The Sp(3,$R$) Sympletic Model: a comparison of exact and approximate matrix elements*

The Sp(3,$R$) symplectic model has a close physical connection to both the microscopic shell model and the collective deformation and rotational degrees of freedom, and it is a natural extension of the Elliot SU(3) model from single-shell to multi-shell dynamics. The Sp(3,$R$) Lie algebra --- which contains the angular momentum operators, the quadrupole and vibrational momentum operators and the quadrupole flow tensor operators --- is the smallest algebra containing both the shell model Hamiltonian and the rotor algebra. In the limit of large number of oscillator quanta, the Sp(3,$R$) algebra contracts to the U(3) boson algebra. For large values of the Casimir operator of the SU(3) subalgebra, the sp(3,$R$) algebra further contracts to the algebra of the collective coupled rotor-vibrator model. The exact Sp(3,$R$) matrix elements, calculated using the vector coherent state method, are compared with approximate matrix elements calculated in the U(3) boson limit.