What Are the Eigenvalues of the Operator σ^.P^ in Quantum Mechanics?

[neelakash]

Homework Statement

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)

Homework Equations

The Attempt at a Solution

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.]

 

[malawi_glenn]

Tell us how you did it instead of just showing the answers:)

the eigenvalues
of (σ^3.P^3) are not something that we all here know by heart ;)

 

[neelakash]

Tell us how you did it instead of just showing the answers:)

the eigenvalues
of (σ^.P^) are not something that we all here know by heart ;)

It was merely a guesswork based only upon the fact that Pauli matrices are somehow related to rotation...and as I told you,possibly it was not a problem from my course.I am trying this to see if any clever trick can solve it.

And we can see none of (a) (c) and (d) give a hint of rotation in a complex plane.This was my basis of guesswork.

 

[malawi_glenn]

Well if you call "guesswork" a trick then ok ;)

I would construct a matrix eq, and find the eigen values for that matrix.

i.e you get the matrix:

Q = \vec{p}\cdot \vec{\sigma }= p_x\sigma _x + p_y\sigma _y + p_z\sigma _z

Find the matrix-representations for the pauli matrices, evaluate the total matrix Q, and then find Q's eigenvalues. I would do this.

 

[neelakash]

Yes,I know that.But when you write Q, you need to know matrix representations of momentum operator;then multiply with each Pauli matrix.Then add...and then you make your task of solving an eigenvalue problem.That is time consuming and may be difficult in an MCQ exam hall.

Therefore,I was searching for a nice trick.My guesswork may not be a right one and I do not know if this answer is at all correct.