Probabilities of the Quantum World (Pt-7)

[To refresh, text in such square brackets is my commentary. Rest of it is a faithful documentation of the most fascinating story ever told of the Quantum Revolution]

[At the end of Part-6, Louis de Broglie had given the most elegant insight into why only certain orbits were stable in the planetary model of the atom.]

[We are now entering the most fascinating era. The Quantum description of the atom is taking shape rapidly and there is light at the end of the proverbial tunnel. Things now begin to get a bit complicated. Simple explanations are becoming difficult to find. This part of the story will need you to stick with the narrative and simply follow the story without sometimes understanding it.]

“The Quantum Theory is very much like some victories — for a month or two you are laughing and then you cry for long years.”

When Niels Bohr announced the coming ‘resolute restructuring’ of the old concepts in 1925, he did not know that this was already underway. And moreover, it was moving in two different corners of Central Europe. Physicists who hadn’t even a chance just to see each other at some congress or seminar had started, independently of one another.

That Bohr did not know about that at that time was due to the inherent property of any intellectual revolution:

at the beginning it passes unremarked and does not recognize itself as a revolution. At first it needs this anonymity.

Moreover, both these ‘revolutionaries’ did not at that moment seek Bohr’s guidance or approval. The explanation was simple: Bohr predicted the coming revolution in conventional concepts even as he started to believe in the reality of the particle-waves, while those two — Erwin Schrodinger and Werner Heisenberg — did not believe at all in the corpuscular-wave duality of the electrons, protons or quanta.

Both creators of the mechanics of the submicroscopic world worked from completely different images when they thought about submicroscopic bodies whose behavior they were attempting to describe by their theories. One thought that only the particles had a physical reality and their wave behavior was a mathematical illusion. The other said that physical reality could be attributed only to the waves, and their corpuscular nature was just a mathematical trick. Therefore, they started their ‘revolutions’ from opposite sides.

Could their results coincide? The answer was obtained in 1926. But they both started in 1925.

We will start with Schrodinger because it is the polite thing to do — he was fourteen years older than Heisenberg and already a professor, while the latter was still an assistant professor.

Schrodinger was a professor in Zurich where in 1900 a young Albert Einstein had graduated from the Polytechnical School. At the beginning of 1925 Einstein published a paper in which he highly praised the wave concepts of de Broglie: “Though it may seem crazy, everything is substantiated.” Einstein’s paper caught the attention of two prominent theorists from Zurich, Peter Debye from the Polytechnical School and Erwin Schrodinger from the university. Both had read de Broglie’s thesis and admitted to each other that they couldn’t understand his concepts; Debye suggested conducting a joint seminar with a report by Schrodinger.

Apparently, that was how Schrodinger started on his quest.

[The unseen hand of Serendipity is seen to be at play, again!]

That year in late winter doctors recommended him to leave the city and spend a few months in the Alpine village of Arosa — he had a lung disease. In this quiet little Arosa cane the first ideas about wave mechanics.

[Personally I find this goosebumpy (totally invented this word just now)! Some years earlier I had a chance to go to Arosa to attend a series of talks on the Upanishads by the world renowned humanitarian and spiritual leader Sri Sri Ravi Shankar. Not only did I not go, I, till now, did not make the connection that that Arosa was the Arosa! Was it by chance that Sri Sri chose this venue for this talks on the insights to the Upanishads? I wonder.]

As soon as Schrodinger had accepted the reality of the ‘matter waves’ of de Broglie, a most natural idea came to his mind — should not there be a similarity between the mechanics of submicroscopic world and the mechanics of waves? It was precisely this idea that led to his success.

It happened that he reached his goal two (!) times. The first time he decided that he had lost his way… To appreciate this better we must recall other happenings of 1925. They did not take place in Zurich and Schrodinger knew nothing about them.

The significance of these events could be given in a few short words: a new quantum number was being born in the physics of the submicroscopic world.

[It is best to take an aside and fill the gap of the various Quantum Numbers associated with this story…]

The principle quantum number was introduced by Bohr quite naturally: the discrete sequence of the stable atomic orbits — integral numbers — first, second, third, …, the nth. They had to count the steps in the energy ladder. This became known as the Bohr quantum number.

Soon more quantum numbers had to be introduced. It was like the apartment blocks in a street are numbered by one sequence of numbers, the storeys in each block are numbered by another and the flats in each storey are numbered by a third series.

Two additional quantum numbers defined a more precise (planetary) address for the electrons zipping around in the atom, that is, the address of their orbits. That was the work of a theorist from Munich, Arnold Sommerfeld. It was known to physicists that if the emitting atoms were placed in an electric or magnetic field, their spectra underwent amazing transformations. The lines split into two, three, four or more separate lines. The spectra acquired a ‘fine structure’.

Arnold Sommerfeld assumed that since the electrons were similar to planets, they were traveling in ellipses rather than in circles. In addition, since they traveled at great velocity one had better use the relativity theory to describe their motion. Thus, Sommerfeld introduced two improvements — a classical one from Kepler and a non-classical one from Einstein.

According to classical physics, the electron velocity in the elliptically elongated orbit continuously varies not only in direction but also in its magnitude. The electron has one velocity far from the nucleus and another velocity nearer to it and according to the relativity theory, the electron mass varies if its velocity varies.

The result is that after completing a full revolution around the nucleus the electron does not return to the point where it started from; it’s position is displaced. The electron as it were stitches its ellipses around the nucleus. While the electron travels along its orbit the elliptical orbit itself rotates rolling along the plane of the orbit. Therefore the real path of the electron is a beautiful curve known as a rosette — tracing something that looks like a flower with many petals, such as a daisy.

Another way of reasoning is that the electron takes part in two independent rotations. The first, the rotation along the orbit, is quantized — not just any orbits are allowed; the second, the rotation of the orbit itself, must have been quantized too — the rosette cannot have just any petals, they probably make up a discrete sequence allowed by Nature. Then a sequence of integers — 1, 2, 3, …, k, should be needed to number them.

But, that is not all.

[At this point if your eyes are glazing over, get up, take a brief walk, come back and continue. It gets super cool...]

The atom is a three-dimensional entity, while electron orbits are two-dimensional. As the electron travels along the ellipse, and the ellipse rolls along the plane of the orbit, this plane itself can rotate in space! This is the third independent rotation in which the electron must take part. Clearly, it is also quantized: not all positions of the orbital plane are allowed. To number them the third series of integers is needed — 1, 2, 3, …, m.

Thus, the two additional quantum numbers of Sommerfeld were added to the principal quantum number of Bohr in the quantum theory of the atom; they made it possible to give a correct description of the fine structure of the atomic spectra.

[By now one thing must be clear. Physics is simply a quest to unravel what we physically see, measure and observe. Physicists don’t do stuff just because! All Physics is developed to explain what we observe. An apple falls, Newton invents some fancy math called Calculus to explain why an apple falls they way it does, or why a ball rolls the way it does or why the planets move the way they move etc. etc.

But so does Spirituality. It is a quest to unravel who we are! Who is the person doing all this seeing, measuring and observing? That is an inward journey.

Like I said in Part-1, Science and Spirituality are parallel tracks. Like all parallel lines. They meet. At Infinity. That is where the Self is.]

Later, Somerfeld received a letter from Einstein too:

“Your spectral studies gave me one of the most beautiful moments I have experienced in physics.”

There was one feature of the fine structure of the atomic spectra that still could not be interpreted: the anomalous Zeeman effect. The effect consisted, for instance, in the splitting of the yellow spectral lines of Sodium into four or six close lines in a magnetic field. The solution eluded the most gifted theorists for a number of years.

One of them — generally regarded as a genius — recalled later how he had become addicted to the problem when he worked at the Bohr Institute in 1923:

“A colleague who met me when I was aimlessly wandering along beautiful streets of Copenhagen said to me: “You are looking very unhappy”. I answered heatedly: “Can a man look happy if he is thinking about the anomalous Zeeman effect?””

The unhappy young man — just twenty-three — was Wolfgang Pauli.

That very spring of 1925 when Erwin Schrodinger and Werner Heisenberg were nurturing the concepts of the mechanics of the submicroscopic world, Wolfgang Pauli wrote in despair to another, even younger, theorist:

“Physics is too difficult for me and I am sorry that I haven’t become a comic film actor, or somebody of the kind, just so I would never hear anything more about physics.”

It was precisely that spring when Pauli published the historic paper that opened the way to a solution of the Zeeman mystery. Pauli finally found a new quantum number ten years after Sommerfeld had introduced his. This new feature of the electron he called ‘two-valuedness’. It was an abstract notion; he did not build any model and did not attempt to visualize such a feature.

The only question that mattered for him was that the quantum possibilities of the submicroscopic world were at least doubled… The ladder of the allowed energy levels in the atom became even more complicated… The anomalous splitting of the spectral lines could be interpreted correctly… In addition many other phenomena were now open to theoretical explanation.

A twenty-year-old American, Ralph Kronig suggested a semi-classical model for the ‘two-valuedness’ of the electron. He suggested that just like planets, the electron rotated around its own axis, except, the rotation was quantized; if a certain position of the electron is fixed the second possible position would be the opposite one. Other positions are forbidden.

However, Pauli, who had said that his concept was just a ‘clever trick’ and nothing else, rejected Kronig’s model. Then, after Pauli, Bohr himself rejected it too!

Ralph Kronig was too young and too inexperienced; he gave up. He did not dare to send his paper to be published. One of the irrefutable objections against his model was that the speed of the rotating electron at its periphery of 10^-13 cm would be higher than the speed of light! And that was not allowed by the relativity theory.

Nevertheless, two equally young Dutch theorists, Goudsmit and Uhlenbeck, came to the model of the rotating electron independently of Kronig in the same year, 1925. The worked in Leiden.

The chair at Leiden University was occupied at that time by an extremely kind man, Paul Ehrenfest. He was a close friend of both Einstein and Bohr. There was no other in the history of the quantum revolution who was better described as a ‘creative critic’. His criticism was always not only very keen but invariably helpful and kind.

On the rotating electron’s surface speed being greater than the speed of light, Ehrenfest commented that it was ‘either very important or nonsense’. He told the young men to write a short communication to a scientific journal. They did that. After having given this communication to the professor, they gave way to incapacitating doubts. Thirty years later in 1955, when he inherited the chair of Lorentz from Ehrenfest, Uhlenbeck recalled:

“We with Goudsmit felt the maybe it was better to refrain from publication but when we talked to Enhrnfest about that he answered: “I have sent your communication to the journal long ago. You both are young enough to allow yourself a folly or two!””

Thus the concept of ‘spin’ in the physics of the submicroscopic world is forever linked to their names. A fourth quantum number was introduced to physics instead of the vague ‘two-valuedness’ of the electron.

[It was indeed Kronig’s bad luck that he relied on the judgement of Pauli who was at the time too young to act as Ehrenfest did. Take a moment and reflect on your life. How often have we been a Pauli to a Kronig? How much cold water have we poured on potentially brilliant ideas because we are so stuck to our limited points of view? I know for sure I have. I seem to have a bucket of cold water with me all the time!! This has to stop. Like Sri Sri Ravi Shankar says, we need to stop being a football of others’ opinion and we need to support others’ enthusiasm. It’s that simple to comprehend but yet that much more difficult to live it!]

This fourth quantum number we know colloquially as Pauli’s Exclusion Principle that restricts at max two electrons from occupying the same orbit.

[Somewhere else I read, now I don’t remember exactly where, that Pauli’s exclusion principle can be derived quite simply from a mathematical description of the three spatial dimension space that we live in!! Imagine that. The ultra-transcendent language of Mathematics can take Pauli’s exclusion principle and from it predict that we must be living in a space that has three extended spatial dimensions.]

… To be Continued

[In the next part, we will return to Arosa with Schrodinger as he develops his wave mechanics in complete oblivion of the birth of the new quantum number!]

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