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 expand
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 C
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, s
NewpostulateinQuantumMechanics
AgivenspeciesofparticlewillalwaysonlybefoundineitheranStypeoranAtypestate.In
particular,Fermions(particleswithhalfintegerspinsuchaselectrons,protons,positrons,etc.)
area
Nparticlestates
ForanNparticlesystem,therewillbeN!totalpermutationsoftheparticleswhere,iftheyare
identical,giverisetoN!nonequivalentstates.
Thus, we need to construct state for Bosons and for Fermions
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 s
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
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