The Magnetic Quantum Number mₗ

What is the magnetic quantum number?

When solving the Schrödinger equation for the hydrogen atom in spherical coordinates $(r, \theta, \phi)$ , the wavefunction separates into a radial part and an angular part. The angular part is further factored into a polar piece (depending on $\theta$ ) and an azimuthal piece (depending on $\phi$ ). Imposing that the azimuthal wavefunction must be single-valued — meaning it returns to the same value after a full rotation $\phi \to \phi + 2\pi$ — forces the separation constant to be an integer. That integer is the magnetic quantum number $m_\ell$ .

In compact notation the azimuthal factor takes the form

$\Phi(\phi) = e^{i m_\ell \phi},$

and the single-valuedness condition $\Phi(\phi + 2\pi) = \Phi(\phi)$ requires $e^{i m_\ell \cdot 2\pi} = 1$ , which holds if and only if $m_\ell$ is an integer.

The range of mₗ

The polar equation imposes a second constraint: the solutions that remain finite at the poles $\theta = 0$ and $\theta = \pi$ — the associated Legendre polynomials — exist only when

$m_\ell \in \{-\ell,\; -\ell+1,\; \ldots,\; 0,\; \ldots,\; \ell-1,\; \ell\}.$

Here $\ell$ is the orbital angular momentum quantum number ( $\ell = 0, 1, 2, \ldots, n-1$ , where $n$ is the principal quantum number). For a given $\ell$ , there are exactly $2\ell + 1$ allowed values of $m_\ell$ .

| $\ell$ | name | $m_\ell$ values | count | |--------|------|-----------------|-------| | 0 | s | 0 | 1 | | 1 | p | -1, 0, 1 | 3 | | 2 | d | -2, -1, 0, 1, 2 | 5 | | 3 | f | -3, -2, -1, 0, 1, 2, 3 | 7 |

Physical meaning

The $z$ -component of the orbital angular momentum $L_z$ is quantized:

$L_z = m_\ell \hbar.$

The full magnitude is $L = \hbar\sqrt{\ell(\ell+1)}$ , so $|m_\ell| \leq \ell$ reflects the fact that a component of a vector cannot exceed the vector's length. Because $L_z$ can take only $2\ell + 1$ discrete values, we say the orbital angular momentum is space-quantized.

Why mₗ matters for chemistry and quantum computing

Each allowed $m_\ell$ value labels a distinct orbital — a distinct spatial wavefunction — inside a sub-shell. Counting these orbitals gives the degeneracy that underlies the shell structure of the periodic table. For quantum computing, any two-level system (such as the $m_\ell = 0$ and $m_\ell = 1$ states of an atomic transition) can serve as a qubit, so understanding how $m_\ell$ enumerates states is also a bridge to quantum hardware.

Try it

This is a numerical exercise — your code should return a number. For $\ell = 3$ (the f sub-shell), how many distinct values of $m_\ell$ are allowed?

Run your code to see the quantum state.