Description
The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wave functions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators ${\hat{\vec a}}^{+} $ and $\hat {\vec a}$ for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set $\{{\hat {\vec a }}^{+} \}$ times other exponential of the set $\{ \hat{\vec a } \}$. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of the so-called "Tassie-Barker" factors, find certain generating functions for the density and momentum distributions and point out other model-independent results. Within such an approach we are studying a combined effect of center-of-mass motion and short-range nucleon-nucleon correlations on the nucleon density and momentum distributions in light nuclei ($^{4}He $ and $^{16}O$). Their intrinsic ground-state wave functions are constructed in the so-called fixed center-of-mass approximation, starting with mean-field Slater determinants modified by some correlator ({\it e.g.}, after Jastrow or Villars). After this separation of the center-of-mass motion effects we propose additional analytic means in order to simplify the subsequent calculations ({\it e.g.}, within the Jastrow approach or the unitary correlation operator method). The charge form factors, densities and momentum distributions of $^{4}He $ and $^{16}O$ evaluated by using the well known cluster expansions are compared with data, our exact (numerical) results and microscopic calculations.
