For a physical implementation such as a charged spin-½ particle in an external magnetic field, the z -axis of the coordinate system is required to be collinear with the magnetic field to obtain a Hamiltonian in the form above (the σ 3 Pauli matrix corresponds to z -axis). These basis states, referred to as spin -up and spin-down, are hence eigenvectors of the perturbed Hamiltonian, so this level splitting is both easy to demonstrate mathematically and intuitively evident.