Feynman-Kac path integrals and excited states of quantum systems (in English)

Abstract

We use transformation properties of the irreducible representations of the symmetry group of the Hamiltonian and properties of a continuous path to define a "failure tree" procedure for finding eigenvalues of the Schrodinger equation using stochastic methods. The procedure is used to calculate energies of the lowest excited states of quantum systems possessing anti-symmetric nodal regions in configuration space with the Feynman-Kac path integral method. Within this method, the solution of the imaginary time Schrodinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation of a subgroup of the symmetry group of the Hamiltonian and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. As a consequence, the sign problem for fermions is eliminated. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic and helium atoms.