In 1922, Sommerfeld decided to spend a year the University of Wisconsin as a visiting professor, so Heisenberg took the opportunity to transfer to Göttingen to work with Max Born, succeeding Pauli as his assistant. With Born, Heisenberg constructed mathematical methods that furthered knowledge of the relationships of the periodic table. However, he again provoked the disagreement of Bohr and now Pauli, who had relocated to Copenhagen. Meanwhile, Heisenberg completed work for a dissertation on the less controversial topic of hydrodynamics. In particular, he focused on solving convoluted equations for the stability and turbulence of flowing fluids. Though Heisenberg's answers were incomplete, his dissertation was passed on the grounds that his handling was evidence of strong ability, and moreover because the subject had been an unusually difficult one for a student to tackle.
Heisenberg encountered much less sympathy at his oral examination, however. The stubborn experimentalist Willy Wien was shocked at Heisenberg's lack of preparation for questions on experimental physics. Wien perhaps took Heisenberg's incompetence personally, as he had spent much time lecturing on these very basic problems. Wien saw no reason to grant Heisenberg his degree, resulting in one of the great clashes between theory and experiment. Finally, Heisenberg was passed with the grade of III, cum laude, where a IV would have meant passing with no honors. Heisenberg was naturally shocked and embarrassed. He had been accustomed to handling oral examinations and defenses expertly, even in front of the biggest names in his field. He was so distraught that he took the evening train back to Göttingen, though he had planned to spend some time on leave in Munich.
As Heisenberg was completing his doctorate in July of 1923, Germany was facing increasing challenges to its economic and political stability. Inflation had made living conditions difficult for everyone, including the isolated scientists. Many had to turn to American benefactors to support their research. The political situation had a direct effect on physics as well. On two occasions, Heisenberg had lost the opportunity to see and meet Einstein because the latter had had to back out due to anti-Semitic threats. Although these circumstances troubled him, Heisenberg made no active opposition and continued believing that science should rise above politics.
Just as inflation had exploded to the point that a nationwide state of emergency was called, Heisenberg was proposing his groundbreaking Zeeman principle. The new theory was based on Bohr's own quantum postulate of 1913:
hv = E(2) - E(1)
This translates into h (Planck's constant) times v (frequency) equals the difference between the two energy states when an electron jumps from E(2) to E(1). Heisenberg saw that this postulate was inconsistent, for while the energies were calculated by classical Newtonian mechanics, the frequency was related not to the energy of one state but to the difference between two. Heisenberg, therefore, replaced the classical energies with a new function that served as an average, or integral, of the classical energy. From this new principle he was able to reproduce the results of previous findings, while also satisfying the Zeeman effect. More important, after applying it to his original core model–with alterations of course–Heisenberg was able to reconcile other principles he had earlier attempted to discard.
This time, before rushing to publication, Heisenberg first sent a copy to Bohr and asked for feedback from others as well. Bohr responded with an invitation to Copenhagen. Much of their time together was once again spent on philosophical debate, specifically avoiding technical details. Bohr also encouraged Heisenberg to plant himself in the institute library to read physics textbooks, so that his general physics understanding could be improved. As his friend Pauli had hoped, Heisenberg left after two weeks with a broader appreciation for physics.
Shortly afterward, Heisenberg coauthored a paper with Alfred Landé, modifying his Zeeman principle. With its publication, the science faculty at Göttingen voted on July 28, 1924, to habilitate the young scientist, certifying his qualification to lecture at all levels in Germany.