Pauli principle and Schrödinger/Pauli equations

In quantum mechanics state of the system is determined by wave function, which evolves according to the Schroedinger equation, or by Pauli/Dirac equation which are derived from Schroedinger equation for particles with spin. But we have also the Pauli principle, which arose from relativistic quantum field theory and requires wave function to be antisymmetric under exchange of fermions. My question is - how do these two combine? We must just discard solutions of wave function equations that violate Pauli principle?

• Have you ever heard of a Slater determinant? Commented Jul 11, 2020 at 9:11
• @SuperCiocia isn't Slater determinant used in approximations of wave function solutions? My question is more general Commented Jul 11, 2020 at 9:24

Let us consider a system of two identical particles interacting via the Coulomb interaction (for simplicity assuming no magnetic field). The Schrödinger equation is then $$i\partial_t \Psi(\mathbf{r}_1,\mathbf{r}_2,t) = \left[\frac{\nabla_{\mathbf{r}_1}^2}{2m} + \frac{\nabla_{\mathbf{r}_2}^2}{2m} + V(\mathbf{r}_1) + V(\mathbf{r}_2) + v(\mathbf{r}_1,\mathbf{r}_2)\right]\Psi(\mathbf{r}_1,\mathbf{r}_2,t),$$ where $$V(\mathbf{r}_{1,2})$$ is a one-particle potential, whereas $$v(\mathbf{r}_1,\mathbf{r}_2)$$ is the Coulomb interaction term. The symmetry constraint is $$\Psi(\mathbf{r}_1,\mathbf{r}_2,t) = - \Psi(\mathbf{r}_2,\mathbf{r}_1,t) \textrm{ (fermions)},\\ \Psi(\mathbf{r}_1,\mathbf{r}_2,t) = \Psi(\mathbf{r}_2,\mathbf{r}_1,t) \textrm{ (bosons)}.$$ Slater determinant (mentioned in the comments) is a wave of obtaining an antisymmetrized wave function from single-particle orbitals, i.e. in zeroth approximation in particle-particle interaction. Using it results in the Hartree-Fock approximation.