This work aims to examine the power of an energy dependent coordinate mapping to facilitate the solution to Schrödinger equations for screened Coulomb potential systems.The model system is taken as two particle system in only interparticle distance dependent potential which is also spherically symmetric. Since time independent Schrödinger equation of such system can be reduced to relative motion separated out of angular coordinates, the wave function becomes dependent on relative radial variable only and the motion can be investigated more easily. The energy parameter is taken as −v2/2 since only bound states are considered. First an unknown coordinate bending function is defined in terms of the coordinate and energy arguments. After appropriate assumptions the resulting spectral problem is solved and the potential function is obtained in terms of coordinate bending function which we called “Characteristic Potential for the Coordinate Bending”. Then appropriate efficient coordinate bending structures can be sought to somehow match the Characteristic potential with the true given potential and system energy can also be identified through this procedure.
This work aims to examine the power of an energy dependent coordinate mapping to facilitate the solution to Schrödinger equations for screened Coulomb potential systems.The model system is taken as two particle system in only interparticle distance dependent potential which is also spherically symmetric. Since time independent Schrödinger equation of such system can be reduced to relative motion separated out of angular coordinates, the wave function becomes dependent on relative radial variable only and the motion can be investigated more easily. The energy parameter is taken as −v2/2 since only bound states are considered. First an unknown coordinate bending function is defined in terms of the coordinate and energy arguments. After appropriate assumptions the resulting spectral problem is solved and the potential function is obtained in terms of coordinate bending function which we called “Characteristic Potential for the Coordinate Bending”. Then appropriate efficient coordinate bending structures can be sought to somehow match the Characteristic potential with the true given potential and system energy can also be identified through this procedure.
We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at c=0.1, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.
The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
Within the framework of the conventional Nikiforov-Uvarov method and a new form of Greene-Aldrich approximation scheme, we solved the Schrödinger equation with the energy-dependent screened Coulomb potential. Energy eigenvalues and energy eigenfunctions were obtained both approximately and numerically at different dimensions. The energy variations with different potential parameters, quantum numbers and energy slope parameter, respectively were also discussed graphically. The major finding of this research is the effect of the energy slope parameter on the energy spectra, which is seen in the existence of two simultaneous energy values for a particular quantum state. Our special cases also agree with the results obtained from literature, when the energy slope parameter is zero.