In some sources, including my textbook it is cited that an atomic orbital is the 3 dimensional region where the probability of finding an electron is 90-95% . But, Other sources such as Wikipedia state otherwise. To quote Wikipedia, "In quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom" With relevance to the fact that an orbital is always a wave function I agree that it can be regarded as a mathematical function, But what is this reference to "a pair of electrons"This has confused me.What is the the true definition of an orbital? I am new to this forum and Chemistry, please excuse me if there's some misconception. Thanks
The mathematical definition is the most rigorous. The presence of the positively charged nucleus means that there is a pull on a negatively charged electron, which we describe with the potential energy. But electrons can only exist with certain discrete energy values. To do a full analysis you need to find the wavefunction that describes the electron. For a given electron, the wavefunction* at point (x, y, z) gives you the probability of finding the electron at the location; this is non-zero pretty much everywhere in space but is only significantly above zero in a very small region: this is what the first definition is getting at. Once you have the wavefunction you can calculate some properties, like the angular momentum. Since each electron has a discrete energy, each electron can be described by a set of four numbers, derived from the wavefunction: principle number (n), orbital angular momentum (l), magnetic angular momentum ($m_l$) and spin number ($m_s$). The spin number can be either +1/2 or -1/2 but we usually ignore it since normally that value doesn't affect the energy of the electron; this is the source of that "pair of electrons" business.
To summarize: An orbital is a specific state of an electron, characterized by its energy and various angular momenta and rigorously described by a mathematical function that satisfies Schroedinger's equation. This can be represented visually in many ways, such as showing the 95% probability solid, or as a line on a molecular-orbital diagram.
*Really this should be the wavefunction times its complex conjugate.