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My question concerns the tensor product of Hilber spaces and the connection to the real Cartesian space. I begin by presenting what my book says, and then what I have understood so far and what confuses me.

I am following Cohen-Tannoudji: Let us form the tensor product $$\mathcal{E}_{xyz}=\mathcal{E}_{x} \otimes \mathcal{E}_{y} \otimes \mathcal{E}_{z}.$$We note that the basis of $\mathcal{E}_{xyz}$ is built from the bases of $\mathcal{E}_x$, $\mathcal{E}_y$, and $\mathcal{E}_z$, such that $$|x, y, z\rangle=|x\rangle \otimes |y\rangle \otimes |z\rangle.$$ Furthermore, we note that the basis kets of $\mathcal{E}_{xyz}$ are simultaneously eigenvectors of $X$, $Y$, and $Z$ operators. Therefore, we have $$X\ |x, y, z\rangle=x\ |x, y, z\rangle\\Y\ |x, y, z\rangle=y\ |x, y, z\rangle\\Z\ |x, y, z\rangle=z\ |x, y, z\rangle.$$ Therefore, $\mathcal{E}_{xyz}$ coincides with $\mathcal{E}_{\bf{r}}$, the state space of a three-dimensional particle and $|x, y, z\rangle$ with $|\bf{r}\rangle$; thus, the $x$, $y$, and $z$ are precisely the Cartesian coordinates of $\bf{r}$.

My understanding and questions

Tensor products are applied to vector spaces and the result is another vector space. In the first line, we consider three 1D Hilbert vector spaces and build, via the tensor product, a 3D Hilbert space. Similarly, we can build the basis kets of this new Hilbert space from the basis kets of the 1D Hilbert spaces. Moreover, it makes sense that these new basis kets are simultaneous eigenvectors of $X$, $Y$, and $Z$, because the three Hilbert spaces are independent of one another so the three position operators should commute in $\mathcal{E}_{xyz}$ (that is, they should act independently). I am totally lost at the last sentence.

How do we go from Hilbert spaces with potentially infinite basis kets to the real Cartesian space with three axes $x$, $y$, and $z$? Also, am I right to think of all $\mathcal{E}$ spaces being Hilbert spaces? Also, is my reasoning for why $X$, $Y$, and $Z$ commute in $\mathcal{E}_{xyz}$ correct?

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  • $\begingroup$ According to Sakurai (2014), equation (6.10a), the state $|\mathbf{x}'\rangle \equiv |x',y',z' \rangle$ is merely a simultaneous eigenket (of the position operators $X,Y,Z$), and not some tensor product of eigenkets. $\endgroup$ – Henry Oct 20 '18 at 8:28
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Yes, they commute because $X$, $Y$ and $Z$ act independently. For simplicity consider a 2d Hilbert space with states $|x,y\rangle$. Then when we write $X$ we really mean the operator $X \otimes \mathbb{1}$, and likewise $Y$ is shorthand for $\mathbb{1} \otimes Y$. The fact that $X$ and $Y$ commute then follows from the fact that $X$ and $Y$ both commute with the unit operator $\mathbb{1}$.

Cohen-Tannoudji means something very simple. He has probably discussed the Hilbert space $\mathcal{E}_{\mathbf{r}}$ before, which is just the space of states labeled by a three-vector $\mathbf{r}$ instead of a single coordinate $x$ --- it's the Hilbert space of a single particle living in $\mathbb{R}^3$. We all know that $\mathbb{R}^3 = \mathbb{R} \oplus \mathbb{R} \oplus \mathbb{R}$. But it's not a priori clear what the relation between $\mathcal{E}_\mathbf{r}$ and the Hilbert space $\mathcal{E}_x$ of a 1d particle is. Mathematically speaking, the discussion you summarize is just the statement that $$ \mathcal{E}_{\mathbf{r}} = \mathcal{E}_x \otimes \mathcal{E}_y \otimes \mathcal{E}_z$$ with the identification $|\mathbf{r}\rangle = |x,y,z\rangle$ if $\mathbf{r} = (x,y,z)^t$ - it's an isomorphism between two Hilbert spaces.

Of course there's also a bijection from $\mathbb{R}^3$ to $\mathcal{E}_\mathbf{r}$: just send $\mathbf{r} \mapsto |\mathbf{r}\rangle$. But this is not an isomorphism of Hilbert spaces: $\mathbb{R}^3$ has dimension 3 and $\mathcal{E}_\mathbf{r}$ is infinite-dimensional. The above map isn't even linear: $$| \mathbf{r}_1 +\mathbf{r}_2 \rangle \neq | \mathbf{r}_1 \rangle + | \mathbf{r}_2 \rangle.$$ Both spaces also have very different inner products.

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  • $\begingroup$ Thank you for clarifying the commutation question. For the spaces discussion, you're saying that $\mathcal{E}_{\bf{r}}$ is a Hilbert space of all quantum states that are specified by $x$, $y$, and $z$ from $\mathbb{R}^3$ because the particle exists in the real space? $\endgroup$ – Ptheguy Oct 30 '17 at 3:23
  • $\begingroup$ And perhaps we can extend that to say the kets $|x\rangle$,$|y\rangle$, and $|z\rangle$ which specify the basis of the Hilbert spaces take their input from the real space? And therefore, the bases of $\mathcal{E}_{xyz}$ take their input from the real space? $\endgroup$ – Ptheguy Oct 30 '17 at 3:31
  • $\begingroup$ The answer to the first question is yes. The second question I don't fully understand. A state like $|x\rangle$ is what it is, and I'm sure that you have spent a lot of time with such states in your first few months of studying quantum mechanics. A state $|x,y,z\rangle$ or $|\mathbf{r}\rangle$ is indexed/labeled by a vector in real space, if you want to call that an input, sure ("input" isn't really a mathematical term). $\endgroup$ – user159249 Oct 31 '17 at 14:23
  • $\begingroup$ Thank you! I think now I have a better understanding. And yes, I used input as in a cs term! $\endgroup$ – Ptheguy Oct 31 '17 at 17:26

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