries of density matrices which directly translate to measured properties, they focus solely on one total-angular-momentum manifold, neglecting interaction between different manifolds often encountered in the NMR of multispin systems. In this work, we extend applicability of this visualization method and introduce its extension, angular momentum coherence (AMC) surface approach. Three examples of nuclear spin dynamics in two-spin systems are presented and visualized using the AMP/AMC surfaces: (I) spin-lock induced crossing (SLIC) sequence with initial state being the singlet state between two spins; (II) interplay between chemical exchange and spin dynamics during the high-field signal amplification by reversible exchange (SABRE) experiment; (III) zero- to ultralow-field (ZULF) NMR experiment in the presence of magnetic field applied perpendicular to the sensitive axis of the detector. The presented visualization technique extends applicability of AMP surfaces to coupled multispin systems and will facilitate intuitive understanding of spin dynamics during complex NMR experiments as exemplified here by the considered cases. The temporal sequences (“the movies”) of such surfaces show phenomena like interconversion of polarization moments (Auzinsh et al., 2010) and, as particularly relevant in NMR and demonstrated here, polarization transfer between different spins. Such effects are difficult to grasp by looking at (time-dependent) numerical values of density-matrix elements.
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which arise due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about the amount of time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.