Total spin
If the Hamiltonian is independent of spin, then it is clear that the total spin of an
particle system

will be a constant of the motion, but so will the individual spins,
of the individual
particles.
What happens, however, when the Hamiltonian
Total orbital angular momentum
In the hydrogen atom or any system with a spherically symmetric potential
have learned that angular momentum
, we
is conserved. The Hamiltonian will be of the form
and will satisfy
so that is a constant of the motion.
This i
Minor and Major Components Solution
so that in the limit
, and
,
which describes particles moving backward in times. Thus, the interpretation is that
the negative energy solutions correspond to antiparticles, the the components,
and
of
correspond to the
Lowering Operators Spin States
The action of
on
is
However, the second term on the right vanishes because each involves a raising
operator acting on a state
the state. Thus,
Thus, the eigenvalue of
with a corresponding
total spin1 state
for either the fi
Hamiltonian Momentum Form
where
and
Although this is the specific form of the potential for this example, what we will show
will be general for any potential that depends only on
Now, the individual angular momenta
.
are no longer conserved, i.e.,
To see
Group theoretics
The spin1/2 rotation group has a special name. It is known as SU(2). SU(2) is the
group of 2
determinant. The representation of such a matrix as
2 unitary matrices with unit
shows that there are an infinite number of such matrices, since
Dirac equation in matrix form:
or
which yields two equations
From the second equation:
Note, one could also solve the first for
and obtain
Using the first of these, then a single equation for
However,
can be obtained
Hence, we have the condition
Since
, t
Solution of the Dirac equation for a free
particle
The Dirac Hamiltonian takes the form
where
Using
, in the coordinate basis, the Dirac equation for a free particle reads
Since the operator on the left side is a 4 4 matrix, the wave function
actually a f
Continuum and Gaps (Dirac Equations)
There is a continuum for
(turquoise) and for
There is also a gap between
We will show that for
and
, an appropriate solution is to take
If this is the case, then
However,
so that
so that the full solution
.
is
(periwin
Basis Vectors
The last state is
and is obtained by recognizing that it must be composed of
those states that have opposite values of
arbitrary coefficients:
and
. So we include these states with
The coefficients and are then determined by the conditions t
angular momentum addition
Given two spin1/2 angular momenta,
and
, we define
The problem is to find the eigenstates of the total total spin operators
and
and
identify the allowed total spin states.
In order to solve this problem, we recognize that the ei