### Key Concepts and Summary

Macroscopic objects act as particles. Microscopic objects (such as electrons) have properties of both a particle and a wave. Their exact trajectories cannot be determined. The quantum mechanical model of atoms describes the three-dimensional position of the electron in a *probabilistic* manner according to a mathematical function called a wavefunction, often denoted as *ψ*. Atomic wavefunctions are also called orbitals. The squared magnitude of the wavefunction describes the distribution of the probability of finding the electron in a particular region in space. Therefore, atomic orbitals describe the areas in an atom where electrons are most likely to be found.

An atomic orbital is characterized by three quantum numbers. The principal quantum number, *n*, can be any positive integer. The general region for value of energy of the orbital and the average distance of an electron from the nucleus are related to *n*. Orbitals having the same value of *n* are said to be in the same shell. The secondary (angular momentum) quantum number, *l*, can have any integer value from 0 to *n* – 1. This quantum number describes the shape or type of the orbital. Orbitals with the same principal quantum number and the same *l* value belong to the same subshell. The magnetic quantum number, *m _{l}*, with 2

*l*+ 1 values ranging from –

*l*to +

*l*, describes the orientation of the orbital in space. In addition, each electron has a spin quantum number,

*m*, that can be equal to

_{s}No two electrons in the same atom can have the same set of values for all the four quantum numbers.