Two commuting observables can be measured simultaneously, i.e. the measurement of one does not cause loss of information obtained in the measurement of the other. If we measure a complete set of commuting observables (C.S.C.O.), then the state of the system after the measurement is one element of an unique eigenbasis. The results of the measurement specify the state completely.
The expression for the mean value of an observable
A in the normalized state |y> is <A>=<y|A|y>.
If |y>
is not normalized then 
.
The root mean square deviation DA characterizes the dispersion of the measurement around <A>.
Let A and B be two observables (Hermitian operators). In any state of the
system 
An operator A is Hermitian if A=AT. A Hermitian operator satisfies <y|A|f>=<f|A|y>*. An operator A is anti Hermitian if A=-AT.
An operator U is unitary if UUT=UTU=I.
An unitary operator preserves the norm. 
Two orthonormal, continuously labeled bases for the vector space L2x
of square integrable functions are 
, and
.
In three dimensions the bases are 
, and
.
Let us associate the kets |p0> with 
and |r0> with 
.
Then 
denotes the components of |y> in the {|p0>} basis and 
denotes the components of |y> in the {|r0>} basis.
In the {|r>} representation the operator Px
coincides with the differential operator 
.
Similarly, 
. In the {|p>}
representation 
.
The evolution operator is a unitary
operator defined through 
is the infinitesimal evolution operator.
If H does not explicitly depend on time, then 
.