A knowledge of this matrix enables us to calculate the mean value of physical quantity of the system and also the probabilities of various values of such a quantity. In the statistical point of view, a state of quantum ensemble is described by a density operator (or statistical operator) the corresponding matrix is called a density matrix. When the description of the system is incomplete, the state of the system is described by means of what is called a density matrix. We consider a quantum mechanical ensemble of identical systems in situations where the description is incomplete. The results illustrated are slightly different. Comparison of the Helmholtz free energy was derived by the Feynmans technique and the path-integral method. 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). The density matrix and kinetic energy per unit length can be directly evaluated from the solving solutions. 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.
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