# State function of two electrons in 1D box

I would like to know how to derive the wave function of two non-intering electrons in one-dimensional space by accounting their spin and Pauli's exclusion principle.

$$\Psi(x_1, x_2,\sigma_1, \sigma_2)$$ and $$V(x_1, x_2) = V(x_1) + V(x_2)$$

I'm just wondering that if I can use the form of separable wavefunction

$$\Psi(x_1, x_2,\sigma_1, \sigma_2) = \psi(x_1, x_2) \cdot \chi(\sigma_1, \sigma_2) = \psi_1(x_1)\psi_2(x_2) \cdot \chi_1(\sigma_1)\chi_2(\sigma_2)$$

Subtituting it in Schrodinger equation would yeild the same result as the case without spin.

You did not yet account for Pauli's exclusion principle. For this the wave function needs to be antisymmetric on exchanging the particles $$1$$ and $$2$$, i.e. $$\Psi(x_1,x_2,\sigma_1,\sigma_2)=-\Psi(x_2,x_1,\sigma_2,\sigma_1)$$
You can achieve this with the following wavefunction \begin{align} \Psi(x_1,x_2,\sigma_1,\sigma_2) &= \psi_1(x_1)\psi_2(x_2) \cdot \chi_1(\sigma_1)\chi_2(\sigma_2) \\ &- \psi_1(x_2)\psi_2(x_1) \cdot \chi_1(\sigma_2)\chi_2(\sigma_1) \end{align} which satisfies both Schrödinger's equation and Pauli's exclusion principle.
• @TommyWang Yes, $\chi_1$ and $\chi_2$ may be different, Commented Jun 24, 2023 at 13:34