Why Does Electron Angular Momentum Not Align with External Field?

[tasnim rahman]

An electron has intrinsic angular momentum(spin) and orbital angular momentum, which gives rise to the total angular momentum of the electron, let's call it pj. When the electron is placed in an external magnetic field, the pj vector precesses around the magnetic field in one of two states(with or against the field). My question is why doesn't the pj vector align itself parallel to the direction of the external field? Is it because the pj vector can only have particular components in the the direction of the axis of the field, and none of the allowed components are equal to the magnitude of pj?

 

[Vanadium 50]

The eigenvalues of L² are l(l+1), and the eigenvalues of L_z are -m...+m. So you can't simultaneously satisfy those conditions and have all of the angular momentum pointing in the z-direction.

 

[M Quack]

In QM, you can only ever determine the magnitude and the projection of the angular momentum vector on one axis (such as an external magnetic field). These correspond to the quantum numbers L and m as Vanadium pointed out.

This is because the 3 operators L_x, L_y and L_z do not commute.

 

[tasnim rahman]

Thanks M Quack and Vanadium 50. But I am sorry, I am not really familiar with the mathematics involved. But what I believe I understood, is that the projection of the total angular momentum vector on one axis(z-direction:here assumed to be the direction of the magnetic field) can have only particular magnitudes, as set by quantum mechanics. And that the magnitude of the projections, can not be equal to the magnitude of the total angular momentum vector. Right?

 

[tasnim rahman]

Quick help anyone?

 

[M Quack]

What you say sounds correct to me.

 

[tasnim rahman]

M Quack. Thank you very much for the verification.

 

[rafid haque]

Thanks M Quack and Vanadium 50. But I am sorry, I am not really familiar with the mathematics involved. But what I believe I understood, is that the projection of the total angular momentum vector on one axis(z-direction:here assumed to be the direction of the magnetic field) can have only particular magnitudes, as set by quantum mechanics. And that the magnitude of the projections, can not be equal to the magnitude of the total angular momentum vector. Right?

Does this work only for the orbital angular momentum, or the total angular momentum (as in the angular momentum by vector addition of the orbital and spin momentum)?

 

[M Quack]

All of them.

Spin, orbital, total. One electron or many.

 

[rafid haque]

Thanks a lot M Quack.