Probabilities of the Quantum World (Pt-10)

[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-9, Heisenberg had his matrix mechanics and Schrodinger had his wave mechanics model for the submicroscopic world. But, the physical meaning of it all was still elusive!]

[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.]

Soon Schrodinger realized that his dream of building up particles from the psi waves was hopeless. Though the formulae of the wave mechanics delivered the goods, his own interpretation of these formulae ‘could not stand’, in the words of Max Born. The fault lay with the wave packets — they spread out.

Ironically, the first to understand, or at least, to express convincingly the meaning of the psi waves was Max Born back in the summer of 1926. It was none other than Max Born who had identified the Heisenberg square tables as the matrices well known to mathematicians. It was he and his highly talented assistant whom Hilbert had made fun of quite recently, in the spring of 1926, for ignoring his good advice to seek for a wave equation for the matrices. And now it is to him that the credit for discovering the physical meaning of the psi waves goes!

He (Max Born) had the advantage of good and grateful memory. He did not forget one old constructive idea of Einstein’s and it helped him.

In the same year, 1926, the quanta of light — Einstein particles of light — were at last named. The physical chemist Lewis called them photons. The name became widely accepted. It was similar to the names of the submicroscopic particles — electrons and protons — and thus emphasized the corpuscular nature of the quanta. Einstein suggested this concept in 1905 and from the very beginning he had to answer one obvious question — if light consists of particles then what are electromagnetic waves about?

Now Max Born had to answer precisely the same question: what was described by the psi waves with their depression and crests if the behavior of the particles was associated with these waves? He remembered Einstein’s idea and found the answer; on many later occasions max Born gratefully remembered that debt.

The simplest answer would be that the particle is at a given moment to be found at the site of the crest of the psi wave! But, according to Einstein, it was not necessarily so; for instance, was there no light at all at the site of the incline of the electromagnetic wave? No: the intensity of light was simply lower there but even there there were photons. The number is smaller but they are there. Why could one not then assume that the electron could be at the slope of the psi wave? (Or any submicroscopic particle whose behavior is analysed.)

The electron was a whole could be found where the psi wave reached a crest and where it followed a slope: but the chances of finding the electron increase as we approach the crest. There is no chance of finding the electron where the psi wave vanishes — there the probability of finding the electron is zero.

The Schroginder psi waves are the probability waves!

These immaterial waves describe the non-classical motion of the submicroscopic particles. It is as it they replaced the strictly defined classical paths. Max Born theoretically substantiated it in the summer of 1926.

Surprisingly, this realization of Max Born had practically no effect on the physicists belonging to Niels Bohr’s circle in Copenhagen. They had been convinced for a long time that the depths of matter were a world of probability laws. The electron in the hydrogen atom could make any possible jumps along the steps of the energy ladder. The spectral lines demonstrated that all types of allowed quantum jumps were taking place. The more intense the spectral line, the more probable the jump.

By the end of 1926 Max Born received a short but very distressing letter. Einstein’s reaction to the probabilistic interpretation of the laws of the new mechanics was:

“The quantum mechanics deserves high respect. But an inner voice tells me that still it is not what is needed. The theory yields much but hardly takes us closer to the mystery of God. At any rate, I’m convinced that He doesn’t play dice.”

It was ironical that Max Born followed Einstein step by step faithfully but when he reached the goal, Einstein disowned him. The probability concept of the world had started its quest…

The understanding of the second mystery — the strange multiplication of the matrices — could not be credited to a single person. But it was Niels Bohr who did more than anybody else to explain it.

Looking at the nonsensical formula AB not equal to BA Bohr immediately understood that A and B could not be numbers. Things are different if A and B are not observed quantities but operations on the observed quantities. If the order of the operations of different kind is changed then the result need not be the same!

Say, the operation A is anesthesia and the operation B is the extraction of a tooth; then the combination AB is tolerable while the reverse combination BA is unthinkable! No experiments are needed to to prove that AB not equal to BA.

A most natural operation on the observable quantities in the submicroscopic world is their observation or, in other words, their measurement. But we can measure nothing in the unseen and soundless atomic world without receiving back from it an answering signal in response to our laboratory question. And a signal requires an expenditure of energy and time, in short, action. The smallest or the weakest of all the possible signals is the Plank quantum of action h. However small it is its magnitude has a real significance on the scale of the submicroscopic world.

The electron or the atom changes its state even after sending such a negligible signal. The measurement violates their being, each time differently. So can one be surprised that when two operations A and B are performed their order is by no means insignificant? This obvious fact had to be necessarily manifested in a true mechanics of the submicroscopic world. Hence the formula AB not equal to BA.

Bohr, for a long time, had been thinking about the measurements in the submicroscopic world. Bohr was gladdened rather than confused by the non commutativity of the matrix multiplication. Of course, the fundamental laws of nature revealed the probabilistic character in the matrix version of the submicroscopic mechanics just as they did in the wave mechanics. To show this we continue the comparison of the square matrices to the tournament tables.

For a score to be written into the table, a game has to be played. Can we state that the score exists before the game? Before the game only a variety of possible scores could be predicted. Some of them were more probable and others were less probable. Nobody could give an absolute prediction, even computers which are allowed to make errors within an acceptable range.

It is tempting to think that the measured values, for instance, of the position and velocity of the electron, had really existed before the measurement. It is tempting but naive. There is no physical meaning in the belief. The simple question — how to you know it — has no answer! Of course, the mechanics of the observables leaves no room for doubts that the electron exists before and irrespective of our observation (otherwise there would be nothing to measure and nothing to discuss). But quantum mechanics refuses to talk about the exact position of the electron without measurements.

Later in summer of 1926 Sommerfeld held a theoretical seminar in Munich. A report was given by Erwin Schrodinger who came from Zurich. Among the audience was Werner Heisenberg who was spending the remaining days of the summer vacation at his parents house. Thus, the creators of the two versions of the quantum mechanics accidentally met for the first time, face to face.

Later, in September of 1926, Schrodinger and Bhor met. Schrodinger attacked the concept of wave-particles and the concept of quantum jumps. How many times had Bohr heard these arguments. How many times he had repeated them to himself. This faultless logic had one weakness — it was based only on the classical experience accumulated through the centuries. According to Heisenberg, Bohr answered:

“What you are saying is absolutely correct. But it does not at all prove that there are no quantum jumps. It proves only that we can’t imagine them, that the objective concepts of everyday life and experiments of classical physics become inapplicable when we approach the description of the quantum discreteness. And we should not be surprised by that since we realize that the process involved here do not appear directly in the sphere of our being.”

Once Schrodinger has exhausted all the defensive arguments, he cried out in despair:

“If these damned quantum jumps indeed are retained in physics I will not be able to forgive myself that I had something to do with the quantum theory!”

Bohr answered politely:

“But we are all extremely grateful to you for what you have done! Your wave mechanics brought with itself such a mathematical clarity and simplicity that it proved to be an enormous step forward…”

Heisenberg lived in the garret of the institute’s building and Bohr lived in a cottage nearby. They had endless discussions; The only goal of these was to understand how quantum mechanics could produce true results in spite of a strange illogicality quite inconsistent with classical physics. They had understood already the meaning of the non-commutative multiplications and the probabilistic meaning of the psi waves but Bohr was not yet satisfied. He insisted that they had missed something fundamental, that they failed to grasp something of universal significance.

“Mathematics is clever enough to do everything by itself without physicists’ speculations.”

The white tracks of fog in cloud chambers allowed scientists to trace accurately the motion of the electron in space and time, did it not? When cloud chambers were placed in a magnetic field, the tracks of light electrons were transformed into circles reminding one of the electron orbits in the atomic model of Rutherford and Bohr. All this could be seen with the naked eye! But the matrix mechanics was based on the assumption that the orbits and any paths of electrons were un-observable!

Two leading theorists were asking each other plain questions and could not answer them. Heisenberg could not appreciate Bohr’s ideas in full. They both dove into unknown waters but to different depths.

“None of us could understand how to reconcile the mathematical language of quantum mechanics with such elementary phenomenon as the electron path in the cloud chamber…”

The ascent to the summit started in February of 1927 when the desperate search in the dark led to a quarrel between Bohr and Heisenberg. They had planned a trip to Norway together hoping that cross-country skiing would cool their tempers. One night Bohr cut short their discussions. Next morning he went North alone leaving Heisenberg behind. Heisenberg told historians later: “He wanted to be alone, to think alone, and I think he was quite right.”

Heisenberg strolled along frozen lanes of a Copenhagen park trying to understand what physicists really saw when they looked at the white tracks of charged particles on the photographs taken in the cloud chamber. The diameter of the smallest drop of moisture in the cloud chamber is a thousandth of a millimeter (10^-4 cm). The electron is smaller still by a factor of a billion (diameter of 10^-13 cm). If we enlarge the electron to the size of a fly (1 cm), the droplet will be scaled up to the size of a planet. Imagine a fly inside a hollow sphere of the size of the Earth — that is what the electron looks like inside the droplet of fog. Where is is precisely and where does it travel? To ask about the electron path by looking at its track in the cloud chamber is the same as asking about the path of the fly by looking at the motion of the Earth along its orbit. Clearly, different questions must be asked:

“Can quantum mechanics describe the fact that the electron is only approximately present at a given point and travels at a given velocity only approximately, and how far can we reduce this approximateness?”

Heisenberg denoted the ideal — most precise — measurements of the position (A) and the velocity (B) of the electron. He now started analyzing not the quantities themselves but the possible approximateness (uncertainty) of their determination — Delta_A and Delta_B. He decided to find out what happens to these uncertainties according to the laws of his mechanics — could they both vanish in the process of the electron motion? The limit dictated by Nature had to be found:

Classical physics said that time was absolute, and it proved to be relative.
Classical physics said that the physical velocity could be as large as desired but it proved to have the upper limit — the velocity of light in vacuum.
Classical physics said that in Nature the action could be as small as desired, but it proved to have the lower limit — the quantum of action.
Classical physics said that waves were only waves and particles were only particles but…

Undoubtedly, a moment came when Bohr in Norway recalled the strange formula AB not equal to BA, as Heisenberg in Copenhagen. If the order of operations is important the obvious conclusion is that they could not be performed simultaneously. If they could, then their order would not have any significance — since simultaneity means the absence of order, there is no ‘before’ and ‘after’ in it.

Thus a new queer feature feature of the submicroscopic world was revealed — it had observable quantities which could not be measured simultaneously.

The formula for non-commutativity of multiplication first appeared in quantum mechanics precisely for the measurement of the coordinate (position) A and the velocity B in the studies of Heisenberg, Bohr and Dirac.

This meant that one could not specify with arbitrary precision simultaneously both the position and the velocity of an electron.

It means that Nature in its depths does without absolute determinancy; it is probabilistic.

One morning in the second half of February 1927, an extremely excited Heisenberg put on paper the short formula relating two uncertainties:

Now it became clear why the orbits in the planetary model were unobservable. We could determine separately either the position, or the velocity of the electron-planet, but if we attempted to measure one of these quantities precisely then immediately the second one became quite indefinite.

As he had done in 1925 after his stay at Heligoland, Heisenberg decided to first of all tell his old pal Wolfgang Pauli about his new discovery. Heisenberg received the answer from Hamburg before the return of Bohr. Pauli was atypically exultant “Let there be light in quantum mechanics.”

It was precisely that fundamental law which Bohr had been groping for in his long quarrel with classical determinism and that he had almost reached in those days in Norway.

[They say that “truth” is born of discussions. Bohr and Heisenberg, without their long and seemingly hopeless discussion lasting several months, could neither of them reached the solution. But the contrary statement is also true — “truth” dies in discussion. Disputes continuously disturb the concentration of each opponent.]

Bohr, after he had glimpsed the outlines of the uncertainty relationship, he came to what later became known as the principle of complementarity. Heisenberg said that Bohr has brought it from Norway.

Bohr immediately saw some slips in Heisenberg’s derivation. The slips were caused by his continuing neglect of the wave-like nature of particles.

But he needed wave theory, for instance, for performing mentally the perfect experiment on the most precise determination of the position and velocity of the electron. Imagine a super-sensitive microscope. We can illuminate it with the shortest possible wavelength. The rays pinpoint the electron and determine its position. The accuracy of determination increases when we decrease the wavelength of illumination. But, as illumination wavelength decreases, its energy increases. This high energy causes higher disturbance to the electron motion. Its position is measured but its velocity is changed unpredictably.

Clearly, the mathematical treatment of this theoretical experiment with illuminating waves has to employ both the corpuscular and the wave-like features of particles. Heisenberg did not like to think about wave theory!

Apparently, it was during those interminable hours under pressure from Bohr that Heisenberg heard the word “complementarity” from him and realized that new understanding Bohr has brought back from Norway.

Bohr kept on thinking about pairs of the observables which for some reason could not be simultaneously determined. Indeed, what was the reason? True knowledge was brought at the price of Nonsense — the inconsistent combination. Why had one to pay such a price? According to Bhor:

“The term ‘experiment’ can in fact be applied only to those activities for which we can tell others what we have done and what we have learned as a result.”

One must bear in mind continuously that submicroscopic physics deals with objects of a dual nature. The experimental work with them is somewhat similar to using binoculars — one cannot look into the eyepieces from both sides at the same time. Observations must be separate from each side of the eyepieces. We see a ‘diminished’ world looking into binoculars from one end and a ‘magnified’ world from the other end, and both images are equally real. The conclusion is that what we see depends on the method of observation.

The dual concept of particle-waves helps our imagination at least to get used to the strangeness of the submicroscopic world just because we cannot imagine it; while the wave and particle are individually quite imaginable.

The strangeness of the grammer of the submicroscopic world lies precisely in its acknowledgement that the classically incompatible concepts or images are given by Nature the right to complement, rather than exclude each other: the vice of incompatibility taken to the extreme of total conflict is transformed into the virtue of complementarity — such were Bohr’s views when he returned from Norway in February of 1927 and this is why he unrelentingly criticized Heisenberg’s sins. Soon Heisenberg’s tears dried.

To make a full description of the variable probabilistic submicroscopic reality we have to consider the incompatible pictures of it as being complementary to one another. Otherwise, we would not be able to understand the whole.

[The Principle of Complementarity continues to amaze me! Bohr arrived at it from purely a Physics point of view. It had to be so otherwise we are unable to understand Nature as a whole. The same principle is offered as a sutra in the Art of Living Foundation’s Happiness Program which is taught under the guidance of the World Spiritual Leader and Humanitarian Sri Sri Ravi Shankar. Sri Sri frames the sutra as “Opposite Values are Complementary”. It is precisely the same answer, except, Sri Sri arrives at it from the Spiritual side. Without this key sutra our understanding of the world is never complete!]

… To be Continued

[In the next part, we will conclude our Journey into the Probabilities of the Quantum World being a fly on the wall at the Fifth Solvay Conference where a photo was taken: 28 out of the 29 in the picture were Noble Prize Winners.]

[Oh ya… for the Medium uninitiated, please leave a couple of “claps” below if you like this and would be interested in the subsequent part(s). Thanks.]