page_title: Energy level splitting
hierarchical_section_title: Energy level splitting
caption_attribution_description: English: Graph showing the Zeeman splitting in Rb-87, the energy levels of the 5s orbitals, including fine structure and hyperfine structure. Here, the quantum number F = J + I, where I is the nuclear spin. (for Rb-87, I = 3/2). The original splitting is due to zero-field fine structure + hyperfine splitting. Graph was produced in MATLAB by plotting the Breit-Rabi equation. The code can be viewed here: https://github.com/delton137/Breit_Rabi_Zeeman_Plotter
context_page_description: 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 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: Ĥ₀ = E₀ I, where I is the 2 × 2 identity matrix. Eigenstates and eigenvalues of a perturbed Hamiltonian
  will be:
  |0⟩: the E₀ + ε level, and
  |1⟩: the E₀ − ε level,
  so this degenerate E₀ eigenvalue splits in two whenever ε ≠ 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.
context_section_description: 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: Ĥ₀ = E₀ I, where I is the 2 × 2 identity matrix. Eigenstates and eigenvalues (energy levels) of a perturbed Hamiltonian
  will be:
  |0⟩: the E₀ + ε  level, and
  |1⟩: the E₀ − ε  level,
  so this degenerate E₀ eigenvalue splits in two whenever ε ≠ 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 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.
caption: Splitting of atomic energy levels in rubidium
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