# Which properties does tensorial product have with respect to scalar product?

Are the associative and distributive properties preserved?

In Cohen-Tannoudji's Mécanique quantique vol. I, the scalar product in the $$\epsilon=\epsilon_1 \otimes\epsilon_2$$ space is defined as follows:

$$\gamma.$$ The scalar product in $$\mathscr E$$

The existence of the scalar products in $$\mathscr E_1$$ and $$\mathscr E_2$$ permits us to define one in $$\mathscr E$$ as well. We first define the scalar product of $$|\phi(1)\chi(2)\rangle = |\phi(1)\rangle \otimes |\chi(2)\rangle$$ by $$|\phi'(1)\chi'(2)\rangle = |\phi'(1)\rangle \otimes |\chi'(2)\rangle$$ by setting: $$\langle\phi'(1)\chi'(2)|\phi(1)\chi(2)\rangle = \langle\phi'(1)|\phi(1)\rangle \ \langle\chi'(2) |\chi(2)\rangle \tag{F-8}$$ For two arbitrary vectors in $$\mathscr E$$, we simply use the fundamental properties of the scalar product [equations (B-9), (B-10) and (B-11)], since each of these vectors is a linear combination of tensor product vectors.

With $$|\phi(1)\rangle$$ expressed in the orthonormal basis $${|u(1)\rangle}$$ of the space $$\epsilon_1$$ and $$|\chi(2)\rangle$$ expressed in the orthonormal basis $${|u(2)\rangle}$$ of the space $$\epsilon_2$$.

I ask this to know if (F-8) equation can be deduced or not from those properties. Could the following be done in order of this?

$$\begin{cases} |\psi\chi\rangle=|\psi\rangle⊗|\chi\rangle\\ \\ |\psi'\chi'\rangle=|\psi'\rangle⊗|\chi'\rangle \end{cases}$$

(For the sake of simplicity, I omit the $$(1)$$ and $$(2)$$ labels) \begin{align} \langle\psi'\chi'\big|\psi\chi\rangle & = \big\langle(|\psi'\rangle⊗|\chi'\rangle)\big|(|\psi\rangle⊗|\chi\rangle)\big\rangle \\ & = \big(\langle\psi'|⊗\langle\chi'|\big)\big|\big(|\psi\rangle⊗|\chi\rangle\big) \\ & = \Big[\langle\psi'|\big(|\psi\rangle⊗|\chi\rangle\big)\Big]⊗\Big[\langle\chi'|\big(|\psi\rangle⊗|\chi\rangle\big)\Big] \\ & = \Big[\langle\psi'|\psi\rangle⊗\langle\psi'|\chi\rangle\Big]⊗\Big[\langle\chi'|\psi\rangle⊗\langle\chi'|\chi\rangle\Big] \end{align}

• Could you please clarify. Do you ask, wether one may derive F-8 by using F-9 and the expansions of $$|\phi(j)\rangle = \sum_i \alpha_i |u_i(j)\rangle$$, where $j\in\lbrace 1,2\rbrace$? Apr 25 '19 at 12:30
• I have just edited the question, I hope now it is a little clearer. Apr 26 '19 at 9:26
• You can't take inner product between different vector spaces, like you do in the last line. Apr 26 '19 at 9:35
• Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. Apr 26 '19 at 10:15