the noble gases for the same value of n. This led Pauli to realize that the complicated numbers of electrons in closed shells can be reduced to the simple rule of one electron per state if the electron states are defined using four quantum numbers. For this purpose he introduced a new two-valued quantum number, identified by Samuel Goudsmit and George Uhlenbeck as electron spin. Connection to quantum state symmetry In his Nobel lecture, Pauli clarified the importance of quantum state symmetry to the exclusion principle: Among the different classes of symmetry, the most important ones (which moreover for two particles are the only ones) are the symmetrical class, in which the wave function does not change its value when the space and spin coordinates of two particles are permuted, and the antisymmetrical class, in which for such a permutation the wave function changes its sign...[The antisymmetrical class is] the correct and general wave mechanical formulation of the exclusion principle. The Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric with respect to exchange. If and range over the basis vectors of the Hilbert space describing a one-particle system, then the tensor product produces the basis vectors of the Hilbert space describing a system of two such particles. Any two-particle state can be represented as a superposition (i.e. sum) of these basis vectors: where each is a (complex) scalar coefficient. Antisymmetry under exchange means that . This implies when , which is Pauli exclusion. It is true in any basis since local changes of basis keep antisymmetric matrices antisymmetric. Conversely, if the diagonal quantities are zero in every basis, then the wavefunction component is necessarily antisymmetric. To prove it, consider the matrix element This is zero, because the two particles have zero probability to both be in the superposition state . But this is equal to The first and last terms are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey: or For a system with particles, the multi-particle basis states become n-fold tensor products of one-particle basis states, and the coefficients of the wavefunction are identified by n one-particle states. The condition of antisymmetry states that the coefficients must flip sign whenever any two states are exchanged: for any . The exclusion principle is the consequence that, if for any then