In the history of physics, "On the quantum-theoretical reinterpretation of kinematical and mechanical relationships" , also known as the Umdeutung (reinterpretation) paper, was a breakthrough article in quantum mechanics written by Werner Heisenberg, which appeared in Zeitschrift für Physik in September 1925.
In the article, Heisenberg tried to explain the energy levels of a one-dimensional anharmonic oscillator, avoiding the concrete but unobservable representations of electron orbits by using observable parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states.: 153
Mathematically, Heisenberg showed the need of non-commutative operators. This insight would later become the basis for Heisenberg's uncertainty principle.
This article was followed by the paper by Max Born and Pascual Jordan of the same year, and by the 'three-man paper' (German: drei Männer Arbeit) by Born, Heisenberg and Jordan in 1926. These articles laid the groundwork for matrix mechanics that would come to substitute old quantum theory, leading to the modern quantum mechanics. Heisenberg received the Nobel Prize in Physics in 1932 for his work on developing quantum mechanics.