Quantum Number


Principal Quantum Number $n$

The principal quantum number describes the electron shell, or energy level, of an electron.


Orbital Quantum Number $l$

The azimuthal quantum number, also known as the (angular momentum quantum number or orbital quantum number), describes the subshell, and gives the magnitude of the orbital angular momentum through the relation.

$$ \large L^2 \ = \ \hbar^2 l (l + 1) $$

where

  • $l = 0$: s orbital

  • $l = 1$: p orbital

  • $l = 2$: d orbital

  • $l = 3$: f orbital

The value of $l$ ranges from $0$ to $n − 1$, so the first p orbital ($l = 1$) appears in the second electron shell ($n = 2$), the first d orbital ($l = 2$) appears in the third shell ($n = 3$), and so on.

$$ \large l = 0, 1, 2, \cdots, n − 1 $$

Magnetic Quantum Number $m_l$

The magnetic quantum number describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis.

$$ \large L_z \ = \ m_l \hbar $$

The values of $m_l$ range from $-l$ to $l$, with integer intervals.

The s subshell ($l = 0$) contains only one orbital, and therefore the $m_l$ of an electron in an s orbital will always be 0. The p subshell ($l = 1$) contains three orbitals, so the $m_l$ of an electron in a p orbital will be −1, 0, or 1. The d subshell ($l = 2$) contains five orbitals, with $m_l$ values of −2, −1, 0, 1, and 2.


Spin Quantum Number $m_s$

The spin quantum number describes the intrinsic spin angular momentum of the electron within each orbital and gives the projection of the spin angular momentum $S$ along the specified axis.

$$ \large S_z \ = \ m_s \hbar$$

In general, the values of $m_s$ range from $−s$ to $s$, where $s$ is the spin quantum number, associated with the particle's intrinsic spin angular momentum.

$$ \large m_s = −s, −s + 1, −s + 2, \cdots, s − 2, s − 1, s. $$

An electron has spin number $s = \tfrac{1}{2}$, consequently $m_s$ will be $\pm \tfrac{1}{2}$, referring to "spin up" and "spin down" states.


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