Energy level splitting

Splitting of atomic energy levels in rubidium

In quantum physics, energy level splitting or a split in an energy level of a quantum system occurs when a perturbation changes the system. The perturbation changes the corresponding Hamiltonian and the outcome is change in eigenvalues; several distinct energy levels emerge in place of the former degenerate (multi-state) level. This may occur because of external fields, quantum tunnelling between states, or other effects. The term is most commonly used in reference to the electron configuration in atoms or molecules.

The simplest case of level splitting is a quantum system with two states whose unperturbed Hamiltonian is a diagonal operator: $Ĥ 0 = E 0 I$ , where $I$ is the $2 \times 2$ identity matrix. Eigenstates and eigenvalues (energy levels) of a perturbed Hamiltonian

{\hat {H}}_{\varepsilon }={\hat {H}}_{0}+\varepsilon \sigma _{3}={\begin{pmatrix}E_{0}+\varepsilon &0\\0&E_{0}-\varepsilon \end{pmatrix}}

will be:

|0⟩

: the

E 0 + ε

level, and

|1⟩

: the

E 0 - ε

level,

so this degenerate $E 0$ eigenvalue splits in two whenever $ε \neq 0$ . Though, if a perturbed Hamiltonian is not diagonal for this quantum states basis ${|0⟩, |1⟩}$ , then Hamiltonian's eigenstates are linear combinations of these two states.

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.

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.