Energy Level Splitting

Energy level splitting occurs in physics when the degenerate energy levels of two or more states are split because corresponding Hamiltonian's eigenvalues become different. This may occur because of external fields, quantum tunnelling between states, or other effects. The term is most commonly used in quantum theory in reference to the electron configuration in atoms or molecules.

The simplest case of level splitting is a quantum system with two states those unperturbed Hamiltonian is a diagonal operator: Ĥ₀ = E₀ I, where I is the 2 × 2 identity matrix. Eigenstates and eigenvalues (energy levels) of a perturbed Hamiltonian

will be:

• |0⟩: E₀ + ε  level, and
• |1⟩: E₀ − ε  level,

so this degenerate E₀ eigenvalue splits in two when ε ≠ 0. Though, if a perturbed Hamiltonian is not diagonal for that quantum states basis {|0⟩, |1⟩} , then Hamiltonian's eigenstates are linear combinations of that two states.

For such physical implementation as a charged spin-½ particle in an external magnetic field the z-axis of the coordinate system 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 eigenvalues 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, that level splitting may appear counter-intuitive, as in examples from chemistry below.