The method of many-body point transforms previously developed by the author is modified to eliminate the interacting particle cluster formalism. The cutoff terms in the transformation, and thus in the Hamiltonian, are therefore removed. For the hard-core many-body problem, as in the previous work, a Hamiltonian is obtained which is Fourier analyzable, Hermitian and amenable to ordinary perturbation and variational techniques while still being equivalent to the original Hamiltonian. No approximations are made. It is demonstrated that the use of the standard zeroth-order approximation of the ground state Bose system, i.e., a constant, for the transformed wavefunction gives a negative expectation value of the energy for the hard core system. This discrepancy arises from the extended range of the new potential generated by the transformation, which necessitates explicit consideration of boundary conditions not satisfied by the above wavefunction.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
In this work a unitary correlation operator is presented that explicitly describes the short-ranged central and tensor correlations in the nuclear many-body system. These short-ranged correlations are induced by the repulsive core and the pronounced tensor force of realistic nucleon-nucleon interactions and cannot be described by the simple many-body states of a mean-field or shell model approach. The unitary correlation operator is discussed for the Argonne V18 and the Bonn-A interactions. Applying the correlation operator onto the Hamiltonian a common effective interaction for low momenta is obtained. Calculations for 4He using the one- and two-body part of the correlated Hamiltonian compare favorably with exact many-body methods. Calculations for 16O and 40Ca which are not possible with exact methods are performed using harmonic oscillator shell model states. The observed deviation from the experimental binding energies and radii is attributed to the missing three-body forces. The correlated interaction in a basis-free representation is used in the Fermionic Molecular Dynamics model to calculate the nuclei of the p- and the sd-shell. In this model the uncorrelated many-particle state is given as a Slaterdeterminant of Gaussian wave-packets with spin and isospin degrees of freedom which allows a consistent description of spherically symmetric, intrinsically deformed and clustered nuclei.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid
He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements
Hmn,
Nmn of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the
Hmn are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (
Hmn) and (
Nmn), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (
Hmn). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any
F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid
He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.
The Goldstone expansion is rederived by elementary time-independent methods, starting from Brillouin-Wigner (BW) perturbation theory. Interaction energy terms $\ensuremath{\Delta}E$ are expanded out of the BW energy denominators, and the series is then rearranged to obtain the linked-cluster result. Similar algebraic methods lead to the linked expansions for the total wave function (Hugenholtz) and the expectation value of a general operator (Thouless). Starting again with a degenerate version of BW perturbation theory, these methods are used to obtain the Bloch-Horowitz energy expansion, as well as the corresponding wave function, expectation-value, and transition-amplitude expansions. A "reduced" form of the Block-Horowitz expansion is described, and also a "completely linked" version. The latter is suggested as a tool for investigating superfiuid phenomena in nuclear matter, and for establishing contact with the Landau theory of Fermi liquids. The physical interpretation of these expansions is carefully studied, especially with regard to nuclear applications, to determine how they handle such "physical" features as antisymmetry, self-energy effects, wave-function renormalizations, and the distinction between "true" and "model" single-particle occupation probabilities. The problem of a correct theoretical definition for the shell-model potential is carefully examined, and a specific theory is presented. These expansions are seen to form a convenient and very powerful set of tools for studying the structure of actual nuclei.