The restriction of the energy levels of the electrons is explained in terms of the wavelike properties of their motions: electrons occupy only those orbits for which their associated wave is a standing wave (i.e., the circumference of the orbit is exactly equal to a whole number of wavelengths) and thus can have only those energies that correspond to such orbits. Moreover, the electrons are no longer thought of as being at a particular point in the orbit but rather as being spread out over the entire orbit. Just as the results of relativity approximate those of Newtonian physics when ordinary speeds are involved, the results of the quantum theory agree with those of classical physics when very large "quantum numbers" are involved, i.e., on the ordinary large scale of events; this agreement in the classical limit is required by the correspondence principle of Niels Bohr. The quantum theory thus proposes a dual nature for both waves and particles, one aspect predominating in some situations, the other predominating in other situations.

- Introduction
- Relationship of Energy and Matter
- Dual Nature of Waves and Particles
- Evolution of Quantum Theory
- Bibliography

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.