Worked Spin Problem
Consider a unit vector
, measured an angle
from the positive
axis in the
plane and angle
from the positive
axis.
Let the components of the spin vector
along
be
.
a.) Solve the resulting eigenvalue equation. (
)
Which, in matrix form (using the definitions of
) looks like:
And define
.
So that:
And the eigenvalue equation becomes:
Which has nontrivial solutions
For
Which means that:
And, for
So that
b.) Verify that the two resulting eigenvectors are orthogonal.
Show that
Where, from above,
So that,
As expected.
c.) Show that these vectors satisfy the closure relation.
This amounts to showing that
So:
And thus,
While
And
So, clearly, we can get: