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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}

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  • $\begingroup$ 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 $j\in\lbrace 1,2\rbrace$? $\endgroup$ – denklo Apr 25 at 12:30
  • $\begingroup$ I have just edited the question, I hope now it is a little clearer. $\endgroup$ – Quaerendo Apr 26 at 9:26
  • $\begingroup$ You can't take inner product between different vector spaces, like you do in the last line. $\endgroup$ – Ryan Thorngren Apr 26 at 9:35
  • $\begingroup$ 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. $\endgroup$ – Emilio Pisanty Apr 26 at 10:15
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No.

Equation (F-8) in the text you've quoted is a definition, and as such it cannot be derived from anything. If things go well, and you have previous structures that you want it to relate to, then you can justify the definition, but it is still a novel statement about what structure you're imposing on the new object (i.e. the tensor-product space) you've just defined.

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