Properties of the antisymmetriser

The properties of the antisymmetriser that are useful in deriving the Hartree-Fock equations are:

To proof eq63, we substitute eq62 in to give .



Show that


When acts on , we have  permutated terms. then acts on each of these terms, resulting in sets of permutated terms, with the terms in one set being identical to the terms in another set. An illustration with is given below:

where the top row is the result of , and each column is the outcome of  acting on each term in the top row.

In other words,


Therefore, , which when substituted with eq62 gives eq63.

Eq64 states that  is Hermitian. To prove that , we note that

Finally, eq65 states that commutes with . This is because

 and correspond to the same state with energy because the way we label identical particles cannot affect the state of the system.


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