The brilliant leap made by Heisenberg's quantum mechanics was made possible by a number of precursors. The BKS (Bohr, Kramers, Slater) theory provided the model that had to be disproved. An American scientist named John Slater had proposed the idea of a virtual radiation field that carried no energy but that was continually emitted and absorbed by virtual oscillators in an atom. Bohr and his assistant, Kramers, took this idea to suggest that a quantum jump could be induced by a virtual field without any real energy transference. Such a theory required the abandonment of the sacred laws of energy and momentum conservation, and violated a basic idea of causality, as absorption and emission of energy were no longer necessarily correlated.

Heisenberg himself was skeptical at first, but soon found the theory quite captivating and hoped to connect it with his own work. Back in Göttingen, Born was using a formula, derived by Kramers in connection with the BKS theory, to progress in quantum mechanics. However, not long after, when Heisenberg met Einstein for the first time, he was discouraged by the master's objections to the BKS theory. Einstein refused to accept the abandonment of certain essential principles, and he wrote to Bohr that if they had to be given up, "then I would rather be a shoemaker or an employee in a gambling casino than a physicist."

As Born was to be away in 1924, Bohr arranged for a longer Copenhagen retreat for Heisenberg. Soon, tension grew between Heisenberg and Kramers, mostly due to competition for Bohr's attention and approval. Nevertheless, their research coincided on one point, and Bohr pushed them to write a joint paper on the topic. The paper advanced a quantum theory of dispersion, which treated light as a wave rather than as quanta. The theory made particular use of the virtual oscillators; the point was to show how they correspond to harmonics in classical theory.

Soon afterward, in late 1924 and early 1925, Pauli would once again take on Heisenberg's core model, this time using a relativistic approach. The mass of an object increases with speed according to relativity, and the electrons in Heisenberg's model would be traveling fast enough to require consideration of this factor. But Pauli found no evidence of any mass change, and argued convincingly against Heisenberg's core model. Rather than feeling frustrated, Heisenberg praised Pauli's insight and looked for ways to build from it. Soon after Pauli's discovery, the BKS theory too was being disproved by experimental evidence.

Heisenberg's achievement came at a time when all of his colleagues were working on different tangents, and with no systematic approach to the answers they all desired. Nevertheless, their topics of research often complemented each other in unexpected ways. While Born and his new assistant Pascual Jordan were working on the quantum theory of aperiodic systems, Heisenberg returned to the problem of virtual oscillators in the atom. The amplitude of the oscillations could be broken down to a Fourier series, but what Heisenberg recognized was that this function had continued to use classical relationships. Assuming that the basic Fourier function held true on the quantum level, he then set about reinterpreting the frequencies and amplitude with quantum principles in mind, as he had in formulating the Zeeman principle.

Since the amplitudes of classical motions could be squared to find the intensity of the emitted radiation, Heisenberg determined to find a corresponding multiplication rule for the amplitude of virtual oscillators, which would yield the intensity of the spectral lines that had long given physicists trouble. The multiplication rule that Heisenberg devised looked familiar to Born as he critiqued the paper. What Born recognized was that the rule involved the same principle used in multiplying matrices. Before long Born, Jordan, and Heisenberg wrote the groundbreaking paper that expressed quantum physics in matrices. Their equations satisfied the prior principles of quantum physics while accomplishing the long-sought goal of quantifying the discrete energy states of an atom.