In quantum mechanics is momentum a vector or scalar? In quantum mechanics, is momentum a vector quantity?
 A: In quantum mechanics, momentum is a vector operator, but there is a subtlety. If we have a state $|\alpha \rangle$ rotated to $|\alpha' \rangle = \mathcal{D}(R)|\alpha\rangle$, for momentum to be a vector operator we must have that,
$$\langle \alpha | p_i |\alpha\rangle \to \langle \alpha'|p_i|\alpha'\rangle = \sum_j R_{ij}\langle \alpha | p_j |\alpha\rangle$$
under the rotation; in words, the expectation value transforms as an ordinary vector by a rotation matrix $R_{ij}$. It can be shown by considering an infinitesimal rotation,
$$\mathcal{D}(R)=1-\frac{i\epsilon \mathbf{J}\cdot \hat{\mathbf{n}}}{\hbar} $$
that a totally equivalent definition of a vector operator is that it satisfies the commutation relations, 
$$[p_i,J_j] = i\hbar \sum_k\epsilon_{ijk}p_k.$$
A: In QM momentum is an operator.
It is generally a vector operator of 3 or more dimensions depending on exactly what theoretical framework you are using.
A: It is a vector operator but when you take its expectation value it becomes a usual vector, that is transforming like a vector.
