The properties of the antisymmetriser that are useful in deriving the Hartree-Fock equations are:
To proof eq63, we substitute eq62 in to give
.
Question
Show that
Answer
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.
