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Revista Brasileira de Ensino de Física

A smooth path to plot hydrogen atom via Monte Carlo method

Pedro Henrique Fernandes Lobo, Everaldo Arashiro, Alcides Castro e Silva, Carlos Felipe Saraiva Pinheiro

Published Jan 1, 2019• 0 citations
Publisher

Abstract

In this paper, we show how to build a basic computer program using the Monte Carlo method to display the hydrogen atomic orbitals. For this, in a heuristic way, we applied a von Neumann acceptance-rejection method in simple problems of potential wells, and we end with the hydrogen orbitals representation. In this technique, we spread points uniformly on the 1D and 2D charts of probability density distributions, then we filtered points under these “curves or surfaces”, and we extended this logic to 3D cases. Throughout the work, we also made some comments to help beginner students better understand the term “wave function” present in the Schrödinger equation. Also, we made all source code available at a third-party platform, for any purpose under the MIT license.

DOI
10.1590/1806-9126-rbef-2019-0073

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In this paper, we show how to build a basic computer program using the Monte Carlo method to display the hydrogen atomic orbitals. For this, in a heuristic way, we applied a von Neumann acceptance-rejection method in simple problems of potential wells, and we end with the hydrogen orbitals representation. In this technique, we spread points uniformly on the 1D and 2D charts of probability density distributions, then we filtered points under these “curves or surfaces”, and we extended this logic to 3D cases. Throughout the work, we also made some comments to help beginner students better understand the term “wave function” present in the Schrödinger equation. Also, we made all source code available at a third-party platform, for any purpose under the MIT license.
A point-like picture of the Schrödinger solution for hydrogen atom is worked to emphasize that "point-like particles" may describe as "probability wave function". In each case, the three-dimensional shape of the |Ψnlm(rn, cos θ)|2 is plotted and the paths of the point-like electron (it is better to say reduced mass of the pair particles) are described in each closed shell. Finally, the orbital shape of the molecules are given according to the present simple model. In our opinion, "interpretations of the Correspondence Principle", which is a basic principle in all elementary quantum text, seems to be reviewed again!
This chapter begins with the simple solution of the Schrödinger equation for hydrogen atom for which the energy value has been derived. The Bohr radius of H atom has also been computed. After this, this chapter provides the generalised solution of the Schrödinger equation for H atom/hydrogen-like species in which F equation, R equation, and T equations have been derived in a simple and lucid way. This chapter also deals with the Laguerre differential equation, Laguerre polynomials, Rodrigues formula for the Laguerre polynomials, Laguerre associated equation and its solution, associated Laguerre polynomials, Rodrigues formula for the associated Laguerre polynomials, and solution of the radial equation. It also provides normalisation of the radial wave function and complete wave function; for the hydrogen atom, hydrogenic atomic orbital; electron density of 1 s electron; mathematical background of shape of orbitals; the most probable distance of electron from nucleus of hydrogen atom, and average distance of electron from the nucleus of hydrogen atom. The end of this chapter describes the references and adequate computational problems (solved problems and question on concepts).
In the framework of the quantum model for hydrogen atom, we use the wave functions provided by the solutions of the Schrödinger equation with Coulomb potential. Considering the radial and angular densities of the localization probability, we identify the representative quantum states that play a high-probability role in transitions for H-I spectra. Besides that, we obtain some other complementary results. So, we present the numerical tables for the radial distributions, the electron density functions and some observables like θ nodal angles, nodal surfaces, along with the ordered sequences of the average radii in the sub-shells associated with this n: ⟨r⟩4f = 18a0 < ⟨r⟩4d = 21a0 < ⟨r⟩4p = 23a0 < ⟨r⟩4s = 24a0 (a0 - the first Bohr radius). Further, we have added the significant scientific 3D-visualization of the orbitals shapes and describe them via the nodal values for θ angles.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
While in the process of studying hydrogen atom wave function in quantum mechanics course,some frequently-asked questions about the radial probability distribution,tree-dimensional angular distribution,and movement trace are raised by beginners.The visualized interpretations for those questions are made by the three dimension model set up in computer,which can not only help students to establish the correct quantum images about the movement of the particles,but also provide a reference for visualization about other quantum phenomena.