The Pauli Exclusion Principle (Preview)

Why two electrons cannot share the same quantum state

Throughout this module we have assigned four quantum numbers to each electron in a hydrogen-like atom: the principal number $n$ , the orbital angular-momentum number $\ell$ , the magnetic number $m_\ell$ , and the spin projection $m_s = \pm\tfrac{1}{2}$ . A natural question arises when we move from one electron to many: can two electrons occupy the same single-particle state, i.e. have identical values of all four quantum numbers?

Experiment — and a deep principle of quantum mechanics — gives an unambiguous answer.

The Pauli Exclusion Principle

Wolfgang Pauli stated the rule in 1925, before the full formalism of quantum mechanics was in place:

No two electrons in the same atom can have an identical set of quantum numbers $(n,\, \ell,\, m_\ell,\, m_s)$ .

More precisely, electrons are fermions — particles with half-integer spin. The many-body wave function of a system of identical fermions must be antisymmetric under the exchange of any two particles. If we label the two electrons $1$ and $2$ and their single-particle states $\alpha$ and $\beta$ , the two-particle state is

\Psi(1,2) = \frac{1}{\sqrt{2}}\bigl[\psi_\alpha(1)\,\psi_\beta(2) - \psi_\alpha(2)\,\psi_\beta(1)\bigr].

Setting $\alpha = \beta$ makes both terms identical, so $\Psi(1,2) = 0$ . A zero wave function is not normalizable and describes no physical state: the configuration does not exist. The antisymmetry requirement is therefore exactly equivalent to the exclusion principle.

Particles with integer spin — bosons — obey the opposite rule (symmetric wave function) and face no such restriction; many bosons can and do condense into the same state.

Consequences for atomic structure

The exclusion principle is not a minor detail; it is the reason matter is stable and has the structure it does.

Consider filling the $n = 1$ shell. The values available are $\ell = 0$ , $m_\ell = 0$ , and $m_s = +\tfrac{1}{2}$ or $m_s = -\tfrac{1}{2}$ . That gives exactly two distinct states. Once those two states are occupied — as in helium — a third electron cannot enter the $n = 1$ shell. It must begin the $n = 2$ shell instead. This sequential filling is captured by the Aufbau principle and directly explains the periodicity of the chemical elements.

For a shell with principal quantum number $n$ , the allowed values are $\ell = 0, 1, \ldots, n-1$ , and for each $\ell$ there are $2\ell + 1$ values of $m_\ell$ and two values of $m_s$ . The total number of electron states in the $n$ -th shell is therefore

\sum_{\ell=0}^{n-1} 2(2\ell+1) = 2\sum_{\ell=0}^{n-1}(2\ell+1) = 2n^2.

(The inner sum $\sum_{\ell=0}^{n-1}(2\ell+1) = n^2$ follows from the identity for consecutive odd integers.) So the $n = 1$ shell holds $2$ electrons, $n = 2$ holds $8$ , $n = 3$ holds $18$ , and so on — numbers that appear directly in the rows of the periodic table.

Spin and the fourth quantum number

The spin quantum number $m_s$ is what makes the $2n^2$ count work out. Without spin, each shell would hold only $n^2$ states and the periodic table would look entirely different. Pauli introduced the idea of a two-valued degree of freedom for the electron in the same 1925 paper, shortly before Uhlenbeck and Goudsmit gave it the physical interpretation of intrinsic angular momentum.

The four quantum numbers $(n, \ell, m_\ell, m_s)$ together form a complete set of quantum numbers for a single electron in a central potential. The exclusion principle says: in any multi-electron system, no two electrons may share all four values simultaneously.