The_Spectrum_of_the_Hydrogen_Atom - The Spectrum of the...

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The Spectrum of the Hydrogen AtomXIAOTANG WANGSOCIAL AND HUMAN SCIENCESUNIVERSITY OF SOUTHAMPTONAbstractIn this report, the time-dependent and time-independent Schrödinger equations are stated andexpressed in spherical polar coordinates. The separation of variables technique is review and used forsolving the Schrödinger equation for a hydrogen atom. The general solution is the wave function.Then, the spectrum and energy levels for the hydrogen atom are discussed.IntroductionA hydrogen atom consists of a positive charged proton and a negative charged electron. It is thesimplest but the most important atom for many applications in quantum mechanics[1]. The spectralfrequencies and energy levels of the hydrogen atom were first discovered by Niels Bohr in 1913. Thesame results were obtained by the Schrödinger equation which is published in 1926. However, the1
Schrödinger equation gives more details on how the energy of the hydrogen atom changes with timein its ground state[2].A study of the energy levels of the hydrogen atom is carried out by solving the time-independentSchrödinger equation in spherical polar coordinates. The solution of the Schrödinger equation isseparated into two ordinary differential equations, called the radial equation and the angular equation[3].Some useful units and symbols relevant to this report are defined as below,ħreduced Planck constant meelectron mass of a particle Vcentral potentialwave functionEtotal energy of an moving electronnprinciple quantum number = 1, 2, 3,…lorbital angular momentum quantum number = 0, 1, 2,…mmagnetic quantum number = 0, , ,…,The Schrödinger EquationThe time-dependent Schrödinger equation is[4]where Ĥ, the Hamiltonian, is the total energy operator of the kinetic and potential energies[5].In three dimensions, the time-independent Schrödinger equation is[6](1.2) whererepresents the Cartesian coordinates (x, y, z).By substituting equation (1.2) into (1.1), it takes the form[7](1.3)This is known as the Schrödinger equation for a particle in three dimensions.The Schrödinger Equation for a Hydrogen AtomA hydrogen atom consists of one electron and one proton. “The electric potential at a distance r froma positive charge e of a hydrogen nucleus is e/r. The potential energy of an electron with charge –e, atthis distance, is V= –e (e/r) = –e2/r.[8]The Schrödinger equation for the hydrogen atom with mass me

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Term
Fall
Professor
DanielNucinkis

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