As we have seen, 
is sufficient to compute the distributions
for
position 
and momentum 
through (3), (13) and (14)
and
thus is sufficient to specify or represent the quantum state
of a system where x and p are the only basic
observables.
However, through (12) and (13) we see that
and 
provide equivalent information and so
either
is an equally valid way of representing the state 
. If we
wish, we can, in principle, compute all of our physical quantities using
in the position representation, or using 
in the momentum representation and get equally valid results.
We have already seen an example of this in Parseval's Theorem. We may
compute the Hermitian Inner product between two states
and 
, either in the position or momentum
representation and get the same result
Because this quantity is the same regardless of which representation
we choose, it depends only on the states 
and
and we thus give it a special symbol independent of the
choice of representation to remind us of this fact
When we write 
it is clear what is meant. To
evaluate it, the reader is responsible for picking a convenient
representation of his own choosing and then performing the
appropriate integration in (15).
The Hermitian Inner product integral itself has many interesting mathematical properties and so it is useful to have a notation for it as well. We shall write
and
So that 
just means form the product 
of the two
functions and integrate over their argument. (15) then is written
compactly as
We note that the mathematical structure of the Hermitian Inner product
(16)
as a sum over the product of values taken from two ``lists'' is
very similar to that of the dot product between two vectors
. It is therefore not
surprising that they share many properties. The most important of these
(whose proof we defer to problem set 6) is the Cauchy-Schwartz
inequality.
For vectors we have
For the Hermitian Inner product, you will prove
