A Method to Calculate a Delta Function of the Hamiltonian by the Suzuki-Trotter Decomposition

Abstract

We propose a new method to calculate expectation values of a delta function of the Hamiltonian, < Ψ\mid δ(\hat{H} - E)\mid Ψ>. Since the delta function can be replaced with a Gaussian function, we evaluate < Ψ\mid \sqrt{\fracβπ} e^{-β(\hat{H} - E)^2} \mid Ψ> with large βadopting the Suzuki-Trotter decomposition. Errors of the approximate calculations with the finite Trotter number N_t are estimated to be O(1/N_t^K) for the Kth-order decomposition. The distinct advantage of this method is that the convergence is guaranteed even when the state \mid Ψ> contains the eigenstates whose energies spread over the wide range. In this paper we give a full description of our method within the quantum mechanical physics and present the numerical results for the harmonic oscillator problems in one- and three-dimensional space.