Conservation of symmetrization in quantum mechanics

Question

I recently read about the symmetrization requirement, which my book states is axiomatic of quantum mechanics: $$ \psi(\mathbf r_1, \mathbf r_2) = \pm \psi(\mathbf r_2, \mathbf r_1). \tag{*} $$ It further states that if a system starts out in such a state, then it will remain in such a state. Is this conservation of symmetrization also an axiom, or can this be proven (like the conservation of normalization)? Moreover, does the time-dependent wave function $\Psi(\mathbf r_1,\mathbf r_2,t)$ also satisfy the symmetrization requirement? Intuitively, I don't see how it can follow from $(*)$ that $$ \Psi(\mathbf r_1,\mathbf r_2,t) = \sum_n \psi_n(\mathbf r_1, \mathbf r_2) e^{-iE_n t/\hbar} = \pm \Psi(\mathbf r_2,\mathbf r_1,t) = \pm \sum_n \psi_n(\mathbf r_2, \mathbf r_1) e^{-iE_n t/\hbar}, $$ and if the time-dependent wave function does not follow the symmetrization requirement, then what good is $(*)$?

Answer 1

If the Hamiltonian is symmetric in $r_1$ and $r_2$, then we can show that its eigenfunctions can be taken to by symmetric or antisymmetric. If we start with a symmetric state at some time $t_1$, then we can expand it over the symmetric eigenstates only. We see that it will remain symmetric at any other time $t$. The same is true if we start with an antisymmetric state, which will remain antisymmetric under time evolution.