A variational principle on certain operators in general and the density matrix in particular

C. E. Campbell, K. E. Kürten, G. Senger +1 more

Masumi Hattori, Huzio Nakano

Quantum variational principle of irreversible processes is generalized to the systems under the uniform magnetic field. For the electron transport in impurity potentials as a typical example, contractions of the density matrix are performed without any approximation in two steps; first the off-diagonal part of the density matrix is eliminated and subsequently the even part with respect to time reversal. In consequence the maximum principle is derived for the odd part of the diagonal elements of the density matrix. This means the contraction of the density matrix brings about irreversibility. The variational principle is shown to be closely related to that for the conventional Boltzmann–Bloch equation and is regarded as its fundamental version. The dependences of the density matrix on the magnetic field and wave number are discussed.

MANFRED L. RISTIG, G. Senger, K. E. Kürten +1 more

Louis J. Kijewski, Jerome K. Percus

Variational principles for lower bounds to the energy, or free energy for T > 0°, of many-body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy—or free energy—for the model Hamiltonian HM. Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H − HM approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs-Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.

H. Flocard

Gil Zumbach, Klaus Maschke

We discuss the one-particle density-matrix functional theory for the N-particle ground-state problem. Using the variational principle we obtain a set of self-consistent equations which could be useful for practical calculations. The derived relation between the eigenvalues of the density matrix and the chemical potential agrees with that of Gilbert. The present theory is compared with the Kohn–Sham and the Hartree–Fock approach to the ground state problem. It is found that both approaches can be obtained from the present theory under certain assumptions.