Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

Abstract

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.

A scheme of numerical solution for three‐dimensional isoelectronic series of hydrogen atom using one‐dimensional basis functions

“…The difficulty of solving the equation exactly for other atoms containing multiple electrons arises from the electronic correlation. Therefore, solution of the many‐body system has been an issue of wide concern since the middle of the last century and a substantial number of methods have been proposed . One of the approaches is to expand the wave function of a many‐body system with a set of chosen basis where is the number of basis, n the number of orbitals, and the coefficients.…”

A microscopic approach to quark and lepton masses and mixings

“…Nevertheless, other choices are possible, e.g. U(r) ∼ r −2 or ∼ r −1 , which has been used by some authors in their studies on the stability of hydrogen like atoms in higher dimensions [72,73,74]. Another possibility is U(r) ∼ exp(−Λ R r)/r 4 in case of the above mentioned massive photon model.…”

A Green's function approach to the nonrelativistic radial wave equation of hydrogen atom

“…The Schrödinger wave equation is one of the fundamental wave equation in nonrelativistic quantum mechanics. Its solution play an essential part in the study of atomic and molecular structure and their spectral behavior . There are various methods available in the literature for its solution such as Fourier transform method, Nikiforov‐Uvarov method, asymptotic iteration method, SUSYQM method, Laplace transform method, ansatz method, and many more.…”

“…Its solution play an essential part in the study of atomic and molecular structure and their spectral behavior. [1][2][3][4][5][6] There are various methods available in the literature for its solution such as Fourier transform method, [7][8][9] Nikiforov-Uvarov method, [10][11][12][13] asymptotic iteration method, [14][15][16][17][18][19][20][21] SUSYQM method, [22][23][24][25][26][27] Laplace transform method, [28][29][30][31][32][33][34] ansatz method, [3,6] and many more.…”

“…In spite of many different solution methods, the exact analytical solution of Schr€ odinger wave equation is only possible in a few well-known cases. [1,3,6] For three and more than three body system, we rely on approximate solution due to the presence of potential term in the Hamiltonian, as the potential term depends on the interaction of all electrons with each other and with the nuclei present in the system. The prominent solution attempts in such cases are perturbation theory, [35][36][37] principle of variation, [38,39] and Hartree-Fock solution [40,41] among others.…”