Calculations of widths in the problem of decay by proton emission

We develop a new fully quantum method for determination of widths for nuclear decay by proton emission where multiple internal reflections of wave packet describing tunneling process inside proton–nucleus radial barrier are taken into account. Exact solutions for amplitudes of wave function, penetrability $T$ and reflection $R$ are found for $n$-step barrier (at arbitrary $n$) which approximates the realistic barrier. In contrast to semiclassical approach and two-potential approach, we establish by this method essential dependence of the penetrability on the starting point $R_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}$ in the internal well where proton starts to move outside (for example, for ${}_{73}^{157}\mathrm{T}\mathrm{a}$ the penetrability is changed up to 200 times; accuracy is $|T+R-1| < 1.5 \cdot 10^{-15}$). We impose a new condition: in the beginning of the proton decay the proton starts to move outside from minimum of the well. Such a condition provides minimal calculated half-life and gives stable basis for predictions. However, the half-lives calculated by such an approach turn out to be a little closer to experimental data in comparison with the semiclassical half-lives. Estimated influence of the external barrier region is up to 1.5 times for changed penetrability.