Antisymmetrization of Electronic Wavefunctions

For the wavefunction itself, this implies that interchange of a pair of electron coordinates can change the wavefunction only by a factor since = 1 for any real number cp. 

This same reasoning applies to a wavefunction of any other kind of indisfinguishable particle, and it turns out that the value of cp depends on fhe infrinsic spin of fhe particles. As will be verified later, for electrons and other half-integer spin particles, cp = t. (For other than half-integer spin particles, such as those with spin = 0,1,2,..cp = 0.) Since e = -1, all electronic wavefunctions must satisfy the following condition  

This is termed an antisymmetrization requirement an electronic wavefunction is antisymmetric (changes sign) with respect to interchange of any pair of electron position coordinates. 

To impose Equation 10.42 on some electronic wavefunction, a special operator, called the antisymmetrizer, can be used. Let us develop the form of this operator, assuming that we have at hand a normalized electronic wavefxmction, O, for some system, and that O has the form of a product of independent, orthonormal functions, u r), of the electron coordinates (e.g., orbitals). 

The interchange of a pair of position coordinates is a well-defined mathematical operation, as we have in Equations 10.41 and 10.42, and we can define an operator that performs such an interchange. Let be an operator that interchanges the coordinates of some electron, x, with the coordinates of some electron, j. That is, 

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