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Quantum wavefunctions representation with python language

Jiang, Jhe-Yua;Lin, Jyh Shinga

Published Nov 15, 2019• 0 citations
OpenAlex

Abstract

The Particle in a box (PIAB) is the most representative model to be used and illustrated for exactly solving the Schrodinger equation in quantum physics. However, as the degrees of freedom of this PIAB model from one dimensional case to three dimensional case the solutions of its wavefunction will become more difficult to visualize and comprehend. Therefore, in this research work we employed the most popular python programming language with its diverse workable modules and its diverse visualizing modules to easily obtain the visualizing representation, that is, isosurface, of solvable wavefunction of PIAB model in three dimensional case. Furthermore, this isosurface of wavefunction enables us to explore its interesting properties of wavefunction based on their different nx, ny and nz quantum numbers. Consequently, we are able to gain a very deep insight into this complex wavefunction of PIAB model in three dimensional case.

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The Particle in a box (PIAB) is the most representative model to be used and illustrated for exactly solving the Schrodinger equation in quantum physics. However, as the degrees of freedom of this PIAB model from one dimensional case to three dimensional case the solutions of its wavefunction will become more difficult to visualize and comprehend. Therefore, in this research work we employed the most popular python programming language with its diverse workable modules and its diverse visualizing modules to easily obtain the visualizing representation, that is, isosurface, of solvable wavefunction of PIAB model in three dimensional case. Furthermore, this isosurface of wavefunction enables us to explore its interesting properties of wavefunction based on their different nx, ny and nz quantum numbers. Consequently, we are able to gain a very deep insight into this complex wavefunction of PIAB model in three dimensional case.
This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.
Abstract The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.
This research aims to develop computational visualization of the wave function and probability distribution of particles in a one dimensional infinite potential well. The wave function is obtained analytically from the time-independent Schrödinger equation and then normalized to give it probabilistic meaning. Numerical simulations are carried out using Python with the NumPy library for computation and Matplotlib for visualization, allowing manipulation of parameters such as box length and quantum number. The visualization results confirm the basic concepts of quantum mechanics, namely quantized energy levels, sinusoidal wave function forms with a specific number of nodes, and probability distributions that have several peaks and are always positive. The simulation also shows the correspondence between wave-function nodes and zero-probability points, as well as the increasing complexity of oscillations and variations in probability distributions as the quantum number increases. These findings are consistent with theoretical predictions and show that the computational approach is effective in visualizing abstract concepts in quantum mechanics, thereby clarifying the relationship between mathematical formulation and its physical interpretation.
A common problem when learning Quantum Mechanics is the complexity in the mathematical and physical concepts, which leads to difficulty in solving and understanding problems. Using programming languages like Python have become more and more prevalent in solving challenging physical systems. An open-source computer algebra system, SymPy, has been developed using Python to help solve these difficult systems. I have added code to the SymPy library for two different systems, a One-Dimensional Quantum Harmonic Oscillator and a Quantum Mapping Gate used in Quantum Computing.
Quantum Mechanics can be an abstract and complex subject. Students often complain of confusion, struggle, and frustration as they try to master the topic. The goal of this book is to reduce the complexity and clarify the abstractions with concrete visual examples driven by simple python programs. It is assumed that the reader is concurrently taking a course in quantum mechanics, or self-studying quantum mechanics, but is looking for supplementary material to help with understanding and visualizing how quantum mechanics works. The focus of this book is writing python programs to visualize the underlying behavior of the mathematical theory. The background needed to understand quantum mechanics is differential equations, linear algebra and modern physics. We need a strong foundation in differential equations and linear algebra because the behavior of quantum systems is governed by equations that are written in terms of these concepts. Modern physics includes concepts such as special relativity and quantum phenomena like the photoelectric effect and energy quantization that the theory of quantum mechanics seeks to explain. This book is also not an introduction to the python programming language, or to numpy, or even to VPython. However its programming examples start simply and grow more complex as the chapters progress, so deep expertise in any of these is not a pre-requisite. Key features: · Provides an accessible and practical guide to the abstractions in quantum mechanics with concrete visual examples driven by simple python programs. · Contains few derivations, equations, and proofs. · For complete beginners of python programming, appendix B serves as a very brief introduction to the main concepts needed to understand the code in this book.
This study presents a computational visualization of the stationary states of a particle confined in a one-dimensional infinite potential well, formulated entirely from analytical solutions and implemented in a cloud-based Python environment. Using Google Colab, wave functions, probability densities, and nodal structures were generated for several quantum numbers to illustrate fundamental quantum-mechanical characteristics, including energy quantization, spatial oscillatory behavior, and the emergence of classical correspondence at higher energy levels. The resulting visualizations offer a clear representation of the spatial distribution of quantum states, thereby supporting a conceptual understanding of the underlying analytical model. As a novelty, this study introduces a Colab-based analytical–numerical combination as a visualization medium, a format rarely explored in depth in quantum-well learning contexts. Beyond instructional applications, the modeled system accurately reflects the essential physical behavior of electrons in semiconductor quantum wells, where quantized subband energies and wave-function profiles significantly influence optical and electronic transitions. These findings demonstrate that analytically derived quantum models, when integrated with computational visualization, can effectively enhance conceptual comprehension while offering insight into the fundamental principles governing modern optoelectronic structures.