# Uncertainty of angular momentum

Is it correct to write for the uncertainty $$\Delta \vec L=\Delta L_x+\Delta L_y+\Delta L_z$$ meaning $L$ is the angular momentum and $L_x,L_y,L_z$ the components. I couldn't find an answer in my textbooks.

• Left hand side is a vector triplet. Right hand side is only one number? – Cosmas Zachos Apr 17 '18 at 19:08
• That was my first thought also...but it was in the question of the exercise, find the uncertainty for $\vec L$..strange..i have to talk to my teacher. – Anastasios Apr 17 '18 at 20:35
• Κύριε Ζάχο, τιμή μου που αφιερώσατε το χρόνο σας σε μια τέτοια ερώτηση..Σας ευχαριστώ! – Anastasios Apr 17 '18 at 20:44

It doesn't really make sense to talk about the vector $\vec L$ in QM because we know from the uncertainty principal that we can only know about one component of the momentum as well as the magnitude of the momentum ($L^2$). Since we can simultaneously diagonalize a component (say $L_z$) and the magnitude, we can talk about the uncertainty in each by calculating the expectation values as we usually would for an operator.
If you want to talk about the uncertainty of $L$, it makes more sense to talk about the uncertainty in the scalar quantity $\sqrt{L^2}$, the operator that represents the magnitude of the momentum.