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EXPONENTIALLY DECAYING WAVE FUNCTION AND QUANTIZED ENERGY FOR NANO SPHERICAL PARTICLES AND HYDROGEN LIKE ATOMS

Ghada E. S. Elammeen Amna E. Musa

Published Mar 16, 2019• 0 citations
Publisher

Abstract

The solution of radial Schrödinger equation for hydrogen like atoms is complex. One needs simple solution the behavior of nano particle needs also to be described. Nano spherical particles are isolated non interacting particles that obey quantum laws. Thus one expects their behavior to resemble that of particles in a box, or hydrogen like atoms. The spherical nano shape requires using spherical coordinate to solve Schrödinger Equation. The solution found that the wave function is diminishing outside the sphere, which agrees with observations. This solution is simpler than the conventional one, where it is a sum of sine and cosine function.

DOI
10.5281/zenodo.2596226

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The solution of radial Schrödinger equation for hydrogen like atoms is complex. One needs simple solution the behavior of nano particle needs also to be described. Nano spherical particles are isolated non interacting particles that obey quantum laws. Thus one expects their behavior to resemble that of particles in a box, or hydrogen like atoms. The spherical nano shape requires using spherical coordinate to solve Schrödinger Equation. The solution found that the wave function is diminishing outside the sphere, which agrees with observations. This solution is simpler than the conventional one, where it is a sum of sine and cosine function.
The lightest and simplest arrangement of protons and electrons is a hydrogen atom. One of the wave characteristics of an atom can be known using the Schrodinger equation. Schrodinger's equations of spherical coordinates consist of radial equations and angular equations. Quantum numbers 4≤ n ≤ 5from the Schrodinger equation sphere coordinates produce different angular wave functions. To find out the angular wave equation using normalization conditions . Based on the normalization conditions for n = 4 a wave function is produced , while for n = 5 a wave function is .
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
Solutions of the Schrödinger equation for central potentials and for cubic systems are the main contents of the chapter. Motion in a central field, energy eigenvalues and wave functions for the hydrogen atom, radial probability density, solutions for the 3D spherically symmetric potential well, and free electron motion in a 3D-box are discussed with the use of algebra that clarifies the essential physics in a simple but appealing manner. Derivation of energy eigenvalues for the hydrogen atom and for a free electron cubic box clearly illustrates the concept of quantization of energy. The number of states and the density of states are introduced for the case of an electron confined to a cubic box. Carefully chosen solved examples further consolidate the understanding of solutions of the Schrödinger equation for central potentials and for a cubic system.
In this work the Schrödinger equation of the hydrogen-like atom is analytically solved. Three sets of analytical solution are obtained if the factor r−l is not neglected. The first solution is the same as the traditional radial wave function; another one diverges; the last one is far different from the traditional solution. On the consideration of the finite size of the nucleus, the third wave function does not diverge while r approaches to zero. Its radial wave function has below characteristics: (1) the angular-momentum quantum number l must be greater than the principal quantum number n; (2) l must not be 0 or 1; (3) the electron-cloud distribution differs from the traditional one; (4) the electron is closer to the nucleus by comparison with that in traditional results. On the other hand, the validity of solutions needs to be verified experimentally. 1.
There are certain prime features of physical systems—notably rotation and its associated angular momentum—that simply do not appear in a one-dimensional world. This chapter introduces the three-dimensional Schrodinger equation and applies it to the analysis of a particle in a three-dimensional box and to the spherically symmetric wave functions of hydrogen. The time-independent Schrodinger equation can be viewed in a way that gives it a family resemblance to many other differential equations of physics, and which expresses its structure in very general terms. A computer solution uses dimensionless variables based on natural units. The Schrodinger theory does not have the limitations of the Bohr theory. Schrodinger theory incorporates the well-established wave nature of matter and can be generalized to analyze systems for which the Bohr theory has proven inadequate: multi-electron atoms, chemical properties of atoms, molecular and solid-state physics and chemistry, as well as more subtle effects in hydrogen itself.