Multipole matrix elements

Multipole matrix elements are quantum-mechanical expectation values between states of operators for charge or current multipole moments. They quantify how states couple to external fields and to radiation.

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Quantum state: A mathematical description containing the probabilities and phases of a quantum system's possible measurement outcomes. A matrix element compares the action of a multipole operator between two quantum states.
Wigner–Eckart theorem: A theorem that factors spherical-tensor matrix elements into angular coefficients and reduced matrix elements. It separates multipole transition strengths into symmetry-fixed factors and state-dependent quantities.
Electric dipole transition: A radiative transition driven by the electric-dipole component of the electromagnetic field. Its strength is set by an electric-dipole matrix element and it dominates many optical transitions.
Electric monopole transition: A transition involving a change in the charge-distribution monopole operator, possible between states of the same angular momentum and parity. Unlike radiative dipole transitions, an electric-monopole transition cannot emit a single real photon.
Matrix element: A complex number formed by placing an operator between two state vectors. Multipole matrix elements are a specialized use of this general quantum-mechanical quantity.
Clebsch–Gordan coefficients: Numbers that describe how two angular-momentum representations combine into states of definite total angular momentum. They supply the angular factors in multipole matrix elements under the Wigner–Eckart theorem.
Magnetic dipole transition: A radiative transition driven by the magnetic-dipole component of the electromagnetic field. Its matrix element describes transitions suppressed or forbidden at electric-dipole order.
Electric octupole transition: A radiative transition associated with the rank-three electric multipole component of the electromagnetic field. Its higher rank illustrates the progressively restrictive angular-momentum rules beyond quadrupole order.
Spherical tensor operator: An operator whose components transform under rotations like the components of an angular-momentum representation. Multipole operators are organized as spherical tensors, making their rotational selection rules explicit.
Selection rule: A constraint determining whether a transition or interaction is allowed by symmetries or conservation laws. Multipole rank and parity impose selection rules on transitions between states.