theory of quantum numbers

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc. Since any quantum system can have one or more quantum numbers, it is a rigorous job to list all possible quantum numbers.

• The principal quantum number (n = 1, 2, 3, 4 ...) denotes the eigenvalue of H with the J² part removed^[ambiguous]. This number therefore has a dependence only on the distance between the electron and the nucleus (ie, the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells.
• The azimuthal quantum number (l = 0, 1 ... n−1) (also known as the angular quantum number or orbital quantum number) gives the orbital angular momentum through the relation . In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. In some contexts, l=0 is called an s orbital, l=1, a p orbital, l=2, a d orbital and l=3, an f orbital.
• The magnetic quantum number (m_l = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue, . This is the projection of the orbital angular momentum along a specified axis.

Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's Rules, which addresses the Pauli exclusion principle. A fourth quantum number with two possible values was added as an ad hoc assumption to resolve the conflict; this supposition could later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern-Gerlach experiment.

• The spin projection quantum number (m_s = −1/2 or +1/2), is the intrinsic angular momentum of the electron. This is the projection of the spin s=1/2 along the specified axis