Energy level splitting

This may occur because of external fields, quantum tunnelling between states, or other effects.

The simplest case of level splitting is a quantum system with two states whose unperturbed Hamiltonian is a diagonal operator: Ĥ0 = E0 I, where I is the 2 × 2 identity matrix.

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 σ3 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.

But in cases where the choice of state basis is not determined by a coordinate system, and the perturbed Hamiltonian is not diagonal, a level splitting may appear counter-intuitive, as in examples from chemistry below.