Heisenberg Uncertainty Inequality and Breaking of Isospin Symmetry in Atomic Nuclei

The Heisenberg uncertainty inequality is used to derive a rigorous lower bound to the amount of isospin impurities, caused by the violation of isospin symmetry in $N=Z$ atomic nuclei, in terms of the difference between the neutron and proton radii and the sum of the charge exchange monopole strengths. We show that the inequality reduces to an identity if one employs a simplified mean field expression for the isovector component of the Coulomb force in first order perturbation theory. The uncertainty inequality is also employed to derive an upper bound to the isovector dipole moment in terms of the amount of isospin impurities, thereby providing an insightful connection between the violation of parity and isospin symmetries.