Translationally invariant matrix elements of general one-body operators

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