The leading theory of the atom when Heisenberg entered the University of Munich in 1920 was the quantum theory of Bohr, Sommerfeld, and their co-workers. Although the theory had been highly successful in certain situations, during the early 1920s three areas of research indicated that this theory was inadequate and would need to be replaced. These areas included the study of light emitted and absorbed by atoms (spectroscopy); the predicted properties of atoms and molecules; and the nature of light itself--did it act like waves or like a stream of particles?
During his work in Munich, Göttingen, and Copenhagen Heisenberg engaged intensively in the theoretical study of all three of these areas of research. By 1924 physicists in Göttingen and Copenhagen were agreed that the old quantum theory had to be replaced by some new "quantum mechanics."
"All of my meagre efforts go toward killing off and suitably replacing the concept of the orbital path which one cannot observe." —Heisenberg, letter to Pauli, 1925
Heisenberg set himself the task of finding the new quantum mechanics upon returning to Göttingen from Copenhagen in April 1925. Inspired by Bohr and his assistant, H.A. Kramers, in Copenhagen, Pauli in Hamburg, and Born in Göttingen, Heisenberg's intensive struggle over the following months to achieve his goal has been well documented by historians. Since the electron orbits in atoms could not be observed, Heisenberg tried to develop a quantum mechanics without them. He relied instead on what can be observed, namely the light emitted and absorbed by the atoms. By July 1925 Heisenberg had an answer, but the mathematics was so unfamiliar that he was not sure if it made any sense. Heisenberg handed a paper on the derivation to his mentor, Max Born, before leaving on a month-long lecture trip to Holland and England and a camping trip to Scandinavia with his youth-movement group. After puzzling over the derivation, Born finally recognized that the unfamiliar mathematics was related to the mathematics of arrays of numbers known as "matrices." Born sent Heisenberg's paper off for publication. It was the breakthrough to quantum mechanics.
"The present paper seeks to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable."
—Heisenberg, summary abstract of his first paper on quantum mechanics
Together with his other assistant, Pascual Jordan, Born worked toward the further development of a quantum mechanics based upon the abstract mathematics of matrices. After Heisenberg returned from his youth-movement travels, the Göttingen work resulted in a famous "three-man paper" setting forth the details of a new matrix-based quantum mechanics, the "matrix mechanics." With the introduction of additional concepts (electron "spin" and Pauli's "exclusion principle"), Heisenberg, Born, Jordan, Pauli, and others showed that the new quantum mechanics could account for many of the properties of atoms and atomic events.
Einstein, however, objected to Heisenberg's approach in which the new theory was based only on observable quantities. Heisenberg recalled a conversation with Einstein on this issue following a lecture Heisenberg delivered in Berlin in 1926.
Independently, and somewhat later, Austrian physicist Erwin Schrödinger proposed another quantum mechanics, an alternative "wave mechanics" in 1926. The wave mechanics appealed to many physicists because it seemed to do everything that matrix mechanics could do but much more easily and seemingly without giving up the visualization of orbits within the atom. This unleashed an intense debate between the followers of the alternative versions of quantum mechanics that formed the background to the later uncertainty relations and the Copenhagen Interpretation.