Showing papers in "Physical Review A in 1976"
TL;DR: In this article, the melting points of small gold particles have been measured using a scanning electron-diffraction technique and the experimental results are quantitatively in good agreement with two phenomenological models.
Abstract: Recently, small particles have been shown to exhibit a melting temperature which depends on the particle size. The various possible experimental methods have been compared and measurements of the melting points of small gold particles have been made using a scanning electron-diffraction technique. This method was applied to particles having diameters down to 20 \AA{}. Consideration of the size distribution over an entire sample makes it necessary to carry out a careful analysis of the experimental results in order to deduce the melting temperature of particles having a well-defined diameter. The experimental results are quantitatively in good agreement with two phenomenological models. The first model describes the equilibrium condition for a system formed by a solid particle, a liquid particle having the same mass, and their saturating vapor phase. The second model assumes the preexistence of a liquid layer surrounding the solid particle and describes the equilibrium of such a system in the presence of the vapor phase. In order to permit a better comparison between both models, a new expression for the thermodynamic equilibrium condition has been derived in the present work. In the case of the first model, the agreement was obtained using only the physical constants of massive gold. In applying the second model, however, one is compelled to assume the existence of a liquid layer having a thickness of about 6 \AA{}.
3,074 citations
TL;DR: In this paper, the concept of two-photon coherent states is introduced for applications in quantum optics, which is a simple generalization of the well-known minimum-uncertainty wave packets.
Abstract: The concept of a two-photon coherent state is introduced for applications in quantum optics. It is a simple generalization of the well-known minimum-uncertainty wave packets. The detailed properties of two-photon coherent states are developed and distinguished from ordinary coherent states. These two-photon coherent states are mathematically generated from coherent states through unitary operators associated with quadratic Hamiltonians. Physically they are the radiation states of ideal two-photon lasers operating far above threshold, according to the self-consistent-field approximation. The mean-square quantum noise behavior of these states, which is basically the same as those of minimum-uncertainty states, leads to applications not obtainable from coherent states or one-photon lasers. The essential behavior of two-photon coherent states is unchanged by small losses in the system. The counting rates or distributions these states generate in photocount experiments also reveal their difference from coherent states.
1,661 citations
TL;DR: In this paper, a numerical study of the Kolmogorov entropy for the H\'enon-Heiles model is presented, based on mathematical results of Oseledec and Piesin.
Abstract: Numerical investigations of dynamical systems allow one to give estimates of the rate of divergence of nearby trajectories, by means of a quantity which is usually assumed to be related to the Kolmogorov (or metric) entropy. In this paper it is shown first, on the basis of mathematical results of Oseledec and Piesin, how such a relation can be made precise. Then, as an example, a numerical study of the Kolmogorov entropy for the H\'enon-Heiles model is reported.
991 citations
TL;DR: In this article, a self-consistent set of equations for an atomic central potential such that the energy given by the orbitals for the potential is minimized is derived, which leads to an effective, local, central exchange potential analogous to the $X\ensuremath{\alpha}$ potential.
Abstract: A self-consistent set of equations is derived for an atomic central potential such that the energy given by the orbitals for the potential is minimized. It is shown that this effective potential behaves like $\frac{\ensuremath{-}{e}^{2}}{r}$ for large $r$ values. The equations have been solved for carbon, neon, and aluminum, and the resulting total energies exceed the Hartree-Fock total energies by less than 0.005%. The theory leads to an effective, local, central exchange potential analogous to the $X\ensuremath{\alpha}$ potential.
925 citations
TL;DR: In this article, the energy levels of the nonzero angular momentum states of the static screened Coulomb potential were calculated using variational trail functions, and it was shown that Hulth-en-like trial functions provide better variational energies and wave functions with fewer parameters than hydrogenic or Slater-type functions for screened-coulomb and similar potentials.
Abstract: Using solutions to a Hulth\`en-like effective potential as variational trail functions we have calculated the energy levels of the nonzero angular momentum states of the static screened Coulomb potential. Our one-parameter results for the $2p$, $3p$, $3d$, $4p$, $4d$, and $4f$ levels are in excellent agreement with earlier, more elaborate calculations. We have also calculated spontaneous emission transition probabilities between several pairs of states and find that our results compare favorably with previous calculations. We conclude that Hulth\`en-like trial functions provide better variational energies and wave functions with fewer parameters than hydrogenic or Slater-type functions for screened Coulomb and similar potentials.
550 citations
TL;DR: In this paper, a study of the dynamics of a discrete two-dimensional system of classical particles is presented, where the equilibrium state is investigated and the Navier-Stokes hydrodynamical equations are derived.
Abstract: A study of the dynamics of a discrete two-dimensional system of classical particles is presented. In this model, dynamics and computations may be done exactly, by definition. The equilibrium state is investigated and the Navier-Stokes hydrodynamical equations are derived. Two hydrodynamical modes exist in the model: the sound waves and a kind of vorticity diffusion. In the Navier-Stokes equations one obtains a transport coefficient which is given by a Green-Kubo formula. The related time correlation function has been calculated in a numerical simulation up to a time of the order of 50 mean free flights. After a short time of exponential decay this time correlation behaves like ${t}^{\ensuremath{-}S}$, the exponent being compared to theoretical predictions.
448 citations
TL;DR: In this article, the electron-ion recombination mechanisms were determined along with the rate coefficients of the important elementary processes which govern the relaxation of the helium afterglow, at room temperature.
Abstract: The electron-ion recombination mechanisms of ${\mathrm{He}}_{2}^{+}$ are determined along with the rate coefficients of the important elementary processes which govern the relaxation of the helium afterglow, at room temperature. The experimental data (atomic- and molecular-ion currents to the walls, atomic and molecular metastable concentrations, electron concentration, elastic electron collision frequency, electron radiation temperature) obtained as a function of time under a wide range of experimental conditions are compared with the solutions of a system of five coupled partial differential equations which includes all the processes occurring in a helium afterglow. A unique set of rate coefficients and constants is found allowing the precise reproduction of all the experimental data obtained at seven pressures from 5 to 100 Torr. The electron energy balance and electron energy distribution function are calculated as a function of time and space. It is shown that the spatial distribution of the electron energy in our cylindrical experimental cell is not uniform and has to be taken into account, as well as the influence of the non-Maxwellian electrons. The recombination rate coefficient for ${\mathrm{He}}_{2}^{+}$, given under the form ${\ensuremath{\alpha}}_{2}=({\ensuremath{\alpha}}_{c2}+{k}_{02}{n}_{0}){(\frac{{T}_{e}}{293\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}})}^{\ensuremath{-}{x}_{2}}+{k}_{e2}{n}_{e}{(\frac{{T}_{e}}{293\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}})}^{{\ensuremath{-}y}_{2}}$, is found to be such that ${\ensuremath{\alpha}}_{c2}l5l{10}^{\ensuremath{-}10}$ ${\mathrm{cm}}^{3}$ / sec, ${k}_{02}=(5\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}$ ${\mathrm{cm}}^{6}$ / sec, ${k}_{e2}=(4.0\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}$ ${\mathrm{cm}}^{6}$ / sec, ${x}_{2}=1\ifmmode\pm\else\textpm\fi{}1$, ${y}_{2}=4.0\ifmmode\pm\else\textpm\fi{}0.5$. These coefficients correspond to a collisional-radiative model for the recombination of ${\mathrm{He}}_{2}^{+}$ with electrons, which strongly depends on pressure, electron concentration, and electron temperature. 70% of the recombined molecular ions produce atomic metastables corresponding to a dissociation in the lower excited states of the molecule. The rate coefficients for ionizing collisions between metastables are found to be ${\ensuremath{\beta}}_{11}=(1.5\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ ${\mathrm{cm}}^{3}$ / sec, ${\ensuremath{\beta}}_{22}=(1.5\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ ${\mathrm{cm}}^{3}$ / sec, ${\ensuremath{\beta}}_{12}=(2.5\ifmmode\pm\else\textpm\fi{}1.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ ${\mathrm{cm}}^{3}$ / sec. The superelastic electron-metastable rate coefficients are ${\ensuremath{\gamma}}_{1}=(4.2\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ ${\mathrm{cm}}^{3}$ / sec and ${\ensuremath{\gamma}}_{2}=(3.8\ifmmode\pm\else\textpm\fi{}0.8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ ${\mathrm{cm}}^{3}$ / sec. All the rate coefficients compare very well with available theoretical data. The method used gives a complete solution of the helium afterglow at room temperature. It can be extended in pure helium to many other experimental conditions and applied to the study of afterglows in other pure gases or mixtures.
413 citations
TL;DR: In this paper, the problem of the interaction between a two-level atom and a quantum electromagnetic field is treated without the use of perturbation theory, without introduction of classical fields or factorization conditions for the states, and without assumptions about loss of memory.
Abstract: The problem of the interaction between a two-level atom and a quantum electromagnetic field is treated without the use of perturbation theory, without introduction of classical fields or factorization conditions for the states, and without assumptions about loss of memory. The calculation is carried out in the Heisenberg picture, without mode decomposition, and the conclusions all refer to physically measurable quantities, such as the fluorescence detected in the far field of the atom. It is shown that in a coherent field of constant amplitude the system always settles down to a quasistationary state, and that the stationarity is a manifestation of the quantum fluctuations. A solution for the growth of the fluorescent light intensity is presented that holds for any coherent exciting field. The two-time correlation function and the spectral density of the fluorescence are calculated, and are found to agree in the long-time limit with earlier results of Mollow. The two-time intensity correlation function of the field is derived, which corresponds to measurable photoelectric pair correlations, and it is found that this reflects several quantum features of the field. It is shown that quantum fluctuations are manifest more explicitly in two-time correlations in the steady state than in transient effects, like spontaneous emission in the vacuum. The measurement of such correlations therefore presents an opportunity for further experimental tests of quantum electrodynamics.
374 citations
TL;DR: In this paper, a complete and detailed treatment of a three-level atom interacting with two near-resonant monochromatic fields is presented, assuming that the only damping mechanism is radiative damping, and that the atoms all have the same resonance frequencies.
Abstract: A complete and detailed treatment of a three-level atom interacting with two near-resonant monochromatic fields is presented. It is assumed that the only damping mechanism is radiative damping, and that the atoms all have the same resonance frequencies, as is the case in an atomic beam. Detailed analytic solutions as well as numerical examples are determined from quantum-electrodynamic equations of motion. In addition, approximation techniques are presented which allow one to get accurate quantitative predictions for strong applied fields from simple rate-equation-like arguments. Absorption and emission spectra are determined, as are the transient and steady-state response of the atom. Two of the more interesting predictions are an emission spectrum containing up to seven components and a prediction of steady-state populations larger than 0.5 in the second excited state.
313 citations
TL;DR: In this paper, a semiclassical treatment of the evolution of an initially inverted system into a superradiant state in an extended, optically thick medium is presented, based on the coupled Maxwell-Schr\"odinger equations.
Abstract: This paper presents a semiclassical treatment of the evolution of an initially inverted system into a superradiant state in an extended, optically thick medium. In this process spontaneous emission and background thermal radiation initiate the collective radiative decay and produce a superradiant output pulse of intensity proportional to the square of the number of radiators. The treatment is based on the coupled Maxwell-Schr\"odinger equations, modified to include a fluctuating polarization source properly constructed to account for the effects of spontaneous emission. Computer results show that for a high-gain system only two parameters significantly influence the evolution process: ${T}_{R}$, the characteristic radiation damping time of the collective system; and ${\ensuremath{\theta}}_{0}$, a function of the conditions which initiate the superradiant process. In this limit one obtains a normalized emission curve and simple analytical expressions for the time delay, pulse width, and peak intensity of the output radiation. These results are in good agreement with experiments. A comparison of our model with previous treatments of superradiance is given.
286 citations
TL;DR: In this paper, the angular distribution of photoelectrons ejected on absorption of left or right circularly polarized light by a mixture of randomly oriented dextrorotatory and levorotatory optical isomers was analyzed.
Abstract: It is shown that the angular distribution of photoelectrons ejected on absorption of left or right circularly polarized light by a mixture of randomly oriented dextrorotatory and levorotatory optical isomers behaves as $A\ifmmode\pm\else\textpm\fi{}Bcos\ensuremath{\theta}+C{cos}^{2}\ensuremath{\theta}$, where $\ensuremath{\theta}$ is the angle between the photoelectron momentum and photon direction of incidence and $\ifmmode\pm\else\textpm\fi{} Bcos\ensuremath{\theta}$ is introduced by the presence of unequal numbers of each isomer. All coefficients of ${cos}^{n}\ensuremath{\theta}$ are of order $\ensuremath{\alpha}$, and analysis shows that $B$ can be as large as $A$ and $C$.
TL;DR: In this article, the static dipole polarizabilities are calculated from self-consistent field and highly correlated wave functions for the ground states of the atoms Li through Ne, and the correlation contributions to the polarizability are found to vary between -16% for Be and + 14% for F.
Abstract: Static dipole polarizabilities are calculated from self-consistent-field and highly correlated wave functions for the ground states of the atoms Li through Ne. The correlation contributions to the polarizabilities are found to vary between -16% for Be and + 14% for F. The polarizabilities as obtained from the coupled-electron-pair approximation are expected to be accurate to about 2%.
TL;DR: In this paper, a density-functional theory appropriate to nonuniform simple classical fluids is developed, which yields surface tensions and profiles in very good agreement with the results of Monte Carlo calculations.
Abstract: A density-functional theory appropriate to nonuniform simple classical fluids is developed. Unlike most previous theories, the theory is in principle exact in the linear-response regime. For practical applications a small number of well-defined approximations, including that of Percus and Yevick, are made. Given only the Lennard-Jones 6-12 potential as input, the theory yields surface tensions and profiles in very good agreement with the results of Monte Carlo calculations. For a liquid pressed against a hard wall the expected oscillations in density in the vicinity of the wall are obtained. Comparisons with previous theories are made, and points of possible improvement in the formalism are discussed.
IBM1
TL;DR: In this paper, the Hartree-Fock-Slater charge distribution for the target atom with Bonderup and Hvelplund's formulation was calculated for protons and helium ions in each of the elements.
Abstract: Energy straggling has been calculated for protons and helium ions in each of the elements, by using the Hartree-Fock-Slater charge distribution for the target atom with Bonderup and Hvelplund's formulation. The results reveal a ${Z}_{2}$ structure in energy straggling, and afford a ready explanation of some earlier measurements.
TL;DR: In this paper, a reduced coupling matrix-element formula has been developed for application to the class of reactions, where the reduced matrix elements have been combined with an absorbing-sphere model based on the Landau-Zener method to obtain a relatively simple formalism for obtaining the electron transfer total cross sections.
Abstract: A reduced coupling matrix-element formula has been developed for application to the class of reactions ${A}^{+z}+B\ensuremath{\rightarrow}{A}^{+z\ensuremath{-}1}+{B}^{+}$, where $4\ensuremath{\le}z\ensuremath{\le}54$. The reduced coupling matrix elements have been combined with an absorbing-sphere model based on the Landau-Zener method to obtain a relatively simple formalism for obtaining the electron transfer total cross sections. The method assumes a large number of final product channels available for reaction and is applicable for relative velocities ${v}_{\mathrm{rel}}\ensuremath{\le}1\ifmmode\times\else\texttimes\fi{}{10}^{8}$ cm/sec. For such electron transfer reactions, the cross sections at a given velocity are dependent primarily on the charge state of the incident ion and the ionization potential of the target atom; also, for a given colliding pair, the cross section is found to be almost independent of the relative velocity. Cross sections have been calculated for H, He, Ne, Ar, Kr, Xe, and ${\mathrm{H}}_{2}$ targets. For the cases of ${\mathrm{H}}_{2}$ and Ar there are experimental data available; the comparison between theory and experiment is quite favorable.
TL;DR: In this paper, a non-variational equation called the density equation is proposed for the direct determination of the density matrix without using a wave function, which is connected with the Schr\"odinger equation by a necessary and sufficient theorem.
Abstract: 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.
TL;DR: In this article, a statistical-mechanical theory of the elastic constants of a liquid crystal based on the Onsager expansion is reexamined, with emphasis on the terms linear in the gradients of the director.
Abstract: A statistical-mechanical theory of the elastic constants of a liquid crystal based on the Onsager expansion is reexamined, with emphasis on the terms linear in the gradients of the director. This gives a microscopic theory of the piezoelectric effect in nematic liquid crystals, and of the spontaneous twisting of a cholesteric. Model cases of tapered rods, bent rods, and threaded rods are considered.
TL;DR: In this paper, the second-order and fourth-order Uehling and Kall\'en-Sabry vacuum-polarization potentials in configuration space were approximated with nominal accuracies of nine and four figures, respectively.
Abstract: Rational approximations are presented for the second-order (Uehling) and fourth-order (K\"all\'en-Sabry) vacuum-polarization potentials in configuration space. These approximations may be applied to point-charge or finite-inducing-charge distributions. For the second-order potential, two approximations are given, with nominal accuracies of nine and four figures for the range $0\ensuremath{\le}r\ensuremath{\le}\ensuremath{\infty}$. For the fourth-order potential, one approximation of about three-figure accuracy is given for the range $0\ensuremath{\le}r\ensuremath{\le}{\ensuremath{\lambda}}_{e}$.
TL;DR: In this article, the transport properties of molten alkali halides have been calculated in molecular dynamics "experiments" based on rigid-ion pair potentials of the Huggins-Mayer form with parameters proposed by Tosi and Fumi.
Abstract: The transport properties of molten alkali halides have been calculated in molecular dynamics "experiments" based on rigid-ion pair potentials of the Huggins-Mayer form with parameters proposed by Tosi and Fumi. The coefficients of shear viscosity and electrical conductivity are obtained by computing the response of the system to a small applied field of the appropriate type, and the extension of the method to the calculation of current-current correlation functions of acoustic and optic character is also considered. Comparison with experimental data shows that the electrical conductivities are very well reproduced, but systematic discrepancies are found for diffusion coefficients and viscosities.
TL;DR: In this paper, a third-order perturbation calculation of line shapes in laser spectroscopy based on the density-matrix formalism is presented, where a new feature of this theory is the inclusion of the Gaussian spatial structure of the laser beams.
Abstract: We present a third-order perturbation calculation of line shapes in laser spectroscopy based on the density-matrix formalism. The new feature of this theory is the inclusion of the Gaussian spatial structure of the laser beams. We study the linewidth as a function of relaxation and transit times. A shift is found when the wave fronts are not flat. General line-shape formulas are given as well as approximate formulas valid in various domains.
TL;DR: In this article, a Landau-Ginzburg free energy is presented that yields a nematic to smectic-$A$ (NA) transition, a smectric to C$ (NC) transition and a Lifshitz point (at the boundary between the NA and NC transitions).
Abstract: A Landau-Ginzburg free energy is presented that yields a nematic to smectic-$A$ (NA) transition, a nematic to smectic-$C$ (NC) transition, and a Lifshitz point (at the boundary between the NA and NC transitions). Fluctuation enhancements of the Frank elastic constants are calculated. For the NC transition, all three elastic constants ${K}_{1}$, ${K}_{2}$, and ${K}_{3}$ diverge as ${\ensuremath{\xi}}^{2}$, where $\ensuremath{\xi}$ is the correlation length for fluctuations of the smectic order parameter. At the Lifshitz point, ${K}_{1}$ and ${K}_{2}$ diverge as $\mathrm{ln}\ensuremath{\xi}$ whereas ${K}_{3}$ diverges as $\ensuremath{\xi}$.
TL;DR: In this paper, a theory of electron-molecule scattering was developed which was a synthesis of close coupling and adiabatic-nuclei theories, and the theory is shown to be a close coupling theory with respect to vibrational degrees of freedom but is a adi-abatic nuclei theory in respect to rotation.
Abstract: A theory of electron-molecule scattering was developed which was a synthesis of close coupling and adiabatic-nuclei theories. The theory is shown to be a close coupling theory with respect to vibrational degrees of freedom but is a adiabatic-nuclei theory with respect to rotation. It can be applied to any number of partial waves required, and the remaining ones can be calculated purely in one or the other approximation. A theoretical criterion based on fixed-nuclei calculations and not on experiment can be given as to which partial waves and energy domains require the various approximations. The theory allows all cross sections (i.e., pure rotational, vibrational, simultaneous vibration-rotation, differential and total) to be calculated. Explicit formulae for all the cross sections are presented.
TL;DR: Tokamaks, stellarators and related plasma confinement devices have existed in a fruitful and yet usually precarious symbiosis with atomic physics as discussed by the authors, and they have been blamed, usually unjustly, for practically every ailment besetting the plasmas.
Abstract: Tokamaks, stellarators and related plasma confinement devices have existed in a fruitful and yet usually precarious symbiosis with atomic physics. In principle, the only atoms involved in such plasmas are those of hydrogen isotopes, and perhaps eventually helium, but in practice this has not been the case. Other elements have intruded in a variety of ways mostly unexpected, and they have been blamed, usually unjustly, for practically every ailment besetting the plasmas.
TL;DR: In this paper, the thermodynamic properties of one-component ionic plasmas in a responding (polarized) background of fully degenerate electrons are calculated over a wide range of temperatures and densities.
Abstract: The thermodynamic properties of one-component ionic plasmas in a responding (polarized) background of fully degenerate electrons are calculated over a wide range of temperatures and densities The weak and intermediate screening regimes are treated by two complementary methods which take the one-component plasma in a rigid background as a starting point Thermodynamic perturbation theory is used in the weakly screened ionic plasma typical of white-dwarf matter; this leads to simple, analytic expressions for the thermodynamic properties Relativistic effects in the very dense electron gas are considered explicitly A variational method, based on the Gibbs-Bogolyubov inequality, and the physical idea of an effective charge reduction of the ions, yields satisfactory results in the intermediate screening regime typical of the deep interior of Jupiter Our results are in good agreement with the available Monte Carlo data and represent a significant improvement over previous calculations based on a hard-sphere reference system
TL;DR: In this article, the multichannel $J$-matrix technique was used to solve the pseudostate close-coupling equations for photodetachment using only ${L}^{2}$ basis functions and standard configurationinteraction techniques.
Abstract: The multichannel $J$-matrix technique is used to solve the pseudostate close-coupling equations for ${\mathrm{H}}^{\ensuremath{-}}$ photodetachment using only ${L}^{2}$ basis functions and standard configuration-interaction techniques. The problem of linear dependence which arises when the same ${L}^{2}$ basis is used to expand both the target and free electrons is discussed and exactly solved using a matrix partitioning. Results for total and partial photodetachment cross sections are given from threshold to 70-eV photon energy (1664 to 18 nm) with emphasis on the single-photon two-electron ejection cross section, and the $^{1}P$ shape resonance just above the hydrogenic $n=2$ threshold.
TL;DR: The Hartree-Fock equation for the reduced density matrix of order $s is derived in this article, without recourse to the variational method, by assuming that ${ensuremath{\rho}}{s+1} and ${ensemblemath{n+2} are particular functionals of the density matrix.
Abstract: Starting with Schr\"odinger's time-independent equation for $N$ indistinguishable particles, an exact equation is derived for the reduced density matrix of order $s$, ${\ensuremath{\rho}}_{s}$. The equation contains terms dependent on ${\ensuremath{\rho}}_{s+1}$ and ${\ensuremath{\rho}}_{s+2}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\ensuremath{\rho}}_{2}$ and ${\ensuremath{\rho}}_{3}$ are particular functionals of ${\ensuremath{\rho}}_{1}$.