Quantum Numbers ℓ and m

With the spectrum established, we name and organize the two numbers that label every angular-momentum state. These quantum numbers are the bookkeeping language of atomic and nuclear physics.

The orbital quantum number ℓ

The first quantum number $\ell$ fixes the magnitude of the angular momentum through the $\hat{L}^2$ eigenvalue $\hbar^2\ell(\ell+1)$. For orbital angular momentum it takes non-negative integer values

Historically these are given spectroscopic letters: $\ell = 0$ is $s$, $\ell = 1$ is $p$, $\ell = 2$ is $d$, $\ell = 3$ is $f$, and then alphabetical ($g, h, \dots$).

The magnetic quantum number m

The second quantum number $m$ (sometimes written $m_\ell$) fixes the projection of the angular momentum on the $z$-axis through the $\hat{L}_z$ eigenvalue $\hbar m$. For a given $\ell$ it runs in unit steps,

It is called magnetic because it determines how an energy level splits in a magnetic field (the Zeeman effect): each value of $m$ acquires a slightly different energy proportional to $m$.

Multiplicity

Counting the values of $m$ from $-\ell$ to $+\ell$ gives

This is the degeneracy of the multiplet — the number of orthogonal states sharing the same $\ell$. Thus $s$ has 1 state, $p$ has 3, $d$ has 5, $f$ has 7, and so on. The pattern of odd numbers $1, 3, 5, 7, \dots$ is a direct fingerprint of integer angular momentum.

A worked enumeration

For $\ell = 2$ (a $d$ state), the allowed projections are

exactly $2(2) + 1 = 5$ values. Each $|\,\ell=2, m\,\rangle$ is a distinct quantum state; together they form one complete multiplet. The same recipe applies for any $\ell$, including the half-integer values that arise for spin.

Try it

For the $f$-orbital multiplet $\ell = 3$, count the allowed values of $m$ and return that integer (it should equal $2\ell + 1 = 7$).

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