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$$\lvert \langle M \lvert \hat{L}_x+i\hat{L}_y \rvert M-1 \rangle \rvert ^2=c$$.

where ##c## is any old real number. If I undid the modulus square to find:

$$ \langle M \lvert \hat{L}_x+i\hat{L}_y \rvert M-1 \rangle=\pm \sqrt{c} $$ Would I not have to consider ##c## as being negative, positive as well as imaginary negative positive? So:

$$ \langle M \lvert \hat{L}_x+i\hat{L}_y \rvert M-1 \rangle=\pm i \sqrt{c} $$ as well.

I am trying to get to:

$$ \langle M \lvert \hat{L}_x \rvert M-1 \rangle=\frac{1}{2} \sqrt{c} $$

and:

$$ \langle M \lvert \hat{L}_y \rvert M-1 \rangle=-\frac{i}{2} \sqrt{c} $$

From Landau and Lifshitz QM 3ed page 89, and I am not following. Any tips will be appreciated.

Thanks,

KQ6UP