GroupTheory`
GroupTheory`
GTProjectionOperator
GTProjectionOperator[group,ireducible representation,m,n,function,arguments] gives the part of a given function with arguments, which transforms like the m-th row and the n-th column of an irreducible representation.
Details and Options
- The projection operator
is given by 
, where 
is the dimension of the irreducible representation 
with matrix elements 
and 
is the order of the group. The application of the projection operator 
to an arbitrary function will result in a function that transforms like the 
row of the irreducible representation 
. - Typically arguments is a list of Cartesian coordinates:
- arguments = {x,y,z}
- The following option can be given:
-
GOFast GOFastValue Skips the input validation - See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 5.9.
Examples
open allclose allBasic Examples (1)
The example is based on the point group 
. The projection operators concerning to the first row of the irreducible representation 
are applied to a function. First, the point group and character table are installed and the irreducible representation matrices are calculated.
The function 
will be analysed.
The application of the projection operator, related to the first row and the first column of the irreducible representation 
can be calculated by:
The same formalism holds for the second row and the first column:
