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Journal of Physics A Mathematical and Theoretical

The finite difference time domain method for computing the single-particle density matrix

I Wayan Sudiarta, D. J. W. Geldart

Published Jun 25, 2009• 11 citations
Publisher

Abstract

A general method for numerical computation of the thermal density matrix of a single-particle confined quantum system is presented. The Schrodinger equation in imaginary time τ is solved numerically by the finite difference time domain (FDTD) method using a set of initial wavefunctions at τ = 0. By choosing this initial set appropriately, the set of wavefunctions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.

DOI
10.1088/1751-8113/42/28/285002

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A general method for numerical computation of the thermal density matrix of a single-particle confined quantum system is presented. The Schrodinger equation in imaginary time τ is solved numerically by the finite difference time domain (FDTD) method using a set of initial wavefunctions at τ = 0. By choosing this initial set appropriately, the set of wavefunctions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
A general method for numerical computation of the thermal density matrix of a single-particle quantum system is presented. The Schrodinger equation in imaginary time tau is solved numerically by the finite difference time domain (FDTD) method using a set of initial wave functions at tau=0 . By choosing this initial set appropriately, the set of wave functions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation, and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
A quantum system in the thermal equilibrium state is a mixed state consisting of statistical ensembles of several different quantum systems can be represented by a thermal density matrix. In this research, the thermal density matrix is calculated for two-particle system case non-interaction in one-dimensional square well and one-dimensional harmonic oscillator using finite difference time domain (FDTD) method. In addition, thermal density matrix calculations are also performed for the case of two particle systems interacting in a one-dimensional harmonic oscillator. We present results of probability densities, partition functions, and internal energies for three cases: two distinguishable particles, two fermions and two bosons. Validation of numerical results of thermal density matrix and probability density is accurate with analytical solutions. Then, the result of partition function and internal energy the system is strongly effect by temperature. At low temperatures, internal energy the system will lead to the lowest energy or ground state.
A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.