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

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

2000-01-21

arxiv.org/abs/cond-mat/0001304v1

Physics

Condensed Matter

Materials Science

13 pages, 8 figures(eps-files), latex

Scientific paper

We propose a new method to calculate expectation values of a delta function of the Hamiltonian, < \Psi \mid \delta(\hat{H} - E)\mid \Psi >. Since the delta function can be replaced with a Gaussian function, we evaluate < \Psi \mid \sqrt{\frac{\beta}{\pi}} e^{-\beta (\hat{H} - E)^2} \mid \Psi > with large \beta adopting 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 \Psi > 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.

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