# Why can not the total angular momentum of electron align parallel to external field?

1. Mar 29, 2012

### tasnim rahman

An electron has intrinsic angular momentum(spin) and orbital angular momentum, which gives rise to the total angular momentum of the electron, lets 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?

2. Mar 29, 2012

Staff Emeritus
Re: Why can not the total angular momentum of electron align parallel to external fie

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

3. Mar 29, 2012

### M Quack

Re: Why can not the total angular momentum of electron align parallel to external fie

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.

4. Mar 29, 2012

### tasnim rahman

Re: Why can not the total angular momentum of electron align parallel to external fie

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?

Last edited: Mar 29, 2012
5. Mar 29, 2012

### tasnim rahman

Re: Why can not the total angular momentum of electron align parallel to external fie

Quick help anyone?

6. Mar 29, 2012

### M Quack

Re: Why can not the total angular momentum of electron align parallel to external fie

What you say sounds correct to me.

7. Mar 29, 2012

### tasnim rahman

Re: Why can not the total angular momentum of electron align parallel to external fie

M Quack. Thank you very much for the verification.

8. Mar 31, 2012

### rafid haque

Re: Why can not the total angular momentum of electron align parallel to external fie

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)?

9. Apr 1, 2012

### M Quack

Re: Why can not the total angular momentum of electron align parallel to external fie

All of them.

Spin, orbital, total. One electron or many.

10. Apr 1, 2012

### rafid haque

Re: Why can not the total angular momentum of electron align parallel to external fie

Thanks a lot M Quack.