The many body Hamiltonian

The electronic and nuclear degrees of freedom of a system are described by the many body Hamiltonian, 

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The problem is to get a computable expression for the ground state wave function without solving the Schrodinger equation for the many body hamiltonian of (1), obviously an impossible task for any non trivial system. As usual in many body problems, we can resort to the variational principle which states that the energy of any proper trial state ) will be greater or equal to... 

The minimum of E[n] is thus attained for the ground-state density. All other extrema of this functional correspond to densities of excited states, but the excited states obtained in this way do not necessarily cover the entire spectrum of the many-body Hamiltonian... 

In the full Hilbert space the many-body Hamiltonian H is of the form... 

The T =0 time-dependent mean-field theory currently provides the best description of nuclear dynamics at low energies [5,6]. We consider two single-particle operators, Q, P interpreted as a collective coordinate and a collective momentum. Their nature depends on the kind of motion that we want to focus upon. We require that Q Q and P-r — P under time reversal and that IP, Q] 0. We then form a constrained Hartree-Fock (CHF) calculation on the many-body Hamiltonian H by minimizing the functional... 

As a simple application of the many-body hamiltonian in form, we consider the problem of N equivalent mutually attracting (gravitating) bosons. That is, we consider the above with all m,- = m, and all Cij = —g for convenience we set m = 1, but retain g through the calculations. This problem is interesting since it is a true many-body problem which can be solved exactly (and easily) for 2 = 1 and for D — oo, though apparently not for D = 3. Here we will solve the problem in the large-2 limit both exactly and in the Hartree approximation. 

The second way in which atomic interactions profoundly affect the condensate properties is through their effect on the energy. The effect of atom-atom interactions in the many-body Hamiltonian can be parameterized in the T —> 0 limit in terms of the two-body scattering length. This use of the exact two-body T-matrix in an energy expression is actually a rigorous procedure, and can be fully justified as a valid approximation.One simple theory which has been very successful in characterizing the basic properties of actual condensates is based on a mean-field, or Hartree-Fock, description of the condensate wave function, which is found from the equation ... 

Following Anderson (1963), the model Hamiltonian of Eq. (17.12) can be deduced easily by using second quantization. Let us consider a simple model for a system of N electrons described within a basis set of N orthonormal spatial orbitals. Each electron is assumed to be localized on one orbital (site). The many-body Hamiltonian of this model is ... 

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