A useful property of the energy eigenstates 
is that
they are orthogonal, the inner product between the pure
states associated with two different energies is always zero,
. Again the proof we give is completely
general and is valid for any Hermitian operator.
If we agree to normalize our eigenstates properly so that 
we then may write compactly
where 
is the Kronecker 
-function, which is
defined as
With this understood, the determination of the constants 
in the
general expansion
is now simple. To extract 
,
we just take the inner product of 
with 
,
With the 
we now may compute
the probabilities of measuring the value 
in an experiment,
. We may also use them to give an explicit form for the
expansion of an arbitrary state 
in terms of the pure energy
states 
,
