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In quantum mechanics, is momentum a vector quantity?

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  • $\begingroup$ Do you have any reason to believe this differs between classical and quantum mechanics? What kind of research have you done before asking this question? $\endgroup$ – ACuriousMind Sep 25 '17 at 10:14
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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,

$$[p_i,J_j] = i\hbar \sum_k\epsilon_{ijk}p_k.$$

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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.

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It is a vector operator but when you take its expectation value it becomes a usual vector, that is transforming like a vector.

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