solutions of the Schrödinger equation associated with the first problem are the so-called plane waves, which cover a continuum of possible energies, from zero to infinity (we speak in this case of a continuous spectrum). The solutions of the problem with the central force field are instead associated with discrete energy values, given by the well-known formula: $E_n= -E_I/n^2$, where $n=1,2,\dots$, and $E_I\approx 13.6\, eV\approx 22 \times 10^{-19}$ J is the ionization energy of the hydrogen atom. One speaks in this case of a discrete spectrum, corresponding to the known (emission and absorption) spectral lines, observed experimentally.

Now, as regards the possibility of acquiring different positions in space, what really matters is the movement of the center of mass of the hydrogen atom, which, as previously mentioned, evolves according to free evolution. What we are interested in is to calculate the spatial spreading of the wave packet associated with the center of mass variable, since such spreading will provide us a good estimate of the probability of observing the hydrogen atom at a certain distance from the place where it was initially observed, say at time $t = 0$ s.

The spatial spreading of the wave packet at time $t$ 