Spin and the Fourth Quantum Number

By the early 1920s, the hydrogen spectrum was well described by three quantum numbers: the principal number $n$ , the orbital angular-momentum number $\ell$ , and the magnetic quantum number $m_\ell$ . Yet spectroscopists kept finding pairs of closely spaced lines where the theory predicted one. In 1925, Uhlenbeck and Goudsmit proposed that the electron carries an intrinsic angular momentum called spin, introducing a fourth quantum number.

Intrinsic angular momentum

Orbital angular momentum arises from a particle moving through space. Spin is different: it is intrinsic to the particle itself, not tied to any spatial orbit. It has no classical analogue (despite the suggestive name). An electron is a fundamental particle; it does not literally rotate the way a ball does. The magnitude of the electron's spin angular momentum is fixed by quantum mechanics at

$S = \hbar\sqrt{s(s+1)},$

where $s = 1/2$ is the spin quantum number for an electron. This value never changes; every electron in the universe has $s = 1/2$ . Substituting gives $S = \hbar\sqrt{3}/2$ .

The spin magnetic quantum number

While the total spin magnitude is fixed, its projection along any chosen axis (conventionally the $z$ -axis) is quantised. The allowed projections are

$S_z = m_s \hbar, \qquad m_s = -s, -s+1, \ldots, +s.$

For $s = 1/2$ this means only two values are possible:

$m_s = +\tfrac{1}{2} \quad (\text{spin up, } |{\uparrow}\rangle) \qquad \text{and} \qquad m_s = -\tfrac{1}{2} \quad (\text{spin down, } |{\downarrow}\rangle).$

The number of distinct $m_s$ values is $2s + 1 = 2$ , which is why a spin-1/2 particle is called a two-level system — it is the simplest non-trivial quantum system and the physical basis for the qubit.

The complete set of hydrogen quantum numbers

The electron in hydrogen is now described by four quantum numbers:

| Symbol | Name | Allowed values | |--------|------|----------------| | $n$ | principal | $1, 2, 3, \ldots$ | | $\ell$ | orbital angular momentum | $0, 1, \ldots, n-1$ | | $m_\ell$ | magnetic (orbital) | $-\ell, \ldots, +\ell$ | | $m_s$ | spin magnetic | $-1/2, +1/2$ |

A complete specification of a hydrogen eigenstate requires all four. For example, the ground state has $n = 1$ , $\ell = 0$ , $m_\ell = 0$ , and $m_s = \pm 1/2$ — giving two degenerate states with the same energy $E_1 = -13.6\,\text{eV}$ .

Why spin splits spectral lines

Even though spin does not change the energy in the pure Coulomb potential, it interacts with the electron's orbital motion through spin-orbit coupling: the electron's magnetic moment (due to spin) feels the magnetic field generated by its orbital motion around the proton. This coupling adds a small correction to the energy

$\Delta E_{\text{SO}} \propto \vec{L} \cdot \vec{S},$

lifting the degeneracy between states that differ only in $m_s$ . The result is that what appeared as a single spectral line splits into a closely spaced doublet, the fine structure of the spectrum. For the $3p$ level of sodium the splitting is about $0.002\,\text{eV}$ , producing the famous yellow sodium doublet at $589.0\,\text{nm}$ and $589.6\,\text{nm}$ .

Counting states

With spin included, the degeneracy of the $n$ -th energy level of hydrogen is $2n^2$ rather than $n^2$ . The factor of 2 comes from the two spin states available for each spatial orbital. For example:

$n = 1$ : $2 \times 1^2 = 2$ states.
$n = 2$ : $2 \times 2^2 = 8$ states.
$n = 3$ : $2 \times 3^2 = 18$ states.

This counting is essential for the Pauli exclusion principle, which determines how electrons fill orbitals in multi-electron atoms — and ultimately explains the structure of the periodic table.