In quantum physics, which 1925 formulation represented observables as mathematical matrices?

In quantum physics, the 1925 formulation that represented observables as mathematical matrices was matrix mechanics.

Werner Heisenberg developed matrix mechanics in 1925, with important assistance from Max Born and Pascual Jordan. Instead of assigning classical-like orbits to electrons, the formulation used arrays of quantities related to observable transitions between energy states. The noncommuting multiplication of these arrays captured a crucial quantum feature: the order of operations can matter.

Matrix mechanics was the first complete and consistent formulation of quantum mechanics. Soon afterward, Erwin Schrödinger developed wave mechanics. Schrödinger initially presented the two approaches as different, but they were shown to be mathematically equivalent under suitable conditions.

The word “matrix” can make the theory sound like a theory of ordinary numerical tables. Quantum observables are represented by operators, often written as matrices after choosing a basis. Modern quantum mechanics uses this operator language broadly, while matrix mechanics remains historically important as one of the theory’s original forms.