## Posts

Showing posts with the label Mathematical tools

### 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