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

2021 ◽  
Author(s):  
Francisco Marcelo Fernandez
Keyword(s):  
Variational Method ◽  
Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Introductory Course ◽  
Relevant Role ◽  
Secular Equations ◽  
The One ◽  
Computer Algebra Software

Abstract We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ 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.

Construction of Dirac Delta Function from the Discrete Orthonormal Basis of the Function Space

2017 ◽  
pp. 54-57
Author(s):  
Hari Prasad Lamichhane
Keyword(s):  
Function Space ◽  
Orthonormal Basis ◽  
Delta Function ◽  
Hamiltonian Operator ◽  
Dirac Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Simple Basis ◽  
Infinite Potential Well

Orthonormal basis of the function space can be used to construct Dirac delta function. In particular, set of eigenfunctions of the Hamiltonian operator of a particle in one dimensional infinite potential well forms a non-degenerate discrete orthonormal basis of the function space. Such a simple basis set is suitable to study closure property of the basis and various properties of Dirac delta function in Physics graduate lab.The Himalayan Physics Vol. 6 & 7, April 2017 (54-57)

scholarly journals Unusual situations that arise with the Dirac delta function and its derivative

2009 ◽  
Vol 31 (4) ◽  
pp. 4302-4308 ◽  
Author(s):  
F.A.B Coutinho ◽  
Y Nogami ◽  
F.M Toyama
Keyword(s):  
Quantum Mechanics ◽  
Delta Function ◽  
Dirac Delta Function ◽  
One Dimensional ◽  
Dirac Delta ◽  
Underlying Mechanisms ◽  
Usual Formula

There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.

scholarly journals A Numerical Variational Method for Calculating Accurate Vibrational Energy Separations of Small Molecules and Their Ions

1986 ◽  
Vol 39 (5) ◽  
pp. 749 ◽  
Author(s):  
G Doherty ◽  
MJ Hamilton ◽  
PG Burton ◽  
EI von Nagy-Felsobuki
Keyword(s):  
Finite Element ◽  
Variational Method ◽  
Small Molecules ◽  
Vibrational Energy ◽  
Basis Functions ◽  
Variational Procedure ◽  
One Dimensional ◽  
Trial Wavefunction ◽  
Vibrational Energies ◽  
The One

A combination of known methods have been spliced together in order to calculate accurate vibrational energies and wavefunctions. The algorithm is based on the Rayleigh-Ritz variational procedure in which the trial wavefunction is a linear combination of configuration products of one-dimensional basis functions. The Hamiltonian is that due to Carney and Porter (1976). The kernel of the algorithm consists o( the one-dimensional basis functions, which are the finite element solutions of the associated one-dimensional problems.

scholarly journals Two-dimensional Scattering by a Potential without a Classical Analogue

1999 ◽  
Vol 52 (5) ◽  
pp. 859 ◽  
Author(s):  
I. V. Ponomarev
Keyword(s):  
Quantum Chaos ◽  
Matrix Theory ◽  
Delta Function ◽  
Two Dimensional ◽  
Dirac Delta ◽  
S Matrix ◽  
Classical Analogue ◽  
Scattering Potential ◽  
The One ◽  
The Given

A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y-axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S-matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.

scholarly journals Rayleigh-Ritz variational method with suitable asymptotic behaviour

Open Physics ◽  
2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Francisco Fernández ◽  
Javier Garcia
Keyword(s):  
Variational Method ◽  
Asymptotic Behaviour ◽  
Orthogonal Polynomial ◽  
Rate Of Convergence ◽  
Orthogonal Basis ◽  
Basis Sets ◽  
One Dimensional ◽  
Variational Calculations ◽  
Dimensional Models ◽  
The One

AbstractThis paper considers the Rayleigh-Ritz variational calculations with non-orthogonal basis sets that exhibit the correct asymptotic behaviour. This approach is illustrated by constructing suitable basis sets for one-dimensional models such as the two double-well oscillators recently considered by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization.

Vibration of Diatomic System in One-Dimensional Nanomaterials

2011 ◽  
Vol 55-57 ◽  
pp. 545-549
Author(s):  
Jun Lu
Keyword(s):  
Bound States ◽  
Hypergeometric Series ◽  
Basis Set ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Energy Eigenvalues ◽  
Diatomic System ◽  
Analytic Expressions ◽  
Hyperbolic Potential ◽  
The One

By means of the hypergeometric series method, the explicit expressions of energy eigenvalues and eigenfunctions of bound states for a diatomic system with a hyperbolic potential function are obtained in the one-dimensional nanomaterials. The eigenfunctions of a one-dimensional diatomic system, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.

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