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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.

Two-Particle Thermal Density Matrices In One Dimension Using Finite Difference Time Domain (FDTD) Method