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/(4πε0) ). 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 = 2r. 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, h2/me2. (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 1024 years (the age of the Earth is less than 1010 years).