 In 1925, Born and Heisenberg formulated the matrix mechanics representation of quantum mechanics.On 9 July, Heisenberg gave Born a paper entitled "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" ("Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations") to review, and submit for publication.In the paper, Heisenberg formulated quantum theory, avoiding the concrete, but unobservable, representations of electron orbits by using parameters such as transition probabilities for quantum jumps, which necessitated using two indexes corresponding to the initial and final states.