matrix element — from formula anatomyIn quantum mechanics, a matrix element is a complex number that quantifies the probability amplitude for a transition between two quantum states under the action of an operator. It is a central object in perturbation theory, scattering theory, and the calculation of observable quantities like transition rates and cross-sections. The matrix element encodes the 'strength' of a coupling between states, such as the interaction of an atom with an electromagnetic field or the decay of a particle.
The concept was developed in the late 1920s by Max Born, Werner Heisenberg, and Pascual Jordan as part of matrix mechanics, the first complete formulation of quantum mechanics. It was later integrated into Dirac's bra-ket notation, which is now standard.
Formally, a matrix element is written as ⟨ψ_f|Ô|ψ_i⟩, where |ψ_i⟩ and |ψ_f⟩ are the initial and final quantum states, and Ô is an operator representing a physical quantity or interaction. The value is computed by integrating the product of the complex conjugate of the final state, the operator acting on the initial state, over all configuration space. The squared modulus of the matrix element gives the probability of the transition, which is then used in Fermi's Golden Rule to compute rates.
Matrix elements are used to calculate decay rates of radioactive nuclei, scattering cross-sections in particle physics, and absorption/emission spectra in atomic and molecular physics.
In particle physics, the matrix element for a process like electron-positron annihilation is often computed using Feynman diagrams, where each diagram contributes a term to the total matrix element.
| A(ω) | variable | spectral function value at energy ω — spectral function |
| ω | variable | energy variable (frequency) — energy joule |
| n | variable | index labeling excited eigenstates |
| |n⟩ | variable | excited eigenstate of the system — quantum state |
| |0⟩ | variable | ground state of the system — quantum state |
| O | variable | operator whose excitations are probed — operator |
| E_n | variable | energy of excited state n — energy joule |
| E_0 | variable | energy of ground state — energy joule |
| ⟨n|O|0⟩ | operator | inner product of states with operator applied — matrix element |
| δ(...) | operator | selects only contributions at exact energy difference — dirac delta |
| ∑_n | operator | adds contributions from all possible excitations — summation |
| E_n | variable | Perturbed energy of state n — energy joule |
| E_n^{(0)} | variable | Unperturbed energy of state n — energy joule |
| E_m^{(0)} | variable | Unperturbed energy of state m — energy joule |
| H' | variable | Perturbation Hamiltonian — hamiltonian joule |
| |n^{(0)}\rangle | variable | Unperturbed state n — quantum state |
| |m^{(0)}\rangle | variable | Unperturbed state m — quantum state |
| \sum_{m \neq n} | operator | Sum over all states m except n — summation |
| \langle m^{(0)} | H' | operator | Matrix element of perturbation between unperturbed states — matrix element |
| \Gamma_{i\to f} | variable | Transition rate from initial state i to final state f — transition rate per second |
| \psi_i | variable | Initial quantum state — wave function |
| \psi_f | variable | Final quantum state — wave function |
| H' | variable | Perturbation Hamiltonian — perturbation hamiltonian joule |
| \rho(E_f) | variable | Density of final states at energy E_f — density of states per joule |
| \hbar | constant | Reduced Planck constant, relating energy to angular frequency — reduced planck constant = 1.054571817e-34 joule second |
| \langle \psi_f | H' | operator | Matrix element of the perturbation between initial and final states — matrix element |