# Pauli's principle on two-electron system

I just have been introduced to the axiom "defining" bosons and fermions, namely (for fermions): Consider a collection of $$N$$ identical particles moving in $$\mathbb{R}^3$$ and having a (half-integer) spin $$l$$. For each $$\psi \in L^2(\mathbb{R}^{3N};(V_l)^{\otimes N})$$ and each permutation $$\sigma$$, we have $$\psi(x^{\sigma(1)},\ldots, x^{\sigma(N)})=sgn(\sigma)\psi(x^1,\ldots, x^N)$$.

Then, as an example of my book, we have a look at Pauli's exclusion principle for two electrons. It is stated that the function $$\Psi: \mathbb{R}^6 \rightarrow \mathbb{C}^2 \otimes \mathbb{C}^2$$ given by $$\Psi(x^1,x^2)=\psi(x^1)\otimes \psi(x^2)$$ is not a possible state of a two-electron system, since $$\Psi$$ does not satisfy the above property.

Hence, we must have $$\Psi(x^2,x^1)=\psi(x^2)\otimes \psi(x^1)\stackrel{!}{=}\psi(x^1)\otimes \psi(x^2)=\Psi(x^1,x^2) \neq sgn(P_{12})\Psi(x^1,x^2)$$.

It has to be obvious but I don't get why the middle equality holds. This is probably because, as a non-physicist, the only intuition I have behind this tensor product is something like "we define a state of a many-particle system by considering the states each particle is in". I would appreciate a hand with that.

• Which book is this? Commented May 7, 2022 at 2:14
• Quantum mechanics for mathematicians, written by Brian C. Hall. Commented May 12, 2022 at 10:18
• In fact, I think the book does not take the spin part into account, or say only implicitely. The symmetry would then come from this part rather than directly from the position part. Commented May 12, 2022 at 10:20