A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$ -representability conditions, although in such a case some unphysical solution may also occur in the non- $N$ -representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.

Physical Review AJul 1, 1976257 citations

The Density Matrix Equations

Mark Linne

Elsevier eBooksJan 1, 20020 citations

Quantum Theory of a Laser Model

C. R. Willis

We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix $R$ to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where $\mathfrak{N}$ is the number of two-level systems per unit volume, ${r}_{0}$ is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix $R$ to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where $\mathfrak{N}$ is the number of two-level systems per unit volume, ${r}_{0}$ is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.

Physical ReviewJul 8, 196622 citations

Density Matrix Equations:

Princeton University Press eBooksDec 13, 20100 citations

Spinorial density matrix equation and gauge covariance

F. C. Khanna, Ademir Eugenio de Santana, A. Matos Neto +2 more

In this work we apply the Lie group representation method introduced in the real time formalism for finite-temperature quantum-field theory, thermofield dynamics, to derive a spinorial density matrix equation. Symmetry properties of such equation are analysed, and as a basic result it is shown that one solution is the generalised density matrix operator proposed by Heinz, to deal with gauge covariant kinetic equations. In the same context, preliminary aspects of a Lagrangian formalism to derive kinetic equations, as well as quantum density matrix equations in curved space-time, are discussed.In this work we apply the Lie group representation method introduced in the real time formalism for finite-temperature quantum-field theory, thermofield dynamics, to derive a spinorial density matrix equation. Symmetry properties of such equation are analysed, and as a basic result it is shown that one solution is the generalised density matrix operator proposed by Heinz, to deal with gauge covariant kinetic equations. In the same context, preliminary aspects of a Lagrangian formalism to derive kinetic equations, as well as quantum density matrix equations in curved space-time, are discussed.

arXiv:hep-th/9812222Dec 23, 19980 citations

Density Matrices

Klaus Bartschat

Springer handbooksJan 1, 20060 citations

N

-representability conditions, although in such a case some unphysical solution may also occur in the non-

N

-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.

We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix

R

to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where

\mathfrak{N}

is the number of two-level systems per unit volume,

{r}_{0}

is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.