## Quantum Theory of Atoms

James Richard Fromm

In 1905, the German patent clerk and physicist Albert Einstein made two separate, and very significant, contributions to our understanding of the nature of matter. In a paper on the photoelectric effect, he suggested that the Planck equation E(f) = hf described the energy of quanta of light called photons, which were one form in which light could be visualized as existing. If that is true, then the interactions of light and matter are interactions of single photons with single electrons in single atoms.

Conservation of energy in each individual interaction then requires that when a photon of energy hf falls upon a metal surface and ejects an electron the energy of the ejected electron must obey the relationship E(f) = hf = eX + mv2/2.

In this equation m is the mass of the electron, e is the electronic charge, v is the velocity with which the electron is ejected, and X is the work function of the metal. The work function of a metal, more commonly symbolized by the Greek letter phi, is an electrical potential, characteristic of the metal, which must be overcome in order to extract an electron from it. By 1916, experimental studies had accurately confirmed his prediction. The photoelectric effect is shown in the Figure below.

Einstein's other 1905 contribution was his special theory of relativity, of which one consequence is the Einstein equation E = mc2. Energy is conserved for any body of any mass moving at any velocity. The energy of a body of rest mass m0 moving at a velocity v is then given by

E = mc2 = m0c2 + mv2/2

The kinetic energy of a body will be a significant part of the mass-energy total of the body only if the body is moving at an appreciable fraction of the speed of light, c. Only electrons and nuclear particles have been found or made to travel at such speeds.

Example. Let us calculate the mass of an electron moving at one-half the speed of light. The rest mass of an electron is 0.910954 x 10-20 kg. The equation given above rearranges to

m - m0 = mv2/2c2, or 1 - (m0/m) = v2/2c2

At a velocity of 0.5c, 1 - (m0/m) = 0.125 and the ratio m0/m is 0.875. The mass of the electron moving at 0.5c is 1.143 times its rest mass, or 14.3% greater; the mass of the moving electron will be 1.041220 x 10-30 kg.

The relationship of the special theory of relativity to the interaction of light and matter came through the proposal of Louis de Broglie, who in 1923 argued that the Planck equation and the Einstein equation together described all matter. In equation form, the proposal of De Broglie is

hf = mc2. In other words, the properties of matter and the properties of electromagnetic radiation are both properties of the same real phenomenon.

The contribution of de Broglie in linking the understanding and mathematical equations used to describe waves and the understanding and mathematical equations used to describe matter (as particles) was taken up by the Austrian physicist Erwin Schroedinger. Schroedinger considered the mass of an electron as a mass corresponding to a standing wave. In 1925, Schroedinger proposed the complex equation now called the Schroedinger equation. Both this equation, which links the energies and positions of electrons within atoms, and its solutions are well beyond our scope, but the concepts it developed are not.

The Schroedinger equation can be solved exactly only for the simplest structure, the one-proton and one-electron structure of atomic hydrogen. The reason that the Schroedinger equation cannot be solved exactly for an atom which contains more than one electron is a mathematical problem which also appears in other areas such as astronomy: there is no exact solution to the equations describing the motion of more than two mutually interacting bodies. This three-body problem, one of the classical problems of mathematics, applies to the helium atom even if the nucleus is considered to be a single particle since the two electrons will mutually influence each other and both are influenced by the nucleus. No exact solution of the Schroedinger equation is possible for any of the atoms heavier than hydrogen, but methods of successive approximations can be used to obtain very good approximate solutions to the Schroedinger equations which describe the electrons in heavier atoms. Modern digital computers are almost mandatory in the very laborious calculations required to obtain accurate results for many-electron atoms by successive approximations. The Schroedinger equation and its approximate solutions can also be used to describe the behavior of electrons in molecules, as we shall see in the following section.

When solved, the Schroedinger equation gives the energies which can be assumed by an electron in an atom. Since light is emitted or absorbed by an atom when an electron moves from one permitted location to another, knowledge of the energies of the various levels available to an electron also gives us the emission and absorption spectra of the atom. This is similar to obtaining the differences in mountain heights, since the energy of the photon of light must equal the energy difference of the electron in its allowed locations.