Unitary Transformation
Start with the state
. Expanding gives
Only one term gives
, which is clearly
The Clebsch-Gordan coefficients, by special case
and
is just 1, so
as expected.
The state
is expanded as
This time, since
and
However, since
, two terms c
Triple States Matrix Equations
The matrix
can easily be shown to be unitary. Note that the elements of the matrix can be
computed from the following overlaps:
These are examples of what are known as Clebsch-Gordan coefficients. In the next
section, we wil
Special Cases General Formulas
where
, and the summation runs over all values for which all of the
factorial arguments are greater than or equal to 0.
This formula is rather cumbersome to work with, so it is useful to deduce some
special cases. These are
Nparticlestates
ForanNparticlesystem,therewillbeN!totalpermutationsoftheparticleswhere,iftheyare
identical,giverisetoN!nonequivalentstates.
Thus, we need to construct state for Bosons and for Fermions which are simultaneous
eigenstatesofallparticlepermut
Let us first examine some of the properties of the coefficients that are useful in
constructing the unitary transformation:
1.
The Clebsch-Gordon coefficients are real:
2.
Orthogonality:
This can be seen by recognizing that
so that if we insert an identit
The number of basis vectors is generally given by
number is equal to
Also, sometimes one sees the so called `
coefficients. This are denoted as
, and that this
' symbols used instead of Clebsch-Gordan
and are related to the Clebsch-Gordan coefficients by
Arbitrary Amounts Transformations
The problem of adding two arbitrary angular moment
and
amounts to finding a
unitary transformation from the set of basis vectors defined by the tensor products of
the individual eigenstates of
and
and
and
and
to the eigen