Derivation of Projection Operators

Let us assume that we have an orthonormal set of /, functions (p 2,. . . , j l which form a basis for the zth irreducible representation (of dimension /, ) of a group of order h. For any operator, R, in the group we may then, by definition, write 

This equation is then multiplied by [r(/ ) v], and each side summed over all operations in the group, giving 

We note that the (pi s are functions independent of R hence the right side of 6.2-2 may be rewritten as 

we have a series of /, terms, each of which is a multiplied by a coefficient each coefficient is itself expressed as a sum of products over the operations R in the group. These coefficients, however, are governed by the 

Thus all except have coefficients of zero, and only when / = j and / = / can even the term in / survive. Thus 6.2-2 simplifies to 

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