Derivation of microscopic unified Bohr-Mottelson rotational model

In a previous article, we derived a microscopic version of the phenomenological Bohr-Mottelson unified rotational model for rotation about a single axis. In this article, we generalize the derivation to that for rotation about all the three axes. As in the previous derivation, we apply the nuclear Hamiltonian directly to the rotational-model wavefunction instead of using the usual canonical transformation. In this way, we avoid using redundant coordinates or imposing any constraints on the rotationally-invariant rotational-model intrinsic wavefunction. We show that, in the transformed nuclear Schrödinger equation, the Coriolis coupling term vanishes exactly only for a choice of the rotational-model Euler angles that is consistent with angle-angular momentum commutation relation and rotational invariance of the intrinsic wavefunction. For this choice of the Euler angles, the kinematic moment-of-inertia tensor, collective-rotation velocity field, and flow vorticity have the rigid-flow characteristics. This quantum rigid flow reduces to irrotational free-vortex flow in the limit of a single particle. We derive a microscopic effective rotation-intrinsic unified Schrödinger equation for the states of a rotational band that reduces to the phenomenological, unified, tri-axial quantum rigid-rotor model in the limit that the off-diagonal elements of the kinematic inertia tensor operator can be neglected. The model derivation shows that a multi-fermion system with unpaired or paired (quasi) particles rotates rigidly and a single-particle system rotates irrotationally if the intrinsic system is rotationally invariant.