How do we treat vectors in classical mechanics and quantum mechanics? How vectors are treated differently in classical mechanics and quantum mechanics?
Actually I was reading vector atom model. This model is semi classical approach in which l and s vectors are treated classical while their magnitudes quantum mechanically. So I am not able to understand how vectors are treated classical or quantum mechanically.
Hope to get an answer.
 A: Let's look at angular momentum. In Quantum Mechanics, an angular momentum is a triplet $L=(L_x, L_y, L_z)$ of Hermitian linear operators acting on the vector space $H$ of quantum state (often called "kets"). This is the first place where confusion can occur: the kets form a vector space (a Hilbert space to be precise), in the mathematically abstract sense of the expression, and the angular momentum can also be seen as a vector, but in a different space, that of the operators acting on $H$.
Then textbook Quantum Mechanics tells us that $L^2$ and $L_z$ commutes and have a common base of eigenvectors $|l,m\rangle$ such that $L^2|l,m\rangle=l(l+1)\hbar^2|l,m\rangle$ and $L_z|l,m\rangle=m\hbar|l,m\rangle$. Let's move now to the vector atom model formalism. As you wrote, it treats $L$ as a 3-dimensional vector, the usual kind of vector, from high school geometry. But then it requires this vector to satisfy the conditions $L^2 = l(l+1)\hbar^2$ and $L_z=m\hbar$. This is very in a way similar to the way Bohr model treats the position $r=(x,y,z)$ of the electron: this is a normal 3-dimension vector but its possible values are those that would be predicted by QM which treats the position as a triplet of operators acting on $H$.
