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arXiv:2106.10231 · European Journal of Physics

Variational approach to the Schrödinger equation with a delta-function potential

Francisco Marcelo Fernández

Published Dec 1, 2021• 12 citations
arXivHTMLPDF

Abstract

Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

DOI
10.1088/1361-6404/ac3f27

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Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.
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This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
Abstract Finding wavefunctions for even the simplest of interacting particle systems consisting of two particles is extremely difficult. It is therefore highly desirable that an accurate and easily implementable method be available to instructors and students of quantum-mechanics for obtaining wavefunctions for these particles. The usual approach taken to do this is to use parametrized functional form for the wavefunction in conjunction with the variational method to find approximate wavefunction and energy for the ground-state of such systems. In this paper, we employ random numbers to obtain ground-state wavefunctions and energies of two interacting particles in different one-dimensional potentials. The idea behind using random numbers is to search freely for functions that lead to lower and lower energy, converging eventually to its lowest value. The method presented is easily applicable numerically using a simple algorithm, and the wavefunctions obtained are highly accurate. Thus, the method presented makes study of two interacting particles accessible to instructors and students alike in a transparent manner.
Abstract The problem of the harmonic oscillator with a centrally located delta function potential can be exactly solved in one dimension, where the eigenfunctions are expressed as superpositions of the Hermite polynomials or as confluent hypergeometric functions in general. The eigenfunctions obtained exactly are difficult to visualise and hence, to gain more insight, one can attempt to use model wave functions which are explicitly and simply expressed. Here, we apply the variational method to verify how closely one can approach the exact ground state eigenvalues using such trial wave functions. We obtain the estimates of the ground state energies, which are closer to the exact values in comparison to earlier approximate results for both the repulsive and attractive delta potentials.
Abstract The conventional variational method is reformulated within a geometrical approach to quantum mechanics, both for finite and infinite dimensional Hilbert spaces. This geometrical setting allows us to deal not only with approximated eigenvalues and eigenstates, but also with approximated Schrödinger equations by means of Hamiltonian evolution equations associated with expectation value functions. We shall consider some specific examples where we extend our approach also to density matrices.