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, and , where is the nuclear charge. The Schrödinger equation for an attractive Coulomb potential, , 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 is an eigenfunction of the Schrödinger operator.

This of course agrees with the known eigenvalues for the energy of this system: .

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