Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.53 Integral constraints
This note verifies the mentioned constraints on the Coulomb and
exchange integrals.
To verify that , just check their
definitions.
The fact that
is real and positive is selfevident, since it is an integral of a
real and positive function.
The fact that
is real can be seen by taking complex conjugate, and then noting that
the names of the integration variables do not make a difference, so
you can swap them.
The same name swap shows that and are symmetric
matrices; and
.
That is positive is a bit trickier; write it as
with . The part within
parentheses is just the potential of a distribution of
charges with density . Sure, may be complex but
that merely means that the potential is too. The electric field is
minus the gradient of the potential,
, and according to Maxwell’s equation, the
divergence of the electric field is the charge density divided by
:
. So
and the integral is
and integration by parts shows it is positive. Or zero, if
is zero wherever is not, and vice versa.
To show that , note that
is nonnegative, for the same reasons as but with
replacing
. If you multiply out the inner product,
you get that is nonnegative, so
.