# Multiplying vector operators $ABC$ in quantum mechanics

Given vector operators as $$\mathbf{A} = (A_{1}, A_{2} ,A_{3})$$

$$\mathbf{B} = (B_{1}, B_{2} ,B_{3})$$

$$\mathbf{C} = (C_{1}, C_{2} ,C_{3})$$

I know that for two vector operators $$$$\mathbf{Q} \mathbf{P} = \sum_{\alpha = 1}^{3} Q_{\alpha} P_{\alpha}$$$$

What is $$\mathbf{A}\mathbf{B}\mathbf{C}$$ in component form?

Edit: I'm looking for the product which appears as the result of quantisation rule which takes the classical expression to the quantum one.

Suppose the classical quantity we have are $$\mathscr A_1(\mathbf{r,p,t})=\mathbf{r(p.r)}$$,

$$\mathscr A_2(\mathbf{r,p,t})=\mathbf{(r.p)r}$$

What will be the quantum operator $$\mathbf A_1$$,$$\mathbf A_2$$ corresponding to $$\mathscr A_1$$ and $$\mathscr A_2$$?

• What kind of product is it? Commented Jan 9, 2023 at 14:24
• "I know that..." - who uses this convention? I would assume any convention of just putting vectors next to each other meant outer product $QP^{\alpha \beta}=Q^\alpha P^\beta$, (maybe symmetrised over the indices) Commented Jan 9, 2023 at 14:31
• Concerning your edit: do you want the operators to be Hermitian, or not? Have you specified your quantization ordering prescription? Where do these symbols come from? Commented Jan 9, 2023 at 17:17
• @CosmasZachos, Yes I want them to be hermitian. I have not found ordering prescription in my text book. These symbols come from studying how we use classical expression to get the relevant quantum operators. Commented Jan 9, 2023 at 17:54
• Isn't it obvious that $\mathbf A_1^\dagger =\mathbf A_2$, so their symmetrization, $\{ \mathbf A_1 ,\mathbf A_2\}$, is self-adjoint? Commented Jan 9, 2023 at 18:41

Your expression is not really defined.

Your formula your have stated is the definition of the dot product.

$$(\vec{A} \cdot \vec{B}) \vec{C}$$
In which case it is the vector $$(A_{1}B_{1} + A_{2}B_{2} + A_{3}B_{3}) \begin{bmatrix} C_{1}\\C_{2}\\C_{3}\end{bmatrix} = \begin{bmatrix}(A_{1}B_{1}C_{1} + A_{2}B_{2}C_{1} + A_{3}B_{3}C_{1}\\ (A_{1}B_{1}C_{2} + A_{2}B_{2}C_{2} + A_{3}B_{3}C_{2}\\A_{1}B_{1}C_{3} + A_{2}B_{2}C_{3} + A_{3}B_{3}C_{3} \end{bmatrix}$$
$$\vec{A} (\vec{B} \cdot \vec{C})= (B_{1}C_{1} + B_{2}C_{2} + B_{3}C_{3}) \begin{bmatrix} A_{1}\\A_{2}\\A_{3}\end{bmatrix}$$
$$\begin{bmatrix}(B_{1}C_{1}A_{1} + B_{2}C_{2}A_{1} + B_{3}B_{3}A_{1}\\ B_{1}C_{1}A_{2} + B_{2}C_{2}A_{2} + B_{3}C_{3}A_{2}\\B_{1}C_{1}A_{3} + B_{2}C_{2}A_{3} + B_{3}C_{3}A_{3} \end{bmatrix}$$