# $L_y$ in terms of $L_+$ and $L_-$? [closed]

On the 5th slide of http://web.ift.uib.no/~lipniack/phys201_v08/angularmomentum.pdf, they say that: $$L_y=\frac{i}{2}(L_+-L-)$$ And this is not the only place I have seen this (http://quantummechanics.ucsd.edu/ph130a/130_notes/node247.html for instance use this implicitly). However, if I start with $L_{\pm}=L_x\pm i L_y$ and then do $L_+-L_-$ I get: $$L_+-L_-=2iL_y$$ $$L_y=\frac{1}{2i}(L_+-L-)$$ So where do they get the formula:

$$L_y=\frac{i}{2}(L_+-L-)$$ from?

## closed as off-topic by ACuriousMind♦, HDE 226868, Kyle Kanos, Qmechanic♦Aug 16 '15 at 15:31

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You can verify this by noticing that (Using their $L_y$): $$L_+=L_x+iL_y=\frac{1}{2}(L_++L_-)+i\frac{i}{2}(L_+-L_-)=L_-$$ Which is obviously wrong. (Do note however that their matrices are correct)
• How can their matrices be correct since if you do: $$L_y|l=1,m_l=1>=\frac{1}{2i}(L_+-L_-)|1,1>$$ $$=\frac{-\hbar}{\sqrt{2}i}|1,0>$$ and not $$=\frac{\hbar}{\sqrt{2}i}|1,0>$$ as they indicate. – Quantum spaghettification Aug 16 '15 at 9:23