Merits of using density matrices instead of wave functions in the stationary Schrödinger equation for systems with symmetries

Abstract

The stationary Schrödinger equation can be cast in the form , where H is the system’s Hamiltonian and is the system’s density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as SSE, applied to many-body systems with symmetries. For a nondegenerate energy level, the solution of the SSE is merely a projection on the corresponding eigenvector. However, in the case of degeneracy is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, 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 can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE 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 , where is the total spin of N spins 1/2, and is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of .