Merits of using density matrices instead of wave functions in the stationary Schr\"odinger equation for systems with symmetries

J. Phys. A: Math. Theor. 53, 075301 (2020) The stationary Schr\"odinger equation can be cast in the form $H \rho = E \rho$, where $H$ is the system's Hamiltonian and $\rho$ is the system's density matrix. We explore the merits of this form of the stationary Schr\"odinger equation, which we refer to as~SSE$_\rho$, applied to many-body systems with symmetries. For a nondegenerate energy level, the solution $\rho$ of the SSE$_\rho$ is merely a projection on the corresponding eigenvector. However, in the case of degeneracy $\rho$ is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, $\rho$ can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE$_\rho$ can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE$_\rho$ for a general system of spins $1/2$ with Heisenberg interactions, and address simple systems of spins $1$. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem ${\bf S}^2 \rho=S(S+1) \rho$, where $\bf S$ is the total spin of $N$ spins $1/2$, and $\rho$ is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of ${\bf S}^2$.