Why the magnetic moment is zero for occupied levels?

[hokhani]

I can not get convinced why for energy level, filled with an even number of electrons, the net magnetic moment is zero!
If we have a level with the quantum number l, we have two electrons with opposite spins in it. Also magnetic moment is proportional to J_total where J_t= J_1+J_2(1and 2 refer to electrons). Moreover J=L+S. If magnetic moment were to be zero, the J_t must be zero. How can we show that?

 

[Bill_K]

Isn't the correct statement that the magnetic moment is zero only for an atom whose subshells are completely filled. That is, electrons occupy all m values for a given L, so that Σm = 0

 

[hokhani]

Isn't the correct statement that the magnetic moment is zero only for an atom whose subshells are completely filled. That is, electrons occupy all m values for a given L, so that Σm = 0

OK, But if so(Σm = 0) we can only deduce that the z-component of L is zero not the L itself!

 

[Bill_K]

OK, But if so(Σm = 0) we can only deduce that the z-component of L is zero not the L itself!

Along every z-axis. A closed subshell is spherically symmetric, i.e. its total L is zero.

 

[hokhani]

Along every z-axis. A closed subshell is spherically symmetric, i.e. its total L is zero.

Thank you very much. It just remains another question:
By the above statement, do you mean that in all the spherically symmetric systems, If we have a z-component, then we have x and y component? In this case, the uncertainty principle would be invalid!

 

[wotanub]

Thank you very much. It just remains another question:
By the above statement, do you mean that in all the spherically symmetric systems, If we have a z-component, then we have x and y component? In this case, the uncertainty principle would be invalid!

Well if you have 0 total angular momentum which way is the J vector pointing? If you can tell which way it's pointing, how do you know it's projection on the (x,y,z) basis?

Granted the quantum angular momentum vector really doesn't "point" in a specific direction when if is non-zero, but you see my point.

 

[Bill_K]

By the above statement, do you mean that in all the spherically symmetric systems, If we have a z-component, then we have x and y component? In this case, the uncertainty principle would be invalid!

No, of course not, that is not at all what I said. A closed subshell contains 2(2l+1) electrons, an electron pair occupying every possible value of m. The wavefunction is a Slater determinant, and is spherically symmetric, i.e. invariant under rotations. Not just the potential is spherically symmetric, the total wavefunction is spherically symmetric. This means the total L value is zero.

EDIT: Are you thinking that [L_x, L_y] = iħ L_z is an example of the uncertainty principle? The uncertainty principle gives the commutator of variables that are canonical conjugates, which L_x and L_y are not.