Starting from here the charge of the electron is e < 0 . The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong non-uniform magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with L z = −1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of 1 ⁄ 2. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so: H = 1 2 m [ σ σ ⋅ ⋅ ( p − − e c A ) ] 2 + e ϕ ϕ .