Use of the reciprocal basis in neutral meson mixing

In the presence of $\mathrm{CP}$ violation, the effective Hamiltonian matrix describing a neutral meson-antimeson system does not commute with its Hermitian conjugate. As a result, this matrix cannot be diagonalized by a unitary transformation and one needs to introduce a reciprocal basis. Although known, this fact is seldom discussed and almost never used. Here, we use this concept to highlight a parametrization of the Hamiltonian matrix in terms of physical observables, and we show that using it reduces a number of long and tedious derivations into simple matrix multiplications. These results have a straightforward application for propagation in matter. We also comment on the (mathematical) relation with neutrino oscillations.