is a measure of whether or not
two operators commute, and plays a very important role in quantum mechanics

Eigenvalues and Eigenkets (Eigenvectors)
If Q
1> = a_{1 }1>
 i.e. the operator acts on
1> and returns 1> multiplied by a complex number a_{1 }
1> is called an eigenket
of the operator Q and a_{1} is called
the associated eigenvalue
A ket is often labelled
by its eigenvalues, as Q a_{1}>
= a_{1} a_{1}>
The completeness
postulate then says that an arbitrary
ket y > may be expanded in terms of the eigenkets
of Q , as
y
> = c_{1} a_{1}> + c_{2} a_{2}> + c_{3}
a_{3}> + ........
where the c_{i} are called
the expansion coefficients of y > , and as
noted before, c_{1}^{2} = c_{1}*c_{1}
(with proper normalization) gives the probability that if a measurement
of Q is made, the result will be a_{1}

Dual (Bra) Space and Scalar Products
To each ket
A>, there corresponds a
dual or adjoint quantity called by Dirac a bra;
it
is not a ket rather it exists in a totally different space. The generalized
scalar product is defined in analogy with the ordinary scalar product that
you are familiar with as a combination of a bra and a ket
to form a bracket (Diracís pun the reason for his nomenclature)
thegeneralized scalar product is simply a complex number associated
with a pair of kets.
This notion is familiar for the
case of a real vector space, where a real number (the scalar product denoted
(x,y) or x· y) was associated with every
pair of vectors x and y and was required to have certain properties (more
on this below)
You are also probably familiar
with the generalization of this to complex vector spaces, where a single
complex number (x,y) was associated with each pair of complex vectors
The scalar product (as part of
its definition) was required to have the following properties:
(x,y) = (y,x)*
(x,x) > 0 unless x=0
(ax,y) = a* (x,y)
(x,by) = b (x,y)
according to the mathematicians,
this
defines the scalar product
although we physicists are more familiar with evaluating
it in a representation(more
about this later) as
(x,y) º
S x_{i}*y_{i}
_{}
For the generalized scalar product
(bracket), we will assume following the mathematicians that it has
the same abstract properties as the more familiar scalar product
defined above; here too, we will later find out how to evaluate the scalar
product in a representation.

A Bit More About Operators (some
definitions and a theorem):
The Adjoint
Operator: To each operator Q , we define a corresponding
adjoint operator Q ^{+} so that the
bra that corresponds to Q f> is <fQ^{+}
note that here Q ^{+} acts to the left
on the bra <f
Hermitian
(Self Adjoint) Operator:
An operator Q
is called Hermitian, or selfadjoint, if it is its ownadjoint
operator; i.e. if
<gQ
f> = <gQ f>
Every quantum mechanical operator
representing an observable must be Hermitian. The reason for
this is the following famous theorem: The
eigenvalues of a Hermitian Operator must be real.

Linear Operators
In addition, the operators representing
observables must be linear;
this means that
Q
[ a 1> + b 2> ] = a Q 1> + b Q
2>

Orthogonality
Definition: we say that
1> and 2> are orthogonal
if <12> = 0
Another famous theorem: eigenkets
of the same Hermitian Operator having different eigenvalues are automaticallyorthogonal)
(in general, if two kets have
the same eignevalue, while they are not automatically orthogonal,
they may be made orthogonal, provided they are linearly independent)

Normalized Kets
a ket k> is called normalizedif
<kk>
= 1;
two different kets are called
orthonormalif
<kl> = d_{kl}
_{}

Proof of the Two Famous Theorems
Regarding Hermitian Operators:
Let q
be a Hermitian Operator and let 1> and 2> be two eigenfunctions of q
with
respective eigenvalues a_{1} and a_{2}
q
1> = a_{1} 1>
q
2> = a_{2} 2>
Thus it follows that
<2q
1> = a_{1} <21> and
<1q
2> = a_{2} <12> or
<2q
1> = a_{2}*<21>
and, therefore (equating the two
expressions for <2q 1>),
(a_{1}  a_{2}*)
<21> = 0
By setting 2> = 1> the first theorem
follows, and
by assuming a_{1} ?
a_{2} the second theorem results.
Finally, we show that if q
i> = a_{i} i>,
<iq^{+}
= a_{i} <i
one may expand <jq^{+}
= S c_{ja}<a
 and compute
<jq^{+}i>
= S c_{ja}<a
i> = c_{ji}
= a_{i} <ji> =
a_{i} d _{ji}
establishing the desired result.

Expansion of State Functions Again...
Closure
the completeness postulate guarantees
that if Q represents an observable, and
Q
k> = k k>
then an arbitrary state function
y > may be expanded as
y
> = S c_{k} k>
if the eigenkets are orthonormal,
then
<ly
> = S c_{k}
<lk> = S c_{k} d_{lk}
= c_{l}
i.e. the expansion coefficient c_{l}
is the scalar product of l> and the state function y
>
This results in an important
result called closure:
since y
> = S c_{k} k> and c_{k} =
<ky > we have
y
> = S <ky > k>
= S k><ky >
and we can recognize the operator
S
k><k
as the identity
operator, i.e.
I = S
k><k

Probability of Obtaining a Given
Result
We recall that if Q
is measured for a particle described by the state function y
>, the only results possible are the eigenvalues of Q
, k; the probability of observing a particular k is given by
c_{k}^{2}
= <ky >^{2}
assuming that y
> is normalized so that <y y
> = 1. This principle of expressing y
> as a superposition of eigenstates, and the statistical interpretation
thereof is regarded by Dirac as perhaps the fundamental principle of Quantum
Mechanics.

Projection Operators
It is convenient to define yet
one more operator, the projection operator,
P_{k} which operates on y > and "projects
out" or "selects" only that part of y > which
has the eigenvalue k; thus
P_{k} º
k> <k
so that P_{k} y
> = c_{k} k>
It is clear that S
P_{k} = I

Expectation Values:
For a system described by the
State Function y >, if the observable Q
is measured many times, the result will be on the average <y
 Q y >....
the socalled expectation value of Q
; this is exactly what one would expect, since expanding y
> in terms of the eigenstates of Q gives (using
the orthonormality of the eigenstates):
S
c_{k}*c_{k} a_{k}
_{}
the weighted average obtained by
assuming that c_{k}*c_{k} gives the probability of measuring
a_{k}

Generalized Uncertainty Principle:

Whenever the operators representing
two observable A and B do not commute, it is not possible to
exactly specify both of the observables; rather, there is an Uncertainty
Principle which limits the accuracy of the simultaneous measurements,
D A and D B according
to the condition
D
A^{2} D B^{2} => < i
[A , B] >^{2} / 4
see Townsend, sec. 3.5 (p.78)
for the proof of this very important theorem

A Corrolary :

When [A,B] = 0
(i.e. A and B commute) and there is no required uncertainty, one can prove
that one can find kets which are simultaneously
eigenkets of both A and B:
i.e. if
[A,B] = 0 then there
exists a ket a,b> such that
Aa,b> = a a,b> and
Ba,b> = b a,b>