© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1005.0044

Visualization of quantum mechanics on a personal computer

Fredy L. Dubeibe

Published May 1, 2010• 0 citations

Abstract

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.

Related papers

Numerical solution of the time-dependent Schrödinger equation in spherical coordinates by Fourier-transform methods

Christopher E. Dateo, Horia I. Metiu

We present a numerical Fourier-transform method for solving the time-dependent Schrödinger equation in spherical coordinates. The time propagation is accomplished with a novel implementation of a Crank–Nicholson scheme combined with a Feit–Fleck split-operator method. Calculations on model problems give results that are in excellent agreement with those obtained by other methods. The procedure avoids the numerical instabilities associated with the division by the sine of the polar angle. The propagator is unitary, does not require the calculation of any matrix elements, and uses no matrix multiplications other than fast Fourier transforms.

The Journal of Chemical Physics•Nov 15, 1991•44 citations

Modified Crank-Nicolson method for solving the time-dependent Schrödinger equation

A R Kurniawan, Evi Suaebah, Z. I. Supardi

Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it against two well-known numerical methods: the Runge-Kutta method and the Split-Step Fourier method. Quantum tunneling is selected as the case study due to its relevance in quantum physics applications. Comparative analysis is based on numerical error metrics. The proposed method can be applied to other physical phenomena, expanding its potential application in physics research and technology industries.

Journal of Physics Conference Series•Nov 1, 2024•0 citations

An Accurate Pentadiagonal Matrix Solution for the Time-Dependent Schrödinger Equation

Ankit Kumar

One of the unitary forms of the quantum mechanical time evolution operator is given by Cayley's approximation. A numerical implementation of the same involves the replacement of second derivatives in Hamiltonian with the three-point formula, which leads to a tridiagonal system of linear equations. In this work, we invoke the highly accurate five-point stencil to discretize the wave function onto an Implicit-Explicit pentadiagonal Crank-Nicolson scheme. It is demonstrated that the resultant solutions are significantly more accurate than the standard ones. We also discuss the resolution of bipartite wavepacket dynamics and derive conditions under which a product state from the laboratory perspective remains a product state from the center-of-mass point of view. This has profound applications for decoupling complicated bipartite dynamics into two independent single-particle problems.

arXiv:2205.13467•May 26, 2022•1 citation

Fast and stable method for simulating quantum electron dynamics

Naoki Watanabe, Masaru Tsukada

A fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time-dependent Schrodinger equation. This method is a real-space-real-time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method, and an operator named adhesive operator. This method conserves the norm of the wave function, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.

arXiv:physics/0011068•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 2000•83 citations

Numerical Solution Of The Time-Dependent Schrodinger equation

Bernardine Renaldo Wong, Swee-Ping Chia, Kurunathan Ratnavelu +1 more

We implement a method to numerically solve the Time‐Dependent Schrödinger Equation (TDSE) in one dimension. The time‐dependence is modeled using the Crank‐Nicholson formula. The discrete transparent boundary conditions are employed using a Numerov approximation formula. A normalized Gaussian wavefunction is propagated across a potential step and across a potential well of modified Poschl‐Teller form, respectively. Preliminary results will be presented.

AIP conference proceedings•Jan 1, 2009•2 citations

Numerical solutions of the Schrödinger equation with source terms or time-dependent potentials

W. van Dijk, F. Masafumi Toyama

We develop an approach to solving numerically the time-dependent Schrödinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.

arXiv:1412.1802•Physical Review E•Dec 17, 2014•12 citations

In recent decades a lot of research has been done on the numerical solution of the time-dependent Schrodinger equation. On the one hand, some of the proposed numerical methods do not need any kind of matrix inversion, but source terms cannot be easily implemented into this schemes; on the other, some methods involving matrix inversion can implement source terms in a natural way, but are not easy to implement into some computational software programs widely used by non-experts in programming (e.g. Mathematica). We present a simple method to solve the time-dependent Schrodinger equation by using a standard Crank-Nicholson method together with a Cayley's form for the finite-difference representation of evolution operator. Here, such standard numerical scheme has been simplified by inverting analytically the matrix of the evolution operator in position representation. The analytical inversion of the N x N matrix let us easily and fully implement the numerical method, with or without source terms, into Mathematica or even into any numerical computing language or computational software used for scientific computing.

We present a numerical Fourier-transform method for solving the time-dependent Schrödinger equation in spherical coordinates. The time propagation is accomplished with a novel implementation of a Crank–Nicholson scheme combined with a Feit–Fleck split-operator method. Calculations on model problems give results that are in excellent agreement with those obtained by other methods. The procedure avoids the numerical instabilities associated with the division by the sine of the polar angle. The propagator is unitary, does not require the calculation of any matrix elements, and uses no matrix multiplications other than fast Fourier transforms.

Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it against two well-known numerical methods: the Runge-Kutta method and the Split-Step Fourier method. Quantum tunneling is selected as the case study due to its relevance in quantum physics applications. Comparative analysis is based on numerical error metrics. The proposed method can be applied to other physical phenomena, expanding its potential application in physics research and technology industries.

One of the unitary forms of the quantum mechanical time evolution operator is given by Cayley's approximation. A numerical implementation of the same involves the replacement of second derivatives in Hamiltonian with the three-point formula, which leads to a tridiagonal system of linear equations. In this work, we invoke the highly accurate five-point stencil to discretize the wave function onto an Implicit-Explicit pentadiagonal Crank-Nicolson scheme. It is demonstrated that the resultant solutions are significantly more accurate than the standard ones. We also discuss the resolution of bipartite wavepacket dynamics and derive conditions under which a product state from the laboratory perspective remains a product state from the center-of-mass point of view. This has profound applications for decoupling complicated bipartite dynamics into two independent single-particle problems.

A fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time-dependent Schrodinger equation. This method is a real-space-real-time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method, and an operator named adhesive operator. This method conserves the norm of the wave function, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.

We develop an approach to solving numerically the time-dependent Schrödinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin expansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.