Summarizing Quantum Numbers and Rules

Quantum Numbers and Rules Summary

• Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number \(n\) labels the basic states of a system and is given by

  \(n=1,\phantom{\rule{0.25em}{0ex}}2,\phantom{\rule{0.25em}{0ex}}3,\text{.}\text{.}\text{.}.\)

• The magnitude of angular momentum is given by

  \(L=\sqrt{l(l+1)}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}(l=0, 1, 2, ...,\phantom{\rule{0.25em}{0ex}}n-1),\)

  where \(l\) is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the \(z\)-axis is given by

  \({L}_{z}={m}_{l}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}({m}_{l}=-l,-l+1, ...,\phantom{\rule{0.25em}{0ex}}-1, 0, 1, ...\phantom{\rule{0.25em}{0ex}}l-1,\phantom{\rule{0.25em}{0ex}}l)\text{,}\)

  where \({L}_{z}\) is the \(z\)-component of the angular momentum and \({m}_{l}\) is the angular momentum projection quantum number. Similarly, the electron’s intrinsic spin angular momentum \(S\) is given by

  \(S=\sqrt{s(s+1)}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\text{(}s=1/2\phantom{\rule{0.25em}{0ex}}\text{for electrons),}\)

  \(s\) is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the \(z\)-axis is given by

  \({S}_{z}={m}_{s}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}({m}_{s}=-\cfrac{1}{2},+\cfrac{1}{2})\text{,}\)

  where \({S}_{z}\) is the \(z\)-component of spin angular momentum and \({m}_{s}\) is the spin projection quantum number. Spin projection \({m}_{s}\text{=+}1/2\) is referred to as spin up, whereas \({m}_{s}=-1/2\) is called spin down. This table summarizes the atomic quantum numbers and their allowed values.

Glossary

quantum numbers

the values of quantized entities, such as energy and angular momentum

angular momentum quantum number

a quantum number associated with the angular momentum of electrons

spin quantum number

the quantum number that parameterizes the intrinsic angular momentum (or spin angular momentum, or simply spin) of a given particle

spin projection quantum number

quantum number that can be used to calculate the intrinsic electron angular momentum along the \(z\)-axis

z-component of spin angular momentum

component of intrinsic electron spin along the \(z\)-axis

magnitude of the intrinsic (internal) spin angular momentum

given by \(S=\sqrt{s(s+1)}\cfrac{h}{2\pi }\phantom{\rule{1.00em}{0ex}}\)

z-component of the angular momentum

component of orbital angular momentum of electron along the \(z\)-axis