An electron orbital (or simply orbital) is the description of the behavior of an electron in an atom or molecule according to quantum mechanics.
Mostly qualitative introduction
According to quantum mechanics, an electron in an atom or molecule has three curious properties:
1. An electron is most accurately represented not as a particle, but as a wave. The aspect of the electron that "waves", or oscillates, is called its wavefunction and called ψ. The wavefunction is a solution of the Schrödinger equation.
2. An electron's position in space is not determined, but its probability of being at a particular place is. It is given by the square of the value of its wavefunction, ψ2, at that point. (ψ2 is a particular type of probability density function.)
3. An electron can only have certain discrete amounts of energy. The particular amount of energy that it has is determined by its wavefunction.
The combination of an energy level and a probability density function is called an orbital. The name is intentionally similar, but contrasting, to 'orbit'. An orbit determines the energy and the trajectory of a body (such as a planet) according to classical mechanics. An orbital specifies the energy of an electron, and the probability of its being located at any point. An orbital gives no information about an electron's trajectory or precise location, and according to the Uncertainty principle of quantum mechanics, such a trajectory and location do not exist.
Since an orbital specifies the places where an electron is most likely to be found, it is possible to define a region of space within which the electron is likely to exist (with ~90% probability, for example), and outside of which it is not likely to exist. This region of space is also sometimes called an 'orbital'.
Mathematical details
An orbital is an equilibrium state of an electron's wavefunction. More specifically, it is an eigenstate of the Schrödinger equation,
- HΨ = EΨ
where H is the Hamiltonian. When the Schroedinger equation for an atom is written in spherical coordinates, it separates easily into a radial part (a differential equation that depends only on the radius) and a spherical part (which depends only on the angles, the latitude and longitude).
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