137 - Arthur I. Miller [52]
Insight into the exclusion principle
The helium atom has two electrons. Using his quantum mechanics Heisenberg showed how the two sets of spectra arise. To elucidate his result and speed up his calculation of numerical values for the spectral lines, he used Schrödinger’s wave functions—the solutions to the Schrödinger equation—for both the spins and positions of these two electrons. The total wave function is the result of multiplying these two wave functions together. But there are many possible ways of constructing the total wave function for these two electrons.
Heisenberg found that only one sort produced the two distinct groups of spectral lines characteristic of the helium atom. This particular wave function had a unique property. It changed its sign when the spins and positions of the electrons were swapped. It was antisymmetrical, which also meant that it went to zero if the electrons had the same spins or positions.*
What was nature’s selection device for choosing these two sets of wave functions for the two spectra out of the several possible ones? Heisenberg was stumped. Something strange was going on here. Perhaps it related to Pauli’s exclusion principle, according to which no two electrons could have the same spin and position. If they did then one of the two wave functions that make up the total wave function—either for their positions or for their spins—would have to become zero. Perhaps that was the way nature selected the wave function suitable for a particular system of electrons. Thus Heisenberg realized that the exclusion principle was related to the symmetry property of the wave functions for a collection of electrons, in this case two electrons. It was a step forward in exploring its implications beyond making sense of the periodic table of elements.
It was a typical Heisenberg project. He chose a fundamental problem—in this case to understand the spectrum of the helium atom—and then let his intuition lead him into new terrain: the symmetry property of wave functions whether they are symmetric or antisymmetric. Thus he realized how essential the exclusion principle was for quantum mechanics: without it quantum mechanics could not be complete.
There was also the problem that had been Pauli’s original bête noir from his PhD thesis, in which he showed that Bohr’s theory of the atom failed to produce a stable hydrogen-molecule ion, H+2, even though it existed in nature. This problem vexed Born and Heisenberg as well. Pauli wrote to his friend Wentzel, “In Copenhagen sits a gentleman who is calculating H+2 according to Schrödinger.” The “gentleman” was the Danish physicist Øvind Burrau who, as Pauli pointed out, started directly with Schrödinger’s wave mechanics as opposed to starting from the quantum mechanics as Heisenberg had done and used Schrödinger’s wave mechanics only for calculations. As a result he was able to solve the problem simply. Heisenberg wrote to Pauli that, in his opinion, Burrau had straightened out the situation and mentioned the symmetry properties of the wave functions that Burrau had deduced. Perhaps Heisenberg had hoped to find a solution starting from his quantum mechanics. But these once-key problems had become mere calculations now that the correct atomic physics had been worked out.
Although problem after problem that had resisted solution using the old Bohr theory was now being solved by atomic physics, the meaning of the theories used—Heisenberg’s quantum mechanics and Schrödinger’s wave mechanics—was still not understood. And the tension between the two factions was growing.
To the Schrödinger faction Burrau’s successful result, as well as Heisenberg’s for the helium atom (despite his assertion that he had used Schrödinger’s theory merely to facilitate calculations) was proof that Schrödinger’s theory offered a solution to every problem of atomic structure, whereas Heisenberg’s was daunting to use and ugly. This of course greatly pleased Schrödinger.
Heisenberg’s uncertainty principle
In fall 1926, Bohr summoned Heisenberg to his