If P^ is the momentum operator, and σ^ are the three Pauli spin matrices, the eigenvalues of (σ^.P^) are (a) (p_x) and (p_z) (b) (p_x)±i(p_y) (c) ± |p| (d) ± (p_x + p_y +p_z) Pauli matrices are related to rotation.So, (b) looks correct to me. [I am a Bachelor level student and this problem belongs to Masters level.I am trying to do this to see if any tricky method, known to me can be used to solve this.]
HMmmm (b) is not correct as eigenvalues of Hermitian matrix are always real. (c) or (d) is correct I will choose (c) as we cannot possibly multiply pauli matrix (2x2) with a (3x1) momentum operator matrix [I say 3x1 matrix as three components are specified]
I think you have to be careful. I think the equation you have is \(\vec{\sigma}\cdot \vec{p}\) where \(\vec{\sigma} = \left(\sigma_x,\sigma_y,\sigma_z\right)\) is a three vector. So the operator looks like \(\left(\begin{array}{cc}p_z&p_x+ip_y \\ p_x - ip_y&-p_z\end{array}\right)\) Check the minus signs because I always screw up the \(\sigma_y\). Now diagonalize this matrix and you'll find the eigenvalues.
Thanks Ben , The eigen values are C rt, I think for the particular problem it is enough to find the eigen value by just looking at the trace and determinant
The "shortcut" to this problem is to recognize that the eigenvalues of a Pauli operator are always plus or minus 1, regardless of its direction. In other words, the Pauli vector operator dotted into a unit vector has eigenvalues plus or minus 1. Write momentum as its magnitude times a unit vector, and the answer becomes obvious.
