Matrix mechanics

A formulation of quantum mechanics developed by Werner Heisenberg in which physical quantities are represented by matrices and observables need not commute.

Canonical commutation relation: The fundamental quantum relation stating that position and momentum operators have a nonzero commutator. It expresses matrix mechanics’ departure from simultaneous classical position and momentum values.
Werner Heisenberg: German theoretical physicist who formulated matrix mechanics in 1925 and co-founded quantum mechanics. His attempt to ground atomic theory in observable quantities launched the formulation.
Observable: A measurable physical quantity represented in quantum theory by a self-adjoint operator. Matrix mechanics assigns operators to quantities that experiments can measure.
Wave mechanics: Erwin Schrödinger’s formulation of quantum mechanics, in which states are represented by wavefunctions evolving under a differential equation. It describes the same quantum predictions using wavefunctions instead of matrices.
Commutator: An operation measuring the failure of two operators to commute, defined as their product difference in opposite orders. Commutators encode which measured quantities can have sharply defined values together.
Umdeutung paper: Heisenberg’s 1925 paper that recast quantum-theoretical relations using observable transition quantities. This paper introduced the core move that became matrix mechanics.
Operator (physics): A mathematical rule that acts on a state or function, often representing a measurable quantity or transformation. Matrices are concrete representations of the operators central to the theory.
Schrödinger equation: The equation governing the time evolution of a quantum state in wave mechanics. Its wavefunction evolution offers a contrasting representation of quantum dynamics.
Transition amplitude: A complex number whose squared magnitude gives the probability of a quantum transition between states. Heisenberg’s quantities describe transitions between stationary states rather than trajectories.
Max Born: German physicist whose work established the statistical interpretation of quantum mechanics and helped develop matrix mechanics. Born recognized that Heisenberg’s multiplication rules were those of matrices.

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