Step on Weinberg's QM Book (pp. 154-155)

[carlosbgois]

Hello all!

On the problem of taking elements of different (degenerated-)state vectors that do not vanish on the perturbation matrix, Weinberg uses the following approach, when dealing with the Zeeman effect:

We can also avoid the problem without introducing new state vectors in place of $\Psi_{njl}^m$ by simply using a coordinate system in which the 3-axis is in the direction of B.

In this way, he goes from the first to the second equation shown as attachments.

My main source of confusion arises by the fact that I can't see how can you align a (single-direction?) B vector with all the three (supposedly not parallel) axis of the coordinate system used. But I am surely completely missing the point in here.

Can someone help me?
Thanks for your time (:

 

[vanhees71]

Weinberg just puts the $z$-axis in direction of the magnetic field and chooses the joint eigenbasis of the undisturbed hamiltonian, $\vec{L}^2$, and $l_z$.

Note that due to the rotationinvariance of the undisturbed hamiltonian you are allowed to choose any direction as the $z$-axis. For the perturbation it's just convenient to take it in the direction of $\vec{B}$.