An atomic orbital is a one-Electron spatial wavefunction used to describe an Electron's state in a chemical atom. For a one-electron hydrogen atom, orbitals can be exact energy eigenstates; descriptions of many-electron atoms instead use an orbital approximation. A normalized atomic orbital has , and the Born rule gives the position probability density . It is a function extending through space, not a prescribed classical orbit. [ref]
https://openstax.org/books/chemistry-2e/pages/6-3-development-of-quantum-theory. Flowers et al., Chemistry 2e, 6.3: atomic orbitals and the wavefunction description
The linear combination of atomic orbitals combines these single-centre functions to approximate a molecular orbital. The orbital overlap measures their inner product, and their relative quantum phase controls the interference shown in molecular orbital theory.
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An atomic orbital is a mathematical function that describes the wave-like behavior of an electron in an atom. This concept arises from quantum mechanics and is fundamental to understanding the structure of atoms and the arrangement of electrons within them. Here are some key points about atomic orbitals: 1. **Shape and Energy**: Atomic orbitals have distinct shapes (spherical, dumbbell-shaped, etc.) and energy levels. The shape and orientation of an orbital are determined by the quantum numbers that describe it.
In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.
Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.
This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table