U(5) ⊃ O(5) ⊃ O(3) and the exact solution for the problem of quadrupole vibrations of the nucleus

Abstract

Over twenty years ago A. Bohr discussed the quantum mechanical problem of the quadrupole vibrations in the liquid drop model of the nucleus. States of definite angular momentumL could not be obtained exactly except when L=0,3. In the present paper we indicate how we can determine states for arbitrary angular momentumL and definite number of quanta ν in terms of polynomials of the creation operators characterized by irreducible representation (IR) of the chain of groups U(5) ⊆O(3). We furthermore characterize the states by a definite IR λ of O(5) by replacing the creation operators by traceless ones. These states are fully determined by an extra label μ that gives the number of triplets of traceless creation operators coupled to angular momentum zero. We show then how all the wavefunctions of the problem discussed by Bohr can be obtained in a recursive fashion and briefly discuss some of their applications.