We generalize the notion of partial dynamical symmetry (PDS) to a system of interacting bosons and fermions. In a PDS, selected states of the Hamiltonian are solvable and preserve the symmetry exactly, while other states are mixed. As a first example of such novel symmetry construction, spectral features of the odd-mass nucleus
195Pt are analyzed.
Partial dynamical symmetry (PDS) is an algebraic structure in which a symmetry is neither exact nor fully broken: a subset of eigenstates keeps good quantum numbers and stays solvable while the rest mix. It is currently identified from spectroscopy. We ask whether it also leaves a purely structural mark in the eigenstates, and find, for the
sd-IBM Hamiltonians studied here, that it does, though not in the magnitude of entanglement. The diagnostic is the variance of a symmetry Casimir, the label variance $\Var\,\C[G]$, which shares its zero with a block-coherence entropy and a block impurity: all three vanish when a state occupies a single Casimir sector, a single irreducible representation once conjugate representations are resolved by the cubic Casimir. Resolved state by state, the variance is numerically zero on the symmetry-selected subset and large on the mixed states, isolating the PDS states from the wavefunctions alone; the separation does not close as the boson number grows and persists at Leviatan's first- and second-order critical points, whereas the magnitude of bipartite
s∣d entanglement does not separate the two families. We anchor the analysis in
168Er, reproducing Leviatan's fitted $\su3$ admixtures and his Hamiltonian-parameter-independent
γ→g B(E2) ratios. The diagnostic also has an exact dynamical form, a Casimir Loschmidt echo rigid on the symmetry-pure states, and we demonstrate its readout on an IBM superconducting processor at
N=3, where a single solvable/mixed exemplar reproduces the predicted pattern within finite-shot uncertainties.
Microscopic models, which embody the simplicity and significance of a dynamical symmetry approach to nuclear structure, are reviewed. They can reveal striking features of atomic nuclei when a symmetry dominates and solutions in domains that may otherwise be unreachable. 1. Overview of algebraic fermion models. A theory that invokes group symmetries is driven by an expectation that the wave functions of the quantum mechanical system under consideration can be characterized by their invariance properties under the corresponding symmetry transformations. But even when the symmetries are not exact, if one can find near invariant operators, the associated symmetries can be used to help reduce the dimensionality of a model space to a tractable size. Throughout the years, group-theoretical approaches have identified fundamental symmetries in light to heavy nuclei and achieved a reasonable reproduction of experimental data (for a review of fermion models, see 1). In addition, they provide theoretical predictions for nuclear systems including heavy unstable
The quantum mechanics of the hydrogen atom can be solved algebraically by the dynamical symmetry. In this note we introduced firstly Pauli's derivation of the complete relations among the various basic states with the same eigenvalue of the Hamiltonian. Secondly we also derived the same results by using algebraic method and some relations among the associated Legendre functions. As a byproduct, we also obtained a recursive relation of the radial wave functions. By using this recursive relation, we can derive the general radial wave function from the most simple radial wave function which has only an monomial term, much like deriving a general state from the highest weight state by using ladder operators.