Phy5646/Grp3SpinProb

From PhyWiki
Jump to navigation Jump to search

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: