Home

Staff
     
Consulting
               
Mathematica Tools

Customer Quotes
        
Contact

Older Mathematica Items

 

 

 

 

 

Scientific Arts

Single Electron Atom Wave Functions

Page 1 | Page 2 | Page 3


Schrödinger Equation

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

[Graphics:../Images/hwavefunctions_gr_30.gif]

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  

[Graphics:../Images/hwavefunctions_gr_35.gif]

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.

[Graphics:../Images/hwavefunctions_gr_36.gif]

Now define a Schrödinger operator,

[Graphics:../Images/hwavefunctions_gr_37.gif]

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

[Graphics:../Images/hwavefunctions_gr_38.gif]

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

[Graphics:../Images/hwavefunctions_gr_40.gif]
[Graphics:../Images/hwavefunctions_gr_41.gif]

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

<<Previous Next>>


For further information on our services send email to info@scientificarts.com .
Contents of this web site Copyright © 1999-2011, Scientific Arts, LLC.

s