Probabilities of the Quantum World (Pt-9)

[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-8, Schrodinger had published his development of matter-wave mechanics. We now shift our focus to the other theorist who, simultaneously with Schrodinger, was taking a different approach to understand de Broglie’s work.]

[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.]

In contrast to the other physicists, the second scientist who started developing the mechanics of the submicroscopic world, the young Werner Heisenberg felt no joy but was rather confused when in the spring of 1926 he first learned about Schrodinger’s success. Heisenberg developed his version of quantum mechanics earlier than Schrodinger though he has started almost simultaneously with him. And he also published it earlier, back in 1925.

Moreover, he could not suspect that the professor from Zurich had suggested a better theory — the differences between them were too glaring! The theorist from Gottingen was struck by a sudden suspicion:

“Both of us are completely wrong…”

Both of them wrong! That was what terrified him: Everything that they had done was nonsense.

Let us return to the spring of 1925 and to Gottingen in Germany. As a lung disease exiled Schroginger to the mountains, a complaint of another kind sent Heisenberg to the seashore. Fate also had provided him with a period of solitude. The spring blooming of flowers brought on an acute attack of hay fever; his face became swollen, his eyes were bloodshot. Max Born, his superior in Gottingen, did not hesitate in giving him a leave of absence. he advised him to go to the island of Heligoland with its health-giving sea breezes, and where there were only bare rocks and no irritating pollen.

Heisenberg gazed at the beach and saw precipices in rocks, deep gaps between sea waves… He thought of the electron-particle. He regarded the quantum discreteness as being given by Nature — they did not need any justification in terms of continuous waves. The quantum jumps did not terrify him by their inexplicability; neither he nor his student friend in Munich, Wolfgang Pauli, were scared off by them.

Both Heisenberg and Pauli (who was just a year older than him) felt that they owed incomparably more spiritually to Neils Bohr; they both had spent time as his assistants in the Copenhagen Institute. Going on his trip to Heligoland he already knew what he wanted to find… His starting idea was simple: should not the mechanics of the submicroscopic world deal only with observable quantities?

Why only observable quantities? Because this strange submicroscopic world has quantities which are, in principle, unobservable. It would be purposeless to include such quantities in a description of the events in this world — the description would get out of control. Even worse, it would be physically meaningless because it would be unknown what was described by them.

The quantum jumps are the main events in the intra-atomic mechanics. But they constitute a clear violation of the continuity of electron motion. Therefore any attempts to describe these electron jumps in traditional terms — as displacements in time and space — are obviously doomed to failure.

[This is clear thinking. What can we measure? What can we observe impartially? Use only those things in developing a theory. Any theory is simply a rational attempt at explaining cause and effect; observable effects with observablecauses.]

Heisenberg thought that the history of physics in the 20th century was supporting his ideas. Did not Einstein refuse to recognize absolute time — a common time for all moving bodies — just because no, even imaginary, experiment could prove its existence?! Rutherford’s concept of the electron-planets possibly and even probably is no more than an illusion. What is observable? Only that the atom varies its energy discretely. This discreteness testifies to the existence of a ladder of allowed energy levels in the atom. The indivisible jumps along this ladder are indicated by the emission of light in entire portions — the quanta. This is genuine knowledge.

What can be measured here? — The frequencies and amplitudes of the oscillatory processes which take place in some way in the atoms and give rise to the radiation quanta. The spectral lines describe their frequencies and amplitudes by their color and their intensity. The frequencies describe the energy of the quanta — the higher the frequency the higher the energy of the quantum. The amplitudes describe the probability of the emission of the quanta — the higher the amplitude the higher the probability of emission (therefore the line is more intense). This is genuine knowledge.

Heisenberg started working on his theory of the atom even before the flowers came into bloom. He started with the simplest atom — Hydrogen. But he did not get any results in Gottingen. At first he lost his way.

Apparently it happened at precisely the same time when Schrodinger lost his way in the mountain village of Arosa. But the causes of their failure did not look similar… Schrodinger did not know about a new physica fact; but Heisenberg was not aware of the existence of an old mathematical method for calculating such quantities as discrete sets of observable variables.

[Imagine this for a moment! A mathematical method existed, the physicist was unaware of this, and the physicist lost his way!!]

Physics was not concerned with such things as discrete sets of observable variables! Heisenberg set to working on that. Later, Max Born was reported as having said, admiringly, that one had to be really intelligent ignoramus to not know the proper mathematics but be able oneself to develop the appropriate mathematical method if one needed it!

Heisenberg struck terra firma even before his flight to Heligoland.

In principle, quantum transitions can occur between any two energy levels in the atom. Thus, a unified notation had to cover all the possible quantum jumps along the energy ladder. That seemed like trying to find a format for writing down all the results of a tournament in which everyone plays with everyone. The participants in the tournament are the stationary states. The results of the games between them are the emission or absorption of the quanta. A square tournament table is suitable for writing down at once the results of all the possible games, one table for the frequencies and another table for the amplitudes.

Heisenberg has a day in Heligoland — sea, solitude, quiet — when he saw the light ahead. Evening had fallen. Forty years later Heisenberg told the historians:

“I was extremely excited and it was just early in the morning. I got in a state of great excitement because I saw that it worked out so nicely… I worked all night and I made many slips in the calculations. I decided I would go out for a walk and so I did. I rather half-climbed on one of the cliffs on Heligoland just for excitement and I felt ‘well, now something has happened’. Then I started writing on paper.”

[He started writing on paper!!This is such a profound statement.]

There was another moment when Heisenberg felt that everything was wrong. He saw that the algebra of the square tables did not always satisfy the age-old law, A multiplied by B is equal to B multiplied by A. In nature this commutativity of multiplication was always thought to be self-evident; but when Heisenberg multiplied different observable quantities he found that he could not transpose them without changing the result: AB not equal to BA. He was terribly alarmed by that.

At that moment the future of the mechanics of the submicroscopic world was hanging by a thread just as in the same spring in Arosa. In contrast to Schrodinger however, Heisenberg did drop his method for a few months — and that was why the Heisenberg version of the quantum mechanics appeared only some months earlier than the Schrodinger version.

[What an amazing story! He comes up with a contradiction to the well established commutative law of multiplication and gives up!]

After his initial alarm with the inequality AB not equal to BA the young theorist affected an inexplicable but happily care-free attitude to it:

“But then I said to myself: ‘Fortunately I don’t need it, fortunately it is not very important’”

Little did he think that it was precisely that feature that would prove crucial, and precisely that feature that would reveal what is, perhaps, the most unexpected of all the non-classical laws of Nature!

[He thought his Law Of Uncertainty as it would be called in the future was not important. Amazing!!]

His care-free complacency proved to be extremely useful. Despite alarm signals from his common sense, he continued his work after returning from Heligoland. Yet there was one inhibition he could not overcome — his reluctance to show that he had written to Max Born or send it to Niels Bohr. Born, true to Gottingen traditions, always asked for mathematical rigor. Bohr, true to Copenhagen tradition asked for a reliable physical substantiation. His paper lacked both! Still, he wanted somebody to look at the paper and to give him friendly criticism.

In early July he sent his paper to Wolfgang Pauli in Hamburg. This time Pauli’s judgement was generous. This is the more remarkable since he, in his turn, did not agree with all the ideas of his old friend: “There are many more observable quantities in the atomic world than are dreamed of in your philosophy, Heisenberg.” What was it that attracted this most severe of critics?

It was that Heisenberg dared to reject completely all the traditions of classical description for motion in the submicroscopic world. Pauli saw a crack appearing in the wall of the blind alley. Immediately he stopped envying film actors. In the autumn of 1925 he wrote to Kronig:

“Heisenberg mechanics restored to me hope and joy of life”

Having received Pauli’s approval, Heisenberg finally gave his paper to Born, his superior, asking him to ‘Do with it anything you think proper.’

Max Born read it the same day. It was not easy for him: he thought that it ‘looked rather mystical but was undoubtedly true and profound’. He wrote that to Einstein, and he sent the paper with his recommendation that it be published. However, he was also bothered with the preposterous formula AB not equal to BA … It faintly reminded him of something he had known long ago but he just could not identify what it was:

“One morning I saw the light — I remembered the algebraic theory that I had studied at the university. Such square tables were well known to the mathematicians; they were known as matrices and I saw that the Heisenberg multiplication was nothing else but an element of matrix calculus. Now we could move further on. I was excited as a sailor who had sighted the long-awaited land after a lengthy voyage…”

[Matrix Multiplication!! We now learn this in high school, but, back in 1925, it tripped up the most brilliant physicist!]

In those July days of 1925 Heisenberg was in Cambridge. He did not know that his mechanics had by that time been christened the ‘matrix mechanics’. Niels Bohr received a letter from Heisenberg: “I have written a paper on quantum mechanics which I would like to have your opinion about.” It would have been better if he had dared to do that earlier! The fact was that Niels Bohr was then practically the only theorist who was not puzzled on seeing for the first time the formula of multiplication for matrices… Perhaps he understood the formula AB not equal to BA immediately! At any rate, he soon wrote:

“It can be hoped that a new era has opened for mutual stimulation of mathematics and mechanics. Perhaps, physicists will at first be sorry that in our understanding of the atom we cannot overcome the limitations on the normal methods for describing Nature. But one would like to think that this feeling will be replaced by a gratitude to mathematics which provides us with an instrument for advancement in this field.”

Yes, the matrix method using something that looked like tournament tables was quite unusual. But it was highly promising — and it was precisely tat that time that Schrodinger was close to a successful completion of this wave mechanics.

But what then was the cause of Heisenberg’s panic (“we both are hopelessly lost!”) when he learned from Pauli’s letter about the Schrodinger wave mechanics?

Later, Heisenberg smilingly compared himself and Schrodinger to two mountain climbers searching for a way to the summit in a fog. When the fog cleared, how different were the landscapes before their eyes near their goal! One saw sheer rocks (the quantum jumps) and the other, smooth slopes (the matter waves). Could the both be sure that what they saw was the same mountain? No, they had a strong suspicion that, perhaps, both had lost their way…

The wave mechanics and the matrix mechanics were not at all at odds. They ran parallel courses as if they could be literally translated: it was as if the first was saying to the submicroscopic world ‘I love you’ and the second was saying ‘Je t’aime’…

In the summer of 1925 when the wave mechanics was not yet in existence and the matrix mechanics had just appeared, two theorists from Gottingen went begging to the great David Hilbert, the established head of the Gottingen mathematical school. They asked the world-famous scientist to help them with the matrices. Hilbert listened to them and said something quite remarkable — each time he had to deal with these square tables they appeared in his calculations as a sort of ‘a byproduct’ in the solutions of the wave equations, “So, if you look for the wave equation which has these matrices you can probably do more with that.”

According to the American Edward Condon, the theorists were Max Born and Werner Heisenberg. The episode ended in this way:

“They had thought it was a goofy idea and that Hilbert did not know what he was talking about. So he was having a lot of fun pointing out to them later that they could have discovered Schrodinger’s mechanics six months earlier if they had paid a little more attention to his words.”

[This is a clear example of the blindness of a one-sided approach. At the same time it shows how true was the classically ‘impossible’ concept of the wave-particles.]

The two mountain climbers had seen the same mountain: if it were otherwise, only one of the two versions of the mechanics would be true — either the matrix mechanics or the wave mechanics. How they were naturally amalgamated or, better to say, fused into what is today known as the quantum mechanics of the submicroscopic world.

Nobody could predict in 1926 that:

• in 12 years time the first source of nuclear energy would be discovered in Uranium fission reaction (Germany, 1938);
• in 19 years the first atomic bomb would be tested at the end of the Second World War (USA, 1945);
• and 28 years later the first nuclear power station would be built (USSR, 1954)

The atomic age sprang into human history from the quiet laboratories and low voiced academic discussions. Yes, at first it was doubly unclear. In both versions physicists had to grope in the dark as soon as they crossed the doorstep. In the wave mechanics it was the mystery of the Schrodinger psi waves; in the matrix mechanics it was the enigma of the matrix multiplication. Everything was right mathematically but the physical meaning of both was either puzzling or controversial.

… To be Continued

[In the next part, we will “understand” the physical meaning of the psi-waves and the matrix multiplication leading to the now famous Uncertainty Principle.]

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