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.

The pursuit of science: its motivations

We have been talking about how Feynman influenced our learning of science and towards our scientific imagination – with the help of a mathematical abstract view, Julius Sumner Miller’s exciting science demonstrations … I felt I should continue this stream of thoughts by adding some observations on the pursuit of science. (These are my compilations from the wonderful book “Truth and Beauty” by Professor S. Chandrashekhar, Penguin Books Ltd (1991) – don’t miss to read this book if you happen to come across). I am aware that I am not offering here any concrete thinking on the “pursuit of science” -- but surely it helps to put things in their places, and it invites deeper thinking on this issue.

“The pursuit of science: its motivations” is a difficult topic because of the variety and the range of the motives of the individual scientists; they are as varied as the tastes, the temperaments, and the attitudes of the scientists themselves. Besides, their motivations are subject to substantial changes during the life-times of the scientists. Indeed it is difficult to discern a common denominator! We may consider some examples to get some better ideas on this. Let us think of Albert Michelson: His main preoccupation throughout his life was to measure the velocity of light with increasing precision. His interest came about almost by accident, when the commander of the United states Naval Academy asked him – he was then an instructor at the Academy – to prepare some lecture demonstrations of the velocity of light. That was in 1878 and it had led to Michelson’s first determination of the velocity of light in 1880. On 7th May 1931, i.e., fifty years later and two days before he died he dictated the opening sentences of a paper, posthumously published, which gave the results of his last measurement! Michelson’s efforts resulted in an improvement in our knowledge of the velocity of light from one part in 3,000 to 1 part in 30,000 – i.e., by a factor of 10. But by 1973, the accuracy had been improved to 1 part in 100000000000 -- a measurement that made obsolete, beforehand, all earlier measurements! Were Michelson’s efforts over fifty years in vain? Leaving that question aside, one must record that, during his long career, Michelson made great discoveries derived from his delight in “light waves and their uses”. Thus, his development of interferometry, leading to the first direct determination of the diameter of a star, is breathtaking. And who does not know the Michelson-Morley experiment, which -- through Einstein’s formulation of the special and the general theory of relativity -- changes irrevocably our formulation of the nature of space and time? It is a curious fact that Michelson himself was never happy with the outcome of his experiment!! Indeed, it is recorded that when Einstein visited Michelson in April 1931, Mrs. Michelson felt it necessary to warn Einstein in a whisper when he arrived: “Please don’t get him started on the subject of the ether”!
When Michelson was asked towards the end of his life, why he had devoted such a large fraction of the time to the measurement of the velocity of light, he is said to have replied “It was so much of fun”! There is no denying that “fun” does play a role in the pursuit of science. What are the other factors? Difficult to say! I leave it open to your own jurisdiction now.

Scientific imagination

I have a free afternoon session today – no lab work to take care of and no students came with some bothering questions! So, I decided to post what Harshini’s Fine Man says about scientific imagination – which had fired my imagination for sure!

“Let us try to imagine electric and magnetic fields; you may say, “Professor, All this business of electric and magnetic fields is pretty abstract! What is actually happening? Why can’t you explain it? Please give us an approximate description of the electromagnetic waves, even though it may be slightly inaccurate, so that I too can “see” them!” I am sorry – I can’t do that for you. I don’t know how. I have no picture of this electromagnetic field that is in any sense accurate. The only sensible question is, “what is the most appropriate way to look at their effects?” Some people prefer to represent them as field lines and feel that writing vector E and vector B is too abstract! The field lines, however, are only a crude way of describing a field. Field lines cannot efficiently describe superposition of electromagnetic waves. From the mathematical stand point, on the other hand, superposition is easy – we simply add two vectors to get another vector. The field lines have some advantage in giving a vivid picture, but they also have some disadvantages. So, the best way is to use the abstract field idea. That it is abstract is UNFORTUNATE but NECESSARY!

Our science makes terrific demands on the imagination. It appears that scientific imagination requires mathematical view and then its experimental verification. Now, what is a mathematical view?
For example, from a mathematical view, there IS an electric field vector and magnetic field vector at every point in space. This concept is abstract – true, but in some sense the fields are real, because after we are all finished fiddling around with mathematical equations, we can still make the instruments detect electromagnetic signals.

The whole question of imagination is often misunderstood by people in other disciplines. They try to test our imagination in the following way. If someone asks me: “Here is a picture of some people in a situation. What do you imagine will happen next?” I would say “I can’t imagine (because I don’t have enough facts”. So, people think that I have a weak imagination. They overlook the fact that whatever we are allowed to imagine in science must be consistent with EVERYTHING else we know.

The electric and magnetic fields we talk about are not just some happy thoughts which we are free to make as we wish, but ideas which must be consistent with all the laws of Physics we know. We can’t allow ourselves to seriously imagine things, which are obviously in contradiction to the known laws of nature. One has to have the imagination to think of something that has never been seen before, never been heard before. But creating something new, has to be consistent with everything, which has been seen before, is extremely difficult and requires scientific imagination.”

So, tell me if this caught your imagination (poetic – if not scientific) too!