Application of the variational method to the particle-in-the box problem

Abstract

It is the purpose of this paper to consider a more general application of the variational method to the particle-in-the-box problem with polynomial trial functions.

Exploring the Rayleigh-Ritz Variational Principle

“…In this example, representations of successive approximations for greater values of n give pairs of indistinguishable plots owing to fast convergence. ( 2) , in eq 18 versus the number of terms taken in |S n ͘. The appearance of the points by pairs is due to the null value of the even coefficients of eq 28.…”

“…The simplicity of the mathematics required for the treatment of this problem and the knowledge of the exact solution make it suitable for the study and understanding of the main approximate methods of quantum theory applied to chemistry. In particular, some works published in this Journal (1)(2)(3)(4)(5) use this model to simply illustrate important quantum chemical concepts.…”

“…Figure 2. Logarithmic representation of the quadratic integral error, ε n(2) , in eq 18 versus the number of terms taken in |S n ͘. The appearance of the points by pairs is due to the null value of the even coefficients of eq 28.…”

Dilemma of the “Best Wavefunction”: Comparing Results of the STO-NG Procedure versus the Linear Variational Method

“…The second semester of a quantum-chemistry course is typically devoted to modern ab initio approaches to the calculation of the electronic structure and properties of molecules. Most of the sophisticated ab initio approaches rely upon the self-consistent-field approximation (also called the Hartree–Fock method) as the starting point. − When the Hartree–Fock method employs a set of basis functions for describing the molecular orbitals, it gives rise to the Roothaan equations ,, and to the Pople–Nesbet equations, derived by the Ritz or linear variational method. − At this point, most students assume that the “best wavefunction” is the one constructed with the most basis functions. They do not question the meaning of the term “best wavefunction”.…”

Using the Screened Coulomb Potential To Illustrate the Variational Method

“…Sooner or later chemistry students learn in quantum chemistry or related courses that the elucidation of the structural and dynamic properties of matter at the microscopic, or we should say nanoscopic, scale does necessarily suppose solving the Schrödinger equation, and that, except for a few simple but extremely important models such as the particle in a box, the harmonic oscillator, the rigid rotator, and, indeed, the hydrogen atom, the solutions of the Schrödinger equation have to be found by using approximation methods. The variational method, along with perturbation theory and numerical integration of the Schrödinger equation, is one of the most powerful tools employed in this context, and its study is accordingly mandatory in quantum chemistry courses, as is its presentation in the textbooks covering the subject. − This is also why the variational method has received considerable attention in this Journal . − …”