Evaluation of Density Matrix and Helmholtz Free Energy for Harmonic Oscillator Asymmetric Potential via Feynmans Approach

We apply a Feynmans technique for calculation of a canonical density matrix of a single particle under harmonic oscillator asymmetric potential and solving the Bloch equation of the statistical mechanics system. The density matrix and kinetic energy per unit length can be directly evaluated from the solving solutions. From the evaluation, it was found that both of the density matrix and kinetic energy per unit length depended on the parameter of the value of asymmetric potential , the value of axes-shift potential , and temperature (T). Comparison of the Helmholtz free energy was derived by the Feynmans technique and the path-integral method. The results illustrated are slightly different.