Breaking the curse of dimension for the electronic Schrödinger equation with functional analysis

2018 ◽  
Vol 1142 ◽  
pp. 66-77 ◽  
Author(s):  
James S.M. Anderson ◽  
Farnaz Heidar-Zadeh ◽  
Paul W. Ayers
Keyword(s):  
Functional Analysis ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Curse Of Dimension

Approximate and analytical bound state solutions of the Frost–Musulin potential

2014 ◽  
Vol 92 (1) ◽  
pp. 18-21 ◽  
Author(s):  
A.G. Adepoju ◽  
E.J. Eweh
Keyword(s):  
Functional Analysis ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Short Range ◽  
Bound State ◽  
State Solution ◽  
Centrifugal Term ◽  
Analysis Method ◽  
Bound State Solution ◽  
Short Range Potential

Despite all the attempts made by several authors to investigate the bound state solutions of the Schrödinger equation with various potentials, until now, such investigations have not been conducted for Frost–Musulin diatomic potential. In this study, we obtain the approximate bound state solution of this potential via the functional analysis method. We also numerically solved the Schrödinger equation without any approximation to centrifugal term for the same potential. The comparisons between the results reveal the accuracy of our approximate results for short-range potential.

scholarly journals The Schrödinger Equation with Deng-Fan-Eckart Potential (DFEP): Nikiforov-Uvarov-Functional Analysis (NUFA) Method

2021 ◽  
Vol 3 (5) ◽  
pp. 58-62
Author(s):  
E. B. Ettah
Keyword(s):  
Functional Analysis ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Energy Spectra ◽  
Analysis Method ◽  
Radial Part ◽  
Molecular Physics ◽  
Centrifugal Barrier ◽  
Eckart Potential ◽  
Good Agreement

In this study, the radial part of the Schrödinger equation with the Deng-Fan-Eckart potential (DFEP) is solved analytically by employing the improved Greene and Aldrich approximation to bypass the centrifugal barrier and using the Nikiforov-Uvarov-Functional Analysis method (NUFA). The energy expression and wave function are obtained respectively. The numerical energy spectra for some diatomic molecules have been studied and compared with the findings of earlier studies and it has been found to be in good agreement. The NUFA method used in this study is easy and very less cumbersome compared to other methods that currently exist and it is recommended that researchers in this area adopt this method. The findings of this study will find direct applications in molecular physics.

scholarly journals The application bispherical coordinate in Schrödinger equation for Mobius square plus modified Yukawa potential using Nikiforov Uvarov Functional Analysis (NUFA) method

2020 ◽  
Vol 4 (2) ◽  
pp. 48
Author(s):  
Briant Sabathino Harya Wibawa ◽  
A Suparmi ◽  
C Cari
Keyword(s):  
Functional Analysis ◽  
Wave Function ◽  
Energy Spectrum ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Yukawa Potential ◽  
Radial Part ◽  
Angular Part ◽  
Variable Separation Method ◽  
Bispherical Coordinates

<p class="Abstract">The application bispherical coordinates in Schrödinger equation for the Mobius square plus modified Yukawa potential have been obtained. The Schrödinger equation in bispherical coordinates for the separable Mobius square plus modified Yukawa potential consisting of the radial part and the angular part for the Mobius square plus modified Yukawa potential is solved using the variable separation method to reduce it to the radial part and angular part Schrödinger equation. The aim of this study was to solve the Schrödinger's equation of radial in bispherical coordinates for the Mobius square plus modified Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Nikiforov Uvarov Functional Analysis (NUFA) method used to obtained energy spectrum equation and wave function for the Mobius square plus modified Yukawa potential. The result of energy spectrum equation for Mobius square plus modified Yukawa potential can be shown in Equation (50). The result of un-normalized wave function equation for Mobius square plus modified Yukawa potential can be shown in Table 1.</p>

scholarly journals ANALYTICAL SOLUTION OF SCHRÖDINGER EQUATION IN BISPHERICAL COORDINATE FOR MOBIUS SQUARE PLUS MODIFIED YUKAWA POTENTIAL USING NIKIFOROV UVAROV FUNCTIONAL ANALYSIS (NUFA) METHOD WITH ITS THERMODYNAMICS PROPERTIES FOR DIATOMIC MOLECULES

2021 ◽  
Vol 17 (37) ◽  
pp. 111-134
Author(s):  
Briant Sabathino Harya WIBAWA ◽  
A SUPARMI ◽  
C CARI
Keyword(s):  
Functional Analysis ◽  
Wave Function ◽  
Partition Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Diatomic Molecules ◽  
Radial Part ◽  
Radial Wave ◽  
Vibrational Partition Function ◽  
Bispherical Coordinates

Background: The analytical solution of the Schrödinger equation in bispherical coordinates has attracted a great deal of interest for theoretical physics researchers in the branch of quantum physics. The energy and wave function are solutions of the Schrödinger equation which are very important because it contains all necessary information regarding the behavior of quantum systems. Aim: This study aimed to obtain energy, radial wave functions and thermodynamic properties for diatomic molecules from the radial part of the Schrödinger equation in bispherical coordinates for the modified Mobius square plus Yukawa potential using the Nikiforov Uvarov Functional Analysis (NUFA) method. Methods: The variable separation method was applied to reduce the Schrodinger equation in bispherical coordinates to the radial part and angular part Schrodinger equation. The Schrodinger equation of the radial part in bispherical coordinates was solved using the Nikiforov Uvarov Functional Analysis (NUFA) method to obtain the energy equation and radial wave function. Furthermore, the vibrational partition function 𝑍 was obtained from the energy equation. The vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆 were obtained from the vibrational partition function 𝑍. Results and Discussion: The results showed that the increase of parameters of 𝑛 and 𝛼 caused the decrease of energy, but the increase of parameters of 𝐿 and 𝑚0 caused the increase of energy. The radial quantum number 𝑛 and the potential range 𝛼 had the most effect to the wave functions. The parameters 𝑛𝑚𝑎𝑥, 𝑇, and 𝛼 had effect to the vibrational partition function 𝑍, vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆. Conclusions: From the results of this study, it can be concluded that the energy, radial wave function, and thermodynamic properties for diatomic molecules have been obtained using the Nikiforov Uvarov Functional Analysis (NUFA) method.

Solusi Persamaan Schrödinger Berbasis Panjang Minimal untuk Potensial Coulomb Termodifikasi

2018 ◽  
Vol 2 (2) ◽  
pp. 43-47
Author(s):  
A. Suparmi, C. Cari, Ina Nurhidayati
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Coulomb Potential ◽  
Schrodinger Equation ◽  
Minimal Length ◽  
Energy Spectra ◽  
Atomic Particle ◽  
Approximate Analytical Solutions ◽  
Radial Schrödinger Equation

Abstrak – Persamaan Schrödinger adalah salah satu topik penelitian yang yang paling sering diteliti dalam mekanika kuantum. Pada jurnal ini persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Fungsi gelombang dan spektrum energi yang dihasilkan menunjukkan kharakteristik atau tingkah laku dari partikel sub atom. Dengan menggunakan metode pendekatan hipergeometri, diperoleh solusi analitis untuk bagian radial persamaan Schrödinger berbasis panjang minimal diaplikasikan untuk potensial Coulomb Termodifikasi. Hasil yang diperoleh menunjukkan terjadi peningkatan energi yang sebanding dengan meningkatnya parameter panjang minimal dan parameter potensial Coulomb Termodifikasi. Kata kunci: persamaan Schrödinger, panjang minimal, fungsi gelombang, energi, potensial Coulomb Termodifikasi Abstract – The Schrödinger equation is the most popular topic research at quantum mechanics. The  Schrödinger equation based on the concept of minimal length formalism has been obtained for modified Coulomb potential. The wave function and energy spectra were used to describe the characteristic of sub-atomic particle. By using hypergeometry method, we obtained the approximate analytical solutions of the radial Schrödinger equation based on the concept of minimal length formalism for the modified Coulomb potential. The wave function and energy spectra was solved. The result showed that the value of energy increased by the increasing both of minimal length parameter and the potential parameter. Key words: Schrödinger equation, minimal length formalism (MLF), wave function, energy spectra, Modified Coulomb potential

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

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