GCM Solver (Ver. 3.0): A Mathematica Notebook for Diagonalization of the Geometric Collective Model (Bohr Hamiltonian) with Generalized Gneuss–Greiner Potential

• Published in: Computation 2018-09, Vol.6 (3), p.48
• Main Authors: Ferrari-Ruffino, Fabrizio, Fortunato, Lorenzo
• Format: Article
• Language: English
• Subjects: Algebra; Angular momentum; Bohr Hamiltonian; Bohr–Mottelson model; collective model; Eigenvectors; Energy levels; Frankfurt model; Harmonic oscillators; Kinetic energy; Mathematical analysis; Mathematical models; Matrix methods; Nuclear physics; Operators (mathematics); Phase transitions; Predictions; Publishing; quadrupole; Spectrum analysis
• ISSN: 2079-3197; 2079-3197
• DOI: 10.3390/computation6030048

The program diagonalizes the Geometric Collective Model (Bohr Hamiltonian) with generalized Gneuss–Greiner potential with terms up to the sixth power in β . In nuclear physics, the Bohr–Mottelson model with later extensions into the rotovibrational Collective model is an important theoretical tool with predictive power and it represents a fundamental step in the education of a nuclear physicist. Nuclear spectroscopists might find it useful for fitting experimental data, reproducing spectra, EM transitions and moments and trying theoretical predictions, while students might find it useful for learning about connections between the nuclear shape and its quantum origin. Matrix elements for the kinetic energy operator and for scalar invariants as β 2 and β 3 cos ( 3 γ ) have been calculated in a truncated five-dimensional harmonic oscillator basis with a different program, checked with three different methods and stored in a matrix library for the lowest values of angular momentum. These matrices are called by the program that uses them to write generalized Hamiltonians as linear combinations of certain simple operators. Energy levels and eigenfunctions are obtained as outputs of the diagonalization of these Hamiltonian operators.