Story of spin

Both Harshini and Shruthi were quite excited about GR’s lectures last Sunday -- which, I was told, revolved mainly around “spin”. I told them I will post an historical account of spin and here it is.

This goes back to 1920’s during which Quantum physics was being intensely explored. Specifying a complete list of quantum numbers associated with electrons in atoms had occupied much of interest. A decisive contribution was made by Wolfgang Pauli (he was 25 year old at that time). Like many theoreticians of the day, Pauli was concerned with understanding the spectral lines emitted by atoms. Bohr’s original model worked only for the relatively simple spectral patterns emitted by hydrogen; but heavier, more complex elements were much harder to understand. For example, Cesium, Sr, and Ba (alkaline earth materials) produce spectral lines that are seen to split into two – they were called doublets. In December 1924, Pauli suggested that a complete set of quantum numbers of an orbiting electron would include its energy, angular momentum (l), and its orientation in space (m); in addition, to explain the alkali doublets, he suggested that there had to be a fourth quantum number, which he referred to – rather unhelpfully – as “Zweideutigkeit” (two-valuedness). During the summer of 1925, Samuel Goudsmit, a young Dutch physicist, was trying to explain Pauli’s ideas to another young countryman, George Uhlenbeck. During such afternoon talks, it occurred to Uhlenbeck that Pauli’s “Zweideutigkeit” was not really another new quantum number, but simply another property of an electron. He suggested that perhaps an electron spins about its axis like a toy top – but unlike a toy top, the spin of the electron would be quantized, and so, it could only “spin” at certain specific values. Looking at Pauli’s formulae, Uhlenbeck and Goudsmit realized that if electrons had a second angular momentum, this would perfectly account for “two-valuedness”. of the alkaline earth metals. The amount of “spin” turned out to be ½ hbar. Both men took their idea to Paul Ehrenfest, Uhlenbeck’s teacher, who made them write up a short paper on spin and then told them to take it to H. Lorentz, the grand old man of Dutch physics.
In 1925, Lorentz was 72, retired, but he still taught a class at Lieden every Monday morning at 11.00 AM. After one such class, Uhlenbeck and Goudsmit showed Lorentz their paper, which was only a few paragraphs long. Lorentz said that it was interesting and he would think about it further. Thinking, for Lorentz, was apparently an active occupation. Two weeks later, Lorentz gave a stalk of papers with long calculations to Uhlenbeck: Lorentz had calculated the speed of the spinning electron with ½ hbar angular momentum to be 10 times that of light!! Uhlenbeck and Goudsmit were most unhappy – they went back to Ehrenfest and said “You better not publish that paper, because Lorentz has shown that it is not correct”. But Ehrenfest had already submitted the paper and the paper was expected to be published within a few days! Later, Bohr, dismissed Lorentz’s objections saying that the faster-than-the-speed-of-light problem disappears when the full quantum theory is applied to a structureless point electron – apparently, Lorentz’s calculations were valid for a classical extended particle with spin ½ hbar. As it turned out, Bohr was correct. The eigenvalues, eigenkets of angular momentum and the matrix representation of angular momentum operators was first obtained in a 1926 paper by Max Born, W Heisenberg and P. Jordan (Zeitschrift fur physic, 35 (1926) 557). It was shown, basing the analysis wholly upon the commutation properties of the angular momentum operators, that there are two types of angular momentum, one with eigenvalues that are only integral multiples of hbar and the other, which can assume half odd integral multiples of hbar values also.