I've read that if the wavefunction of a combined state can be represented as the tensor product of the individual wavefunctions, the two particles involved are non-entangled.
Now taking a scenario in which the two particles can only occupy two state, i.e. :
$$ \rvert\psi_a\rangle=a_1\rvert0\rangle+a_2\rvert1\rangle$$
$$ \rvert\psi_b\rangle=b_1\rvert0\rangle+b_2\rvert1\rangle$$
The combined two particle wavefunction is:
$$\begin{align}
\rvert\Psi_{ab}\rangle &=\rvert\psi_a\rangle\otimes\rvert\psi_b\rangle \\
&= a_1b_1\rvert 0\rangle\otimes\rvert 0\rangle+a_1b_2\rvert 0\rangle\otimes\rvert 1\rangle+a_2b_1\rvert 1\rangle\otimes\rvert 0\rangle+a_2b_2\rvert 1\rangle\otimes\rvert 1\rangle\\ &=a_1b_2\rvert 0\rangle\otimes\rvert 1\rangle+a_2b_1\rvert 1\rangle\otimes\rvert 0\rangle \\ &= (a_1b_2-a_2b_1)\rvert 0\rangle\otimes\rvert 1\rangle \\ &= (a_1b_2-a_2b_1)\rvert 01\rangle \\
\end{align}$$
So here, evidently the two particles are not allowed to occupy the same state (among other restrictions).
- Is there a name for this restriction?
- Is there more physically intuitive content behind this picture?
EDIT: The mathematics done above is missing steps, as elucidated below.
Best Answer
This is precisely the Pauli exclusion principle: two indistinguishable fermions are forbidden from occupying the exact same quantum state. The formulation you give is the more rigorous, grown-up version. The effect is fully quantum mechanical and it cannot really be explained in a particularly intuitive fashion. Fermions cannot occupy the same state because that wavefunction is symmetric under exchange, and fermions - by definition - need an exchange- antisymmetric wavefunction.
Note that this effect, while counterintuitive, is perfectly physical. Among many other things, it is directly responsible for electron degeneracy pressure, which is what enables white dwarf stars to survive.