Next: Spinors Up: Angular Momentum Previous: The Spherical Harmonics
Next: Spinors
Up: Angular
Momentum Previous: The
Spherical Harmonics
The electron possesses an internal angular momentum called the spin
which can assume only the values 
and 
in some arbitrarily chosen direction. In fact, all elementary particles
have a spin degree of freedom. Fermions possess half-integral spin
while bosons have integral spin (including zero). In what follows, we will
develop the theory for spin- 
fermions.
Let the spin operator be 
. If 
is a unit vector (pointing in some arbitrary direction), then the Stern-Gerlach
experiment indicates that the eigenvalue of the operator 
has only two values (which turns out to be 
) i.e.
Without loss of generality, one can choose
to point in the z-direction. (Then 
). The eigenvalue equation then takes the form:
where 
corresponds to the spin operator pointing in the positive z-axis direction
(i.e a spin-up state) and 
corresponds to the spin operator pointing in the negative z-axis direction
(i.e. a spin-down state). Since spin is a physical observable, 
is Hermitian and the states belonging to distinct eigenvalues are orthogonal,
that is:
We further normalise them to unity:
The spin-operators satisfy the AM commutation relations:
where
For spin 
, 
has the eigenvalue 
:
truein
truein
and
by the spin matrices
truein
truein
we obtain for them
truein
where I is the 
unit matrix. Show that the above results can be summarised compactly by
the identity
Hence, given arbitrary vectors 
show that
truein
Problem : 18
Spin state space is a two-dimensional Hilbert space 
, spanned by the orthonormal basis vectors 
. In matrix notation, these basis vectors are defined as:
These basis vectors are the eigenvectors of the
, i.e.
with
This basis is complete:
In general, 
, we have:
(Note that in general 
and 
are complex numbers.) In the
basis, the matrix representation of
is diagonal:
while the x- and y-components of the spin operator are:
(denoted by 
) and 
(denoted by 
) when expanded in the basis
are: 
and 
with Cartesian components: 
Show that if 
then:
Then show that in the basis
, the eigenvectors of 
are:
truein