Once Before Time - Martin Bojowald [30]
An atom’s ground state is not of this kind, because it shows strong quantum behavior; otherwise it could not solve the classical problem of instability. Similarly, the near excited states with slightly higher energies, in which quantum jumps are still sizable, are not semiclassical. But when atoms go to higher energies, thus stronger excitations, quantum jumps become milder. This must happen because there are infinitely many energy levels, but only a finite amount of energy is required to remove the electron from the atom. Indeed, atoms can be ionized by adding a sufficient amount of energy, removing one or more electrons from the orbits to leave a positively charged ion behind. All the infinitely many energy levels occupy a finite energy range and must crowd together at some point (figure 6). This is the place where distances between neighboring energy levels become very small, a behavior much closer to the classical one.
Wave functions for an atom in this dense energy range can easily behave classically. As in figure 8, such a wave function can be imagined as a single crest with an extremely sharp top. In time, it follows a trajectory such as the classical equations of motion would have determined. What we have here is a wave function of a certain spread; strictly speaking, the electron’s position is still not precisely defined, but if we do not look too closely, the extension of the wave function can, after a first approximation, be ignored, allowing us to view the wave’s peak position as the “location” of an electron. In this way, the classical theory presents as a limiting case of the more generally valid quantum theory.
8. A wave moving along the dashed curve, spreading out in the course of time (upward).
If we do look closely, even semiclassical waves do not exactly follow the classical laws of motion. Watching the motion of a semiclassical wave function over longer time intervals, we will notice deviations from the classical trajectory, first small but then becoming more and more pronounced. The wave function as a whole is, after all, subject to temporal evolution as given by the Schrödinger equation (again, a differential equation for the change of the wave function during the smallest changes of time). Not only the peak position of the wave but also its width and shape matter. Only in exceptional cases can the peak follow the classical trajectory exactly; otherwise it is yanked away from the classical curve by the changing shape of the whole wave function. As in the case of water waves, an uneven ground influences the velocity of various parts of the wave differently. On one side of the peak it may fall back behind the maximum, and it might seem as if the slope of the wave mountain will come sliding down. The mountaintop is made to move by the changing shape of the whole mountain, in addition to its rigid motion; both contributions are illustrated in figure 9. For a semiclassical wave function, such deviations caused by the shape are small at any given time. Initially, it may seem that the classical trajectory would be realized precisely. Before long, however, deviations do add up and corrections must be incorporated for an exact description. Thanks to those deviations from classically expected curves, effects of quantum theory can often be confirmed very precisely by experiments.
9. Depending on the ground, a wave can move rigidly without changing its shape (left), or it can change its height differently at different positions. The wave crumbles, for instance on its left-hand side, such that its maximum seems to move to the right. Such changes affecting parts of the wave do not occur in classical physics, and yet they have an influence on the position of the maximum. To describe the position precisely, quantum corrections to the classical