^{–15}m.

^{1}It is the wave nature of the electron that prevents this collapse. If the electron spiraled into the proton, it would be confined to a smaller and smaller region of space, which means that its associated wavelength would have to become smaller and smaller. According to the de Broglie equation, smaller wavelength means larger momentum, which in turn means more energy of motion (kinetic energy). This is the crux of the matter. The wave nature of the electron means that it can be confined to a small region only if it has a high kinetic energy. Because of the electrical attraction, the electron “wants” to be near the proton. But in order to have the smallest possible energy, it “wants” to have a very large wavelength and be spread over a large region of space. These two opposing influences—the proton’s force tending to pull it in, its wave nature tending to push it out—reach a point of balance for the electron wave spread over a certain distance; this distance happens to be about 10

^{–10}m and determines the size of the atom.

The argument can be expressed mathematically in the following way. The electrical potential energy of the electron-proton system, negative because the force is attractive, is given by

where *e* is the magnitude of each charge, *r* is the distance between the particles, and *k* is the constant that appears in Coulomb’s law of electrical force (often written 1/(4**πε**_{0}) ). The electron’s kinetic energy is

According to the de Broglie equation, the electron momentum* p* can be replaced by *h/***λ***,* giving for the kinetic energy

Now you may assume that the wavelength of the electron is about equal to the diameter of the atom (the idea of nonlocalizability), or λ = 2*r*. This means that the electron kinetic energy can be expressed in terms of the atomic radius by

This equation states that as the atomic size decreases, the kinetic energy necessarily increases (because of the wave nature of the electron). At a very large radius, this kinetic energy is negligible and the potential energy dominates. Here the electron is drawn inward. At a very small radius, the kinetic energy overwhelms the potential energy and the electron tends to fly outward. The electron takes for its domain of motion a distance such that its kinetic and potential energies are comparable in magnitude:

Solution of this approximate equation for the atomic radius gives

Numerically, the right side of this formula is equal to 2.6 × 10^{–10} m. It is important to be aware that this derivation, although mathematical, is only qualitative. It is an order-of-magnitude calculation. In truth the electron’s average kinetic and potential energies are not exactly equal, nor is the electron wavelength exactly twice the atomic radius. Nevertheless, the derivation is significant, for it shows that the size of an atom is determined by a certain combination of fundamental constants, *h*^{2}/*me*^{2}. (Because of the approximate nature of the derivation, the particular numerical factor 8 appearing in the final formula above is without special significance.) Until Planck’s constant *h *entered physics, there was no possible way to explain the size of atoms.

1^{} The thoughtful reader might translate this situation to the solar system, and wonder why the Earth, which is described by classical laws, does not spiral into the Sun. The answer: It does! The difference, an all-important one, is a matter of time. The electron, according to classical reasoning, should spiral into the proton in about 10^{–8 }s, the Earth into the Sun in about 10^{24 }years (the age of the Earth is less than 10^{10 }years).