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Operator-valued vectors in quantum mechanics

Let \(\boldsymbol{e}_1\), \(\boldsymbol{e}_2\), \(\boldsymbol{e}_3\) be a right-handed triplet of mutually orthogonal unit vectors in a three-dimensional Euclidean space. An operator-valued vector \(\boldsymbol{A}\) is a "vector" whose components along \(\boldsymbol{e}_1\), \(\boldsymbol{e}_2\), \(\boldsymbol{e}_3\) are operators \(A_1\), \(A_2\), \(A_3\), respectively: \[\boldsymbol{A} = \boldsymbol{e}_1 A_1 + \boldsymbol{e}_2 A_2 + \boldsymbol{e}_3 A_3 \,.\] Examples include the position and momentum operators in quantum mechanics: \[\boldsymbol{x} = \boldsymbol{e}_1 x_1 + \boldsymbol{e}_2 x_2 + \boldsymbol{e}_3 x_3 \,,\] \[\begin{align*} \boldsymbol{p} &= \boldsymbol{e}_1 p_1 + \boldsymbol{e}_2 p_2 + \boldsymbol{e}_3 p_3 \\[0.2cm] &= \boldsymbol{e}_1 \frac{\hbar}{i} \frac{\partial}{\partial x_1} + \boldsymbol{e}_2 \frac{\hbar}{i} \frac{\partial}{\partial x_2} + \boldsymbol{e}_3 \frac{\hbar}{i} \frac{\partial}{\partial x_3} \,. \end{align*}\] Let \(\bol