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 in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>N</a:mi> <a:mo>=</a:mo> <a:mi>Z</a:mi> </a:mrow> </a:math> atomic nuclei, caused by the violation of isospin symmetry. The bound is fixed by the difference between the neutron and proton radii and the sum of the charge-exchange monopole strengths. It can be used to check the consistency of advanced many-body calculations accounting for the breaking of isospin symmetry. The uncertainty inequality is also employed to derive an upper bound to the isovector dipole moment in terms of the amount of isospin impurities, providing an insightful connection between the violation of parity and isospin symmetries.