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circular motion, the same reasoning regarding waves applies: Electron energies in different orbits cannot be arbitrarily close because of the fixed wavelength separating the wave function’s crests; changing the energy slightly causes the wave crests to go out of synch. If we want to change the energy, we must do so by a certain finite amount to ensure that crests fall on crests again. As an immediate consequence of the wavelike nature of electrons, this ladderlike form of allowed energies—the discreteness of the energy spectrum—distinguishes many properties of quantum theory (see figure 6). By contrast, in classical physics, which lacks the extra condition for the wavelength, one would expect that simply choosing the radius of the electron’s orbit would allow one to provide any arbitrary energy value.

In particular, energies cannot be arbitrarily small, bringing us back to atomic stability. Classically, the electron, like any electric charge in spherical orbit, would emit all its energy and quickly fall into the nucleus. For a quantum mechanical, wavelike electron, this condition does not arise, owing to the restrictive nature of the additional condition that already gave rise to discrete energies. More intuitively, we can relate the important contributions of quantum theory in this context to the necessity of counteracting forces. At the same time, this will lead to a further visualization of wave functions and their role.

6. For an electron bound in a hydrogen atom, there are infinitely many allowed energy values, which occupy a limited range. They must accumulate at some point, at which the discrete distribution looks like a continuum. Here, the special properties of quantum theory are suppressed—in stark contrast to the low energy values, where long distances between the allowed energy values are realized.

To stabilize a hydrogen atom, with a single proton in its nucleus, we need a new force counteracting the classical energy loss from electromagnetic radiation emitted by the orbiting electron. Exactly such a force arises as a result of the spread of the wave function when the distance between its center and the proton in the nucleus is smaller than the width of the wave. While the main part of the wave function remains concentrated mainly to one side of the proton, a sizable fraction is located by the opposite side. (The proton itself is not a classical point particle either, and must also be described by a wave function. Its spread, however, is much smaller because of the larger mass of the proton and does not play a role here: The heavy proton does not whirr around as wildly as the light-footed electron, and thus has a much more precisely defined position.)

Since the wave function describes the probability for the electron’s position, there is now a possibility for it to change place to the opposite side of the proton. There, it will still be electrically attracted toward the proton, but the force, as seen in figure 7, points in the opposite direction compared to what it was on the original side. Thus emerges a repulsive force, counteracting the classical electric attraction of two point particles to help stabilize atoms. Repulsion is largest for an electron wave function fitted around the proton like a sphere, a configuration that indeed corresponds to the quantum mechanically computed state of lowest possible energy.2 Here we have an example of how quantum theory can lead to new forces, and consequently to stability.

7. The extended wave function of an electron means that at any given time, it does not have to be strictly on one side of the proton. Electric attraction from different sides intuitively implies repulsive forces (opposing arrows) due to the quantum mechanical nature of particles, bringing about the stability of atoms.

In this way, separate states of atoms exist, with clearly distinguished energies not allowed to be arbitrarily small. Among those, the stable state, called the ground state, has the lowest energy; all others are more energetic and excited. From an excited state the atom can