Claude Bloch

Sur la théorie des perturbations des états liés

Abstract Two methods leading to general prescriptions for the construction of the time independent perturbation expansions for bound states to all orders are discussed. The two methods differ by the choice of the zeroth-order wave functions. In both cases, the expansions are simplified by the use of non-orthonormalized zeroth-order wave functions. The terms of the expansions are described by means of diagrams. The accuracy of the expansions is tested by a calculation of two eigen-values of the Mathieu equation.

Sur la détermination des premiers états d'un système de fermions dans le cas dégénéré

Abstract The linked diagram expansion of the ground state energy and wave function of a system of fermions are extended to the case where the ground state of the unperturbed system is degenerate, as, for instance, in a nucleus with an incomplete shell. The state where all the complete shells are filled is taken as vacuum state. It is then shown, to all orders in the perturbation expansion, that the energy of the first levels is given by the sum of two terms. The first term, is the energy of the core alone and is given by a linked diagram expansion of the usual form. The second term is the energy of the particles outside the core. It is obtained by solving an eigenvalue problem in the space of the degenerate unperturbed wave functions. The corresponding eigen-functions are given by linked diagram expansions very similar with the usual ones. A few generalizations of the method, and its relation to the Brillouin-Wigner perturbation method are discussed.

Une formulation unifiée de la théorie des réactions nucléaires

Abstract The determination of the scattering matrix in the theory of nuclear reactions is essentially equivalent to the construction of the Green function for the Schrodinger equation in the internal region of the configuration space with proper boundary conditions at the nuclear surface. This Green function can be expressed as the inverse of an operator involving the sum of the Hamiltonian and of a “boundary value operator” which is different from zero only on the nuclear surface where it has a singularity of the same kind as a Dirac function. A general operator expression for the scattering matrix is derived. This expression can be transformed into a matrix expression by introducing an arbitrary basis of orthonormal functions in the internal region. The Wigner-Eisenbud and the Peierls-Kapur formulations are obtained by an appropriate choice of the internal functions. When a large number of resonances contribute to the cross-section, the expansion of the scattering matrix in terms of resonances of the compound system is not useful, and a more appropriate starting point can be obtained from a perturbation expansion of the scattering matrix which is easily derived from the general operator expression. A simple statistical assumption is proposed in order to determine the dominant terms in such an expansion. It leads to the optical model for the elastic scattering and to the direct interactions for the inelastic scattering.

Sur la détermination de l'état fondamental d'un système de particules

Abstract The linked diagram expansions of the ground state energy and wave function of a system of particles are derived by a method involving the use of the time-dependent as well as the time-independent formulations of perturbation theory. The essential step is the derivation, in the time-dependent formulation, of a simple exponential formula obtained by summation of the disconnected vacuum-vacuum diagrams.

Un développement du potentiel de Gibbs d'un système quantique composé d'un grand nombre de particules III—La contribution des collisions binaires

Starting from an expansion derived in a previous work, we study the contribution to the Gibbs potential of the two-body dynamical correlations, taking into account the statistical correlations. Such a contribution is of interest for low density systems at low temperature. In the zero density limit, it reduces to the Beth-Uhlenbeck expression for the second virial coefficient. For a system of fermions in the zero temperature limit, it yields the contribution of the Brueckner reaction matrix to the ground state energy, plus, under certain conditions, additional terms of the form exp(β|Δ|), where the Δ are the binding energies of “bound states” of the type first discussed by L. Cooper. Finally, we study the wave function of two particles immersed in a medium (defined by its temperature and chemical potential). It satisfies an equation generalizing the Bethe-Goldstone equation for an arbitrary temperature.

Formulation de la me´canique statistique en termes de nombres d'occupation (I)

Abstract A physical interpretation of the generalized free energy F , introduced in part I of this work in connection with the variational principle, is obtained by relating it to the statistical distribution of the occupation numbers. The values of the occupation numbers derived from the variational principle are then interpreted as the most probable values in a grand canonical ensemble of identical systems. As a particular consequence of the theory, it is shown that in the case of an infinite system without Bose condensation, the probability distribution of any occupation number has the same form as in the absence of interactions. The classical limit of the theory is then considered. First, the freat similarity of our results with the classical expansions expressed in terms of the local density is exhibited by rederiving the well known classical expressions with the methods of part I. Finally the classical limit of the Gibbs potential is shown to be identical, term by term, with the classical virial expansion.A physical interpretation of the generalized free energy F, introduced (Nuclear Physics 25: 529(196l)) in connection with the variational principle, is obtained by relating it to the statistical distribution of the occupation numbers. The values of the occupation numbers derived from the variational principle are then interpreted as the most probable values in a grand canonical ensemble of identical systems. As a particular consequence of the theory, it is shown that in the case of an infinite system without Bose condensation, the probability distribution of any occupation number has the same form as in the absence of interactions. The classical limit of the theory is then considered. First, the great similarity of our results with the classical expansions expressed in terms of the local density is exhibited by rederiving the well known classical expressions with previously given methods. Finally the classical limit of the Gibbs potential is shown to be identical, term by term, with the classical virial expansion. (auth)

La fonction densité (strength function) dans la théorie statistique des réactions nucléaires

Abstract It is shown by means of a simple model that the strength function corresponding to a single particle state of a nucleon interacting with a nucleus has a Lorentz shape in agreemnt with the analysis of the complex potential model in terms of giant resonances. The value of the imaginary part of the complex potential obtained from this calculation is not directly related to the second moment of the strength function which can be derived from a sum rule. It may, in fact, be appreciably smaller than the root square of the second moment.