Moments in QM: When Can I Say JLS^2 Approx Equal JL+S?

[Petar Mali]

$$\vec{J}$$ - mechanical moment
$$\vec{L}$$ - orbital moment
$$\vec{S}$$ - spin moment

$$\vec{J}=\vec{L}+\vec{S}$$

When can I say
$$J^2\approx J(J+1)$$
$$L^2\approx L(L+1)$$
$$S^2\approx S(S+1)$$?

 

[xepma]

For large enough J, L and S :)

 

[Meir Achuz]

In your equations the J on the left is an operator, the j on the right should be an eigenvalue (a number). Then your equations always hold for angular momentum states of pure j,l,s.

 

[Petar Mali]

$$J$$ - eigen-value

I'm asking you because the formula

$$g_J=\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}$$

I think that that formula is in the game not just for very large $$J,L,S$$.

 

[Fredrik]

You always have $$\vec J\thinspace ^2|jm\rangle=j(j+1)|jm\rangle$$, but if the left-hand sides of your equations are eigenvalues too, then the equations are obviously only valid when J,L,S are negligible compared to J²,L²,S².

 

[Meir Achuz]

$$J$$ - eigen-value

I'm asking you because the formula

$$g_J=\frac{J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}$$

I think that that formula is in the game not just for very large $$J,L,S$$.

That formula is true for all eigenstates of J,L,S.