Quantum_ Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar [108]
Not long after starting the hunt, Schrödinger thought he had bagged just such an equation. However, when he applied it to the hydrogen atom, the equation churned out the wrong answers. The root of the failure lay in the fact that de Broglie had developed and presented wave-particle duality in a manner consistent with Einstein's theory of special relativity. Following de Broglie's lead, Schrödinger started out by looking for a wave equation that was 'relativistic' in form, and found one. In the meantime, Uhlenbeck and Goudsmit had discovered the concept of electron spin, but their paper did not appear in print until the end of November 1925. Schrödinger had found a relativistic wave equation, but unsurprisingly it did not include spin and therefore failed to agree with experiments.15
With the Christmas vacation fast approaching, Schrödinger began to concentrate his efforts on finding a wave equation without worrying about relativity. He knew that such an equation would fail for electrons travelling at speeds close to that of light where relativity could not be ignored. But for his purposes such a wave equation would do. Soon, however, there was more than just physics on his mind. He and his wife Anny were having another of their sustained bouts of marital turbulence, one that was lasting longer than most. Despite the affairs and talk of divorce, each seemed incapable and unwilling to permanently part from the other. Schrödinger wanted to escape for a couple of weeks. Whatever excuse he gave his wife, he left Zurich for the winter wonderland of his favourite Alpine resort, Arosa, and a rendezvous with an ex-lover.
Schrödinger was delighted to be back in the familiar and comfortable surroundings of the Villa Herwig. It was here that he and Anny had spent the previous two Christmas holidays, but there was hardly time enough over the next two weeks to feel guilty as Schrödinger spent his passion with his mysterious lady. However distracted he may have been, Schrödinger made time to continue the search for his wave equation. 'At the moment I am struggling with a new atomic theory', he wrote on 27 December.16 'If only I knew more mathematics! I am very optimistic about this thing and expect that if I can only … solve it, it will be very beautiful.' Six months of sustained creativity were to follow during this 'late erotic outburst' in his life.17 Inspired by his unnamed Muse, Schrödinger had discovered a wave equation, but was it the wave equation he was seeking?
Schrödinger did not 'derive' his wave equation; there was just no way to do it from classical physics that was logically rigorous. Instead he constructed it out of de Broglie's wave-particle formula that linked the wavelength associated with a particle to its momentum, and from well-established equations of classical physics. As simple as it sounds, it required all of Schrödinger's skill and experience to be the first to write it down. It was the foundation on which he built the edifice of wave mechanics in the months ahead. But first he had to prove that it was the wave equation. When applied to the hydrogen atom, would it generate the correct values for the energy levels?
After returning to Zurich in January, Schrödinger found that his wave equation did reproduce the series of energy levels of the Bohr-Sommerfeld hydrogen atom. More complicated than de Broglie's one-dimensional standing electron waves fitted into circular orbits, Schrödinger's theory obtained their three-dimensional analogues – electron orbitals. Their associated energies were generated as part and parcel of the acceptable solutions of Schrödinger's wave equation. Banished once and for all were the ad hoc additions required by the Bohr-Sommerfeld quantum atom – all the previous tinkering and tweaking that sat uneasily now emerged naturally from within the framework of Schrödinger's wave mechanics. Even the mysterious quantum jumping between orbits by an