Eigenvectors of

As an example, lets take the
direction to be in the
plane, between the positive
and
axes,
30 degrees from the x axis.
The unit vector is then
.
We may simply calculate the matrix
.

We expect the eigenvalues to be as for all axes.

Factoring out the , the equation for the eigenvectors is.

For the positive eigenvalue, we have , giving the eigenvector . For the negative eigenvalue, we have , giving the eigenvector . Of course each of these could be multiplied by an arbitrary phase factor.

There is an alternate way to solve the problem using rotation matrices. We take the states and rotate the axes so that the axis is where the axis was. We must think carefully about exacty what rotation to do. Clearly we need a rotation about the axis. Thinking about the signs carefully, we see that a rotation of -60 degrees moves the axis to the old axis.

This gives the same answer. By using the rotation operator, the phase of the eigenvectors is consistent with the choice made for . For most problems, this is not important but it is for some.

Jim Branson 2013-04-22