A different (but somewhat arduous) proof can be given applying the Lorentz transformation directly to Y₊(x, t), including its non-constant amplitude function ÷Y(±)÷, by applying Lorentz contraction to the wave length l occurring in both, etc. Recall that the Lorentz transformation applied to an exponential (wave) function representing a "free particle", multiplied with an arbitrary wave packet function, destroys the right, by wave theory required, 2:1 relationship between group velocity v_g and phase velocity v, as de Broglie pointed out. This is not the case with the wave functions of quantum particle motions parametrized by Y₊(x, t). They all have v_g = dv/dk = dhv/dhk = d(mv²)/d(mv) = 2v.