# 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 $j\in\lbrace 1,2\rbrace$? – denklo Apr 25 '19 at 12:30
• I have just edited the question, I hope now it is a little clearer. – Quaerendo Apr 26 '19 at 9:26
• You can't take inner product between different vector spaces, like you do in the last line. – Ryan Thorngren Apr 26 '19 at 9:35
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