  Precision tests of the Standard Model and searches for beyond the Standard
Model physics often require nuclear structure input. There has been a
tremendous progress in the development of nuclear ab initio techniques capable
of providing accurate nuclear wave functions. For the calculation of
observables, matrix elements of complicated operators need to be evaluated.
Typically, these matrix elements would contain spurious contributions from the
center-of-mass (COM) motion. This could be problematic when precision results
are sought. Here, we derive a transformation relying on properties of harmonic
oscillator wave functions that allows an exact removal of the COM motion
contamination applicable to any one-body operator depending on nucleon
coordinates and momenta. Resulting many-nucleon matrix elements are
translationally invariant provided that the nuclear eigenfunctions factorize as
products of the intrinsic and COM components as is the case, e.g., in the
no-core shell model approach. An application of the transformation has been
recently demonstrated in calculations of the nuclear structure recoil
corrections for the beta-decay of 6He.
