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Revista Mexicana de Física E

Numerical solution of the Schrödinger equation for radiation-matter interaction models using the Runge-Kutta method implemented in Python

Leonardi Hernández-Sánchez, Ariel Flores-Rosas, I. Ramos-Prieto, Francisco Soto, H. M. Moya-Cessa

Published Jul 1, 2026• 0 citations
Publisher

Abstract

Obtaining exact solutions to the time-dependent Schrodinger equation in complex quantum systems presents significant challenges. In this ¨ context, numerical methods offer powerful alternatives for exploring their dynamics. In this pedagogical article, we present a numerical approach based on the fourth-order Runge-Kutta method, implemented in Python, to simulate radiation-matter interaction models. The methodology is illustrated using the well-known Jaynes-Cummings model, and the numerical results are compared with its exact analytical solution for illustrative purposes. Although only this model is explicitly solved, the numerical framework is general and can be readily extended to more complex Hamiltonians, including time-dependent cases. This makes the approach a practical and accessible tool for exploring a wide range of quantum systems.

DOI
10.31349/revmexfise.23.020202

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Obtaining exact solutions to the time-dependent Schrodinger equation in complex quantum systems presents significant challenges. In this ¨ context, numerical methods offer powerful alternatives for exploring their dynamics. In this pedagogical article, we present a numerical approach based on the fourth-order Runge-Kutta method, implemented in Python, to simulate radiation-matter interaction models. The methodology is illustrated using the well-known Jaynes-Cummings model, and the numerical results are compared with its exact analytical solution for illustrative purposes. Although only this model is explicitly solved, the numerical framework is general and can be readily extended to more complex Hamiltonians, including time-dependent cases. This makes the approach a practical and accessible tool for exploring a wide range of quantum systems.
Obtaining exact solutions to the Schrödinger equation in complex quantum systems poses significant challenges. In this context, numerical methods emerge as valuable tools for analyzing such systems. This article proposes a numerical approach using the fourth-order Runge-Kutta method, implemented in Python, to tackle radiation-matter interaction systems. This methodology is applicable to various Hamiltonians, including that of the Jaynes-Cummings model. The accuracy of the numerical results is validated by comparing them with analytical solutions in simplified cases, demonstrating its effectiveness in studying quantum systems where exact solutions cannot be derived.
We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.
Quantum confinement is a relatively new subject matter in quantum mechanics. Quantum confinement traps the atom in a cavity whose dimensions are small enough to alter its properties. Some physicists have used the time-independent Schrodinger equation to obtain exact results for confinement problems. But this paper focuses on the dynamic of a hydrogen atom when it is moved from captivity. The goal of this project is to study and understand different methods used to predict solutions that cannot be obtained analytically which was done by relying on methods that approximates solutions. To do this, a differential equation with boundary conditions was manipulated with the computer programs C++ and Eclipse using Runge-Kutta, B-Splines, and the Schrodinger equation. While the Runge-Kutta method was utilized as a basis for the boundary problem, its simplicity was not viable for this research which contained so many complex variables. Instead, B-Splines were used to solve the time-dependent Schrodinger equation. B-Spline functions are designed to generalize polynomials for the purpose of approximating arbitrary functions. By taking the summation of the timeindependent Schrodinger equation, we obtained our desired results for the timedependent Schrodinger equation.
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.
We study Magnus expansion (ME) approximation scheme for the interaction between an atom and a single quantized cavity mode (Jaynes-Cumming model) in a closed quantum system in resonance or near resonance for a time-dependent coupling coefficient g(t) in both the interaction and rotating picture by implementing a novel numerical method called MG4 and compare out results to the Runge-Kutta 4th (RK4) order solution to demonstrate the conservation of unitary evolution of the ME. A cursory study of open quantum system is given to encourage the study of ME for dissipative systems. Furthermore, we assume that our time-dependent coupling coefficient g(t) can take on two forms, Gaussian and sinusoidal, which are introduced as pulses to study the behavior and response of the cavity. Our results show that ME is a sufficient approximation scheme in our study of closed quantum systems which may have applications in quantum control.
The Jaynes-Cummings (JC) model is one of the most fundamental models for the interaction between light and matter. In this paper, we first derive the explicit expression for the time evolution operator of the JC model when the light field resonates with a two-level atom. Based on this, we calculate the analytical expressions for the system state and the population of each atomic energy level as a function of time when the two-level atom is initially in a superposition of two energy eigenstates and the light field is in a superposition of three photon number states and a coherent state, respectively. Finally, this paper introduces QuTiP, a useful extension library in Python for quantum mechanics simulation and numerical solution, and uses it to numerically solve the time evolution of this model, obtaining results consistent with the analytical solutions. These computational processes are of great significance in scientific research and university physics teaching.