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?
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$\begingroup$ Have you ever heard of a Slater determinant? $\endgroup$– SuperCiociaCommented Jul 11, 2020 at 9:11
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$\begingroup$ @SuperCiocia isn't Slater determinant used in approximations of wave function solutions? My question is more general $\endgroup$– Lalande 21185Commented Jul 11, 2020 at 9:24
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Let me first point out that Pauli principle applies to systems with at least two particles, whereas Schrödinger and Dirac equations are one-particle equations, i.e. they do not have quantum statistics incorporated in them. Thus, when a many-particle system is described using a Schrödinger or Dirac equations, the Pauli principle is accounted via the constraint on the wave function:
- Fermion wave function is antisymmetric in respect to exchanging the arguments corresponding to any two particles
- Boson wave function is symmetric in respect to exchanging the arguments corresponding to any two particles Note that the bosonic constraint is no less important than the fermionic one, although it is usually not implied in the Pauli principle.
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.
Finally, it is necessary to mention that in the second quantization formalism the particle statistics is taken into account via the operator statistics and ordering of the states in the many-particle wave functions (in order to keep track of the sign).
