Polynomial

In this brief essay describing Heisenberg’s “Uncertainty Principle”, my aim is to highlight the not so obvious workings of nature’s flow… For even I, when originally presented with Heisenberg’s marvel of observation, did not fully grasp the basic principle behind the magic of Quantum mechanics. In fact this basic idea took several years of careful study and checked thought before I accidentally stumbled over the essence of Heisenberg’s ideal in the darkness of my, then, unilluminated mind.

Werner Heisenberg best known for asserting the uncertainty principle of quantum theory.

And what a marvel of an ideal it is… For one of the very first times in Physics, an observer has become aware of a very important and much overlooked fact i.e. that how he/she perceives the environment around him/her, directly affects the way in which he/she measures it… If one is uncertain of one’s observations, even if these uncertainties are nearing such miniscule amounts as to seem almost totally insignificant in the fine-spun scheme of the “seemingly” preceise human world… This overlooked resolution will inevitably breed an error into calculations built from these measurements, and a fallacy will extrapolate further down the line, mixing other erred insights together, until a cascade of awkward “blunders” comes awkwardly into the light.

When dealing with the preceision of universal flow, the dot of an electron truly becomes a point which has a definite position in space, but neither size nor shape. And so, when something is so small and specific, especially in comparision to our somewhat large and cumbersome bodies, we can all too readily overlook the details at play in its fate and think nothing of it. Just as Edward Lorenz had discovered in the winter of 1961, when running a computer simulated weather system; his assumption that 3 decimal place numbers, which had been rounded off from 6 decimal place number enteries, should not make any difference to the output of the simulation… How wrong he was! For there is the crux of the matter; that small errors prove catastrophic to the final results. In 1979 Lorenz entitled a paper he wrote examining this phenomenon, “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” and the title has avidly carried the magic held within Chaos’ own never ending and never repeating flow. Today this sensitive dependence on initial conditions is referred to as “The Butterfly Effect.” For these seemingly minor oversights on current input data tallied up over time to yield vastly different outcomes; outcomes that were/are totally unpredicatable.

So… Could Heisenberg’s “Uncertainty Priciple” be the initial realization that, when something is so small and sensitive to its surrounding environment i.e. the electron, which is sensitive to minute charges and other influential magnetic fields, like the magnetic fields that come from the human body, or from a mains plug, some electric lights, or simply another atom’s atomic charge, etc… Then how could we as humans ever hope to accurately reproduce any of the intricate workings inside the Universal mechanisms of matter without this spill over from the world around us? For all of these charges and electromagnetic fields deeply affect one another in an almost infinitely long chain of cause and effect (something the that Buddhists have duly understood and noted within their theory of Interdependent Origination)… And if the things we are studying are “unimaginably” sensitive to initial conditions, while we seem so robust… Then it is my fear that we may never have the chance to precisely understand Universal harmony.

And this beckon’s the question… If we could never really hope to grasp the abstruse nature of reality (if only because of it’s sheer complexity), then is there any point in studying it as certainty? Or are we doomed to make approximations of reality for the rest of eterntiy? For example, if a swiss watch maker, through flaws in his own perception of time AND/OR errors in his manufacture of watches, could never make an accurate watch, would there be any point in continuing his trade? Perhaps he could continue trading if there was a steady demand for inaccurate timepieces… But these would not suite the purpose of knowing the precise time. But then again, are not all watches only approximations of the perfect ideal of a watch i.e. the perfect watch that keeps perfect time based on the daily rotation of the earth around the sun? But as you have seen in the previous post, not even the heavenly spheres move with a precise certainty or symmetry. And every two years or so, I notice my own watches and clocks, whether digital or analog, drifting out by a few minutes… So as chaos theory predicts, are all things prone to chaotic cycles? Where the details, when observed on the microscopic scales, are really vast expanses away from the perfect ideal?

It is here that Heisenberg’s uncertainty priciple philosophically comes into its own… Because if one cannot know the precise position and momentum of a particle at a given instant, mainly due to its stupendously small size in relation to the observer, then its exact path, and so it’s exact future position, could never be realistically, NOR accurately determined. And when something is not accurate… Then it is in essence marginalized or wholly inaccurate. So in essence, Heisenberg’s argument reiterates what Edward Lorenz’s simulated weather system demonstrated… That every concept only has meaning in terms of the experiments used to measure it. And as these measurements would be imprecise as best; that is, in comparision to the size of the particle being measured, and any other subtle external forces that might effect it’s passage or being in this chaotic world of charge… Then we must agree that things which cannot be measured really or exactly, and thus elude any real bearing on their trajectory or course that they might take here in life… And thus these observations would surely hold no real meaning in physics, as physics, being a science based on exactitude, was designed to yield exact results that could provide workable models of the universe that exists around us. So… It must be noted that the path of a particle has no meaning beyond the precision with which it is observed.

Kurt Gödel and Albert Einstein near the IAS. Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, his ideas had an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Here I ask… Is this not what Kurt Gödel discovered with his incompleteness theorems? That ultimately there is the romantic notion that man imposes on his environment i.e. what he expects it to do, whether using self referential axioms to describe its expectant flow, or construct arbitary statements about what it actually is… Only later to discover that it is nothing like what he imagined it to orginally be? Perhaps we are doomed to make these approximations for all eternity when we observe the Lilliputian and Herculean levels of reality (this is where my idea that the implications of the Mandelbrot set could guide us into better modes of understanding regarding a universal truth, demonstration ever finer, more complex structures within and without, infinitely into and out of “precision” itself). Perhaps Georg Cantor was onto something with his idea of varying degrees of infinity, limited only by scaling factors i.e. the infinitely large might never hope to realize the infinitely small?

Georg Ferdinand Ludwig Phillip Cantor, best known as the creator of set theory, which has become a fundamental theory in mathematics.

But I digress… Once one gains a firm grasp of the basic facts that Heisenberg proposed, I would beckon them to recap on Deng Ming-Dao’s lesson on mindfulness, entitled “Make A Single Point.” Again, I do not want to lecture, but rather encourage the reader to think about the two separate ideas and find their own method of sewing them together… For to continually mend holes in the fabric of people’s perception could be misconstrued as being sanctimonious, which could never be further from my intentions. Rather I am happy to quote certain others’ works who have more pertinently and eloquently touched on aspects of this puzzel over the years, with a hope that the reader’s mind will naturally settle on the splendor lying behind the complex and distracting facades of catechism.

And also… With the hope that, rather than giving a tidbit to an inquisitive mind, perhaps the mind might find a mode for sewing new ideas into firtile folds of the brain’s structure and reap them in future times with reason’s sythe. For to give a man a fish, you might feed him for a day. Teach a man to fish, and you feed him for a lifetime.

The Uncertainty Principle

“The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”

It was in Copenhagen, in February of 1927, that Heisenberg developed his uncertainty principle, while working on the mathematical foundations of quantum mechanics. In his paper about this principle, he used the word “Ungenauigkeit” (imprecision). Heisenberg realized that the uncertainty relations had profound implications. First, if we accept Heisenberg’s argument that every concept has a meaning only in terms of the experiments used to measure it, we must agree that things that cannot be measured really have no meaning in physics. Thus, for instance, the path of a particle has no meaning beyond the precision with which it is observed. But a basic assumption of physics since Newton has been that a “real world” exists independently of us, regardless of whether or not we observe it. (This assumption did not go unchallenged, however, by some philsophers.) Heisenberg now argued that such concepts as orbits of electrons do not exist in nature unless and until we observe them.

Erwin Schrödinger was an Austrian theoretical physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. In 1935, after extensive correspondence with personal friend Albert Einstein, he proposed the Schrödinger's cat thought experiment.

With this idea, Heisenberg drew profound implications for the concept of causality, or the determinacy of future events. Schrödinger had earlier attempted to offer an interpretation of his formalism in which the electron waves represent the density of charge of the electron in the orbit around the nucleus. Max Born, however, showed that the “wave function” of Schrödinger’s equation does not represent the density of charge or matter. It describes only the probability of finding the electron at a certain point. In other words, quantum mechanics cannot give exact results, but only the probabilities for the occurrence of a variety of possible results.

Max Born was a Jewish-German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s. Born won the 1954 Nobel Prize in Physics.

Heisenberg took this one step further: he challenged the notion of simple causality in nature, that every determinate cause in nature is followed by the resulting effect. Translated into “classical physics,” this had meant that the future motion of a particle could be exactly predicted, or “determined,”  from a knowledge of its present position and momentum and all of the forces acting upon it. However… The uncertainty principle denies this, Heisenberg declared, because one cannot know the precise position and momentum of a particle at a given instant, so its future cannot be determined. One cannot calculate the precise future motion of a particle, but only a range of possibilities for the future motion of the particle. (However, the probabilities of each motion, and the distribution of many particles following these motions, could be calculated exactly from Schrödinger’s wave equation.)

Although Einstein and others objected to Heisenberg’s and Bohr’s views, even Einstein had to admit that they are indeed a logical consequence of quantum mechanics. For Einstein, this showed that quantum mechanics is “incomplete.” Research has continued to the present on these and proposed alternative interpretations of quantum mechanics.

One should note that Heisenberg’s uncertainty principle does not say “everything is uncertain.” Rather, it tells us very exactly where the limits of uncertainty lie when we make measurements of sub-atomic events. Heisenberg’s uncertainty principle constituted an essential component of the broader interpretation of quantum mechanics known as the Copenhagen Interpretation.

The Copenhagen Interpretation

Heisenberg formulated the uncertainty principle in February 1927 while employed as a lecturer in Bohr’s Institute for Theoretical Physics at the University of Copenhagen. Bohr, who had been on a skiing vacation, returned to the institute to find Heisenberg’s paper already in draft. Forwarding the paper to Einstein at Heisenberg’s request, Bohr complained to Einstein that Heisenberg’s approach was too narrow and his gamma-ray microscope was flawed, although the result was correct. For Bohr, the uncertainty relations arose not merely from the quantum equations and the use of particles and discontinuity. Waves and particles had to be taken equally into account, and the scattering of light waves by the electron was also crucial. When Heisenberg corrected his thought experiment, it only confirmed the results.

Niels Bohr and Albert Einstein debating quantum theory at Paul Ehrenfest's home in Leiden (December 1925).

In Bohr’s words, the wave and particle pictures, or the visual and causal representations, are “complementary” to each other. That is, they are mutually exclusive, yet jointly essential for a complete description of quantum events. Obviously in an experiment in the everyday world an object cannot be both a wave and a particle at the same time; it must be either one or the other, depending upon the situation. In later refinements of this interpretation the wave function of the unobserved object is a mixture of both the wave and particle pictures until the experimenter chooses what to observe in a given experiment. (Remember that, according to Heisenberg, the path of an object first comes into existence when we observe it.) By choosing either the wave or the particle picture, the experimenter disturbs untouched nature. Such favoritism unleashes a limitation in what one can learn about nature “as it really is.” This limitation is expressed by Heisenberg’s uncertainty relations, which, for Bohr, were related to what he was now calling “complementarity.” Complementarity, uncertainty, and the statistical interpretation of Schrödinger’s wave function were all related. Together they formed a logical interpretation of the physical meaning of quantum mechanics known as the “Copenhagen Interpretation.”

Heisenberg vehemently objected at first to Bohr’s views. Insisting on the primary use of particles and discontinuity, he refused Bohr’s suggestion that he withdraw his paper, which was already in press. He did, however, append a paragraph alerting readers to Bohr’s views and admitting the error regarding the resolution of the microscope. The battle with Bohr grew so intense in the early months of 1927 that Heisenberg reportedly burst into tears at one point, and even managed to wound Bohr with his sharp remarks. Obviously, there was much at stake for the 25-year-old.

By the fall of 1927, matters had completely changed. Heisenberg’s job situation was settled upon his appointment to the University of Leipzig. And Bohr presented to a conference at Lake Como, Italy, his complementarity argument. A month later, in October 1927, Born and Heisenberg, speaking to the Solvay physics conference in Brussels, Belgium, went so far as to declare quantum mechanics to be complete and irrevocable.

Not everyone agreed with the new interpretation, or with Born and Heisenberg’s statement about future work. Einstein and Schrödinger were among the most notable dissenters. Until the ends of their lives they never fully accepted the Copenhagen doctrine. Einstein was dissatisfied with the reliance upon probabilities. But even more fundamentally, he believed that nature exists independently of the experimenter, and the motions of particles are precisely determined. It is the job of the physicist to uncover the laws of nature that govern these motions, which, in the end, will not require statistical theories. The fact that quantum mechanics did seem consistent only with statistical results and could not fully describe every motion was for Einstein an indication that quantum mechanics was still incomplete.

The objections of Einstein and others notwithstanding, Bohr, Heisenberg and their colleagues managed to ensure the acceptance of their interpretation by the majority of physicists at that time. They did this both by presenting the new interpretation on lecture trips around the world and by demonstrating that it worked. The successes of the theory naturally attracted many of the best students to institutes such as Heisenberg’s, some coming from as far away as America, India, and Japan. These bright students, nurtured by the Copenhagen doctrine and educated into the new quantum mechanics, formed a new and dominant generation of physicists. Those in Germany and Central Europe carried the new ideas with them as they dispersed around the world during the 1930s and 1940s in the wake of Hitler’s rise to power in Germany.

Practical Application

The equations developed by Heisenberg, Schrödinger and their colleagues give us all a glimpse into the nature of reality… But that’s not just all. They are also the essential tools of modern work in key areas of practical technology – including the electronics you are using to read this text. Thousands of physicists use the equations of quantum mechanics every day to understand and improve computer components, metals, lasers, the properties of chemicals, and so on and so on. Many important physical effects, from fluorescent lights to the shape of a snowflake, cannot be understood at all without quantum mechanics.

Even the Uncertainty Principle isn’t “merely” philosophy: it predicts real properties of electrons. Electrons jump at random from one energy state to another state which they could never reach except that their energy is momentarily uncertain. This “tunneling” makes possible the nuclear reactions that power the sun and many other processes. Physicists have put some of these processes to practical use in microelectronics. For example, delicate superconducting instruments that use electron tunneling to detect tiny magnetic fields are enormously helpful for safely scanning the human brain…

So when something tends to be used with great success to yield desired results… We know it has some basis for being at least as right as it can be for the moment. Perhaps our brains, when we observe the flow of thoughts through them and atune them to finely honed points of focus, might one day function more precisely than they do presently? Food for thought, no doubt…