Geometric model Potential energy: neglect higher-order termsneglect … depends on 2 internal shape variables A B oblate prolate spherical x y …corresponding tensor of momenta quadrupole tensor of collective coordinates (2 shape param’s, 3 Euler angles ) A Bohr, 1952, G Gneuss, W Greiner 1971,.... See e.g.: JM Eisenberg, W Greiner: Nuclear Theory, vol.1 (North Holland, Amsterdam, 1970). Cartesianpolar Motion in the xy- plane represents evolution of nuclear shape in the principal axis system (PAS). Shape “phases”

Scaling properties 3 independent scales Energy TimeCoordinates => only 1 essential parameter arbitrary choice: (a) A =variable, B,C,K=1 (b) B =variable, A= –1, C,K=1 A=+1, C,K=1 (c) C =variable, A= –1, B,K=1 A=+1, B,K=1 4 external parameters Dynamically equivalent classes determined by the dimensionless parameter (a) (b–) (b+) In the quantum case, energy & time scales connected by the Planck constant => 2 essential parameters

Angular momentum Principal Axes System Special case: Dynamics is fully determined by 2 coordinates and the associated 2 momenta 2D system 5D system

Quantum dynamics Now only the restricted case: J=0 2 physically important quantization options (a) 2D system (b) 5D system restricted to 2D (true geometric model of nuclei) Although both options have the same classical limit, they yield different quantum spectra... with an additional ansatz on the angular wave function (to avoid quasi-degeneracies due to the three-fold symmetry of V )