How did Max Born and Pascual Jordan help develop Werner Heisenberg’s ideas?

Max Born and his assistant Pascual Jordan were instrumental in transforming Werner Heisenberg’s initial ideas into a formal mathematical framework for quantum mechanics.21

In 1925, Heisenberg developed a theory based on observable quantities that relied on noncommutative multiplication. Upon reviewing Heisenberg’s manuscript, Max Born recognized that the mathematical entities Heisenberg used to represent physical quantities—such as position, momentum, and energy—were matrices.23

Born and Jordan subsequently collaborated with Heisenberg to formalize this insight. They demonstrated that Heisenberg’s formalism could be expressed using matrix algebra, resulting in the publication of the seminal paper "Zur Quantenmechanik" ("On Quantum Mechanics"). By November 1925, the three physicists completed a follow-up paper, "Zur Quantenmechanik II," which is widely regarded as the foundational document of the new quantum mechanics. This collaboration successfully established the first working version of quantum theory, known as matrix mechanics.231