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Single Electron Atom Wave Functions

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Schrödinger Equation

The Schrödinger equation describing an electron in a central field is of the form


In the scaled coordinates that we are using, [Graphics:../Images/hwavefunctions_gr_31.gif] and [Graphics:../Images/hwavefunctions_gr_32.gif], where [Graphics:../Images/hwavefunctions_gr_33.gif] is the nuclear charge. The Schrödinger equation for an attractive Coulomb potential, [Graphics:../Images/hwavefunctions_gr_34.gif], is  


The standard add-on package Calculus`VectorAnalysis` has commands for representing the various vector differential operators in a number of coordinate systems. First load the package.


Now define a Schrödinger operator,


Set the default coordinate system for the functions from Calculus`VectorAnalysis` to be spherical polar coordinates:


Now check to see if [Graphics:../Images/hwavefunctions_gr_39.gif] is an eigenfunction of the Schrödinger operator.


This of course agrees with the known eigenvalues for the energy of this system: [Graphics:../Images/hwavefunctions_gr_42.gif].

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