According to Pauli principle, it is impossible for two electrons in the electron gas to have the same state. on the other hand, we say that each spatial state can be filled by two electrons with opposite spins. But my question; Suppose we have two electrons, one spin down and another spin up. In the relation below Ψ(x1,x2)=1/√2(ψ1(x1)ψ2(x2)−ψ1(x2)ψ2(x1)) Each of x1s and x2s shows both position and spin, and wave functions' indices(1 or 2) show the particle states. If we have the same wave functions' indices(say both 1 or both 2), then Ψ(x1,x2)=0 no matter what x1 or x2 (and hence the particles' spins) is. Therefore how can one say that two electrons can seat in the same spatial states?