A general expression for the density of a quantum system at finite temperature is obtained in the form of the variational derivative of the canonical partition function. Cyclic expansions for density are obtained in the absence and in the presence of interparticle interaction. In the first case the density is expressed in terms of single-particle characteristics for a set of increasing temperatures. The successive squaring method for the density matrix is considered; with the use of the cyclic density expansions, the densities of 1, 2,..., and 10 noninteracting fermi-particles with spin 1/2 and spin 0 and of 1, 2, 3,4, and 5 spinless noninteracting particles in the Morse potential are evaluated. From the low-temperature data, 10 states in a harmonic field and all 5 bound states in the Morse potential are reproduced with high accuracy. The system of two spinless fermions with and without the Coulomb repulsion is considered. One- and two-dimensional densities of the system and their energies are evaluated. The recurrent state subtraction method is discussed for the evaluation of excited states of quantum systems. The successive squaring method for the density matrix is used to obtain more than 20 states for the one-dimensional harmonic oscillator and all bound states for the Morse oscillator.

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