Taking the Quantum Leap, page 10
Schroedinger’s free particle : The instant after you find it, it spreads.
Imagine, if you will, the pulse as a closely bunched herd of horses galloping around a racetrack bend. The horses can stay together for only a short time. Eventually, the group spreads out as each horse assumes its own pace. The slowest horses fall to the rear of the group, while the fastest ones move to the front. As time goes on, the distance between the slowest and fastest horses lengthens. In a similar manner, the pulse grows fatter as its slower waves fall out of synchronization with its faster waves.
Though big objects, like baseballs, also were made of waves, the larger the object was initially, the slower its waves spread. Thus a baseball maintained its shape because it was so big to begin with. The Schroedinger pulse describing the baseball was no embarrassment.
But an electron was a horse of a different color. While it was confined within an atom, the electrical forces of the atomic nucleus held its waves in rein. Its waves were only allowed to spread over a region the size of the atom, no further. But when an electron was no longer in such confinement, when it was set free, the waves making up its tiny pulse-particle size would begin to spread at an extremely rapid rate. In less than a millionth of a second, the electron pulse-particle would become as big as the nearest football stadium! But, of course, no one has ever seen an electron that big. All electrons appear, whenever they appear, as tiny spots.
This contradiction between our observations of electrons and Schroedinger’s mathematical description of them uncovered a new problem: what prevented Schroedinger’s pulses from growing so large? Little did anyone realize that question was to open the doors of paradox and mystery and lead us to a quite different picture of the universe. The answer to the question was: human observation kept them from growing so large. We were on the verge of the discovery of a new discontinuity.
A “skinny” Schroedinger pulse gets wider as time marches on.
Chapter 6
No One Has Seen
the Wind
The universe is not only
queerer than we imagine,
but it is queerer
than we can imagine.
J. B. S. HALDANE
God Shoote Dice: The Probability Interpretation
It may be difficult for the nonscientist to imagine how repugnant the idea of a discontinuous movement of matter is to physicists who desire continuity. Starting with Einstein, the discontinuity in the movement of light was connected with a mechanical picture. Light consisted of granules. But then came Bohr and his quantum jumping electron inside of the tiny atom. This concept upset continuists because they could not understand how a particle could behave in this fashion. When de Broglie and Schroedinger appeared with their wave interpretation, the continuists breathed a sigh of relief.
Schroedinger’s picture of the atom, although complicated and dependent upon a nearly unimaginable wave function, was nevertheless quite reasonable. The atom’s electron was a wave. The atom radiated, not because its electrons jumped from orbit to orbit, but because of a continuous process of harmonic beats. The light was given out when the atom “music box” played both the upper energy and lower energy frequencies at the same time. The difference between the two electron matter-wave frequencies, which corresponded in Bohr’s conception of the atom to the difference in the electron’s orbital energies, was exactly the frequency of the light waves observed.
Gradually the upper frequency matter-wave tone quieted down, leaving only the lower harmonic. Thus the atom stopped radiating light. There was no longer a higher harmonic to beat against. The atom simply continued vibrating its electron wave at the lower frequency, which (according to the Planck E = hf formula and the de Broglie p = h/L formula) had to be unobservable and tucked safely away inside of the atom.
Later, Schroedinger’s picture would be destroyed, but his equation, his mathematical law, would remain. And he would express to Bohr, after days of long and arduous discussion, his disgust with ever having been involved in this quantum jumping thing. The problem was that, no matter how the wave shook and danced, there still had to be a particle somewhere. Max Born would be the first to provide an interpretation of this “particle” discontinuity. The wave was not the electron. It was a wave of probability.
In 1954, Professor Max Born was awarded the Nobel Prize for his interpretation of the wave function. The award came nearly thirty years after he first offered the interpretation.1 But then, Nobel Prizes come more slowly for ideas in physics than for experimental discoveries. Born explained his motives for opposing Schroedinger’s picture of the atom.2 He simply had too much connection with experimental work. He knew of the collision experiments being carried out in his own institute at Gottingen, Germany. Sophisticated refinements with vacuum techniques and electrical focusing of beams of electron particles had led to detailed studies of collisions between atoms and electrons. Despite the discovery of electron waves, these collision experiments were convincing evidence that the electron was still very much a tiny particle—literally, a hard nut to crack.
There was no doubt that Schroedinger’s mathematics worked. His equation described correctly all observable atomic phenomena. But how could the Schroedinger equation be used in those very same experimental collision studies at Born’s institute? In other words, what kind of wave function describes an electron beam colliding with a rarefied gas of atoms? Since the electrons in the beam were not confined within any atoms, they moved freely through space toward their eventual target, atoms.
Schroedinger’s pulse describing a single electron was inadequate. It just got too big too quickly. It couldn’t be a real tiny electron, the kind seen every day in Born’s laboratory. But Born knew his mathematics. Wide electron pulses spread slowly. So if a pulse was wide to begin with, it would hardly spread at all as it moved from one end of the apparatus to the other end. However, since the pulses had to be many times wider than an atom, how could the electron fit inside of an atom?
Born realized that, in the experiments at Gottingen, no one was really able to locate a single electron in a beam of electrons. Could it be that the width of the wave pulse was connected somehow with our knowledge of the location of each electron? When Born allowed the pulses in his mathematical equations to be as wide as the dimensions of the beam, he found that the spreads of the pulses virtually vanished.
Born’s efforts suggested that the time had come for a new interpretation of the meaning of the wave. The wave was not the reíd particle. Somehow the wave was connected with our knowledge of electron locations. It was, in fact, a probability function.
Probability functions are familiar today. They are used to describe the distribution of likely occurrences. A typical example is the probability function for a coin that is spinning in the air. As it falls, the probability function for it to land heads up is .50. Once the coin has landed, the probability function changes. If it has landed heads up, the probability function becomes 1. If it has landed tails up, the probability function becomes 0.
Max Born viewed Schroedinger’s wave as probability in space for finding the electron.
Insurance companies use probability functions to describe the distribution of automobile accidents. The stream of motorists driving into San Francisco each day is intense. That means the probability for any one car to collide with another is large. And the greater the intensity of the stream of motorists, the higher the probability of a collision. The situation is less intense, automotively speaking, in San Diego. Therefore, the probability density or distribution is lower for a collision to occur in that area. If we were to view the entire state of California from a satellite and watch all of the cars driving about, it would be quite easy for us to predict where collisions would be most likely to occur. We would simply note those areas where the flow of traffic was greatest—that is, most intense.
Born pictured the flow of electrons in much the same manner. Wherever there was a greater concentration of electrons in the beam, the Schroedinger wave had a greater intensity. By calculating that intensity, Born found he could predict the probability of a collision between an electron and an atom.
Born’s picture made a tremendous impression on his fellow physicists. Again, sighs of relief were heard coming from physics labs all across Europe. But the picture still had a hole in it. Born’s system made sense so long as it was applied to a beam or a concentration of actual numbers of particles. Physicists, like insurance actuaries, were quite used to probability ideas when they were dealing with a large number of practically uncountable events. In the experiments at Gottingen, the events were uncountable. But what about just one electron? One atom? How should the Schroedinger wave be interpreted in that case? Did the wave describe a single electron?
Was there a wave in these isolated cases? Was it, in other words, a real wave? And if the wave was a fundamental part of nature that belonged to each individual particle of nature, then who determined where the electron was to be found? Was nature essentially a probability game? Did God play dice with the universe?
A new interpretation was sought. Something was wrong with the probability picture. But what could replace it? The answer to this question had been brewing in Germany since the end of World War I. A new and revolutionary principle of reality was occurring to one man, a principle that was to completely change our thinking about the physical world.
Heisenberg’s Uncertainty Principle: The End of Mechanical Models
If I had a time machine and could return to any period of time, which period would I choose? I would pick the Roaring Twenties; however, it would not be the United States that I would return to. No, indeed. Instead, I would go to post-World War I Germany. And because I am fascinated by pseudodecadence and café society, you would find me among such contemporaries as Bertold Brecht and Thomas Mann. Bauhaus art and design would be flourishing and outrageous Dada art would be creating “authentic reality” through the abolition of traditional cultural and aesthetic forms by the technique of comic derision. For, during that period, irrationality, chance, and intuition were guiding principles. Freud was out. Jung and Adler were in. Life had a certain cabaret feeling.
Now add the physicists. Though they numbered perhaps less than a hundred, a new breed of young, enthusiastic fellows was making its way into the new physics. Planck was past sixty. Einstein had seen his fortieth birthday. Bohr was a middle-aged thirty-five. These older and wiser moderates were to be the guiding lights for the new breed. It was time for Dada physics, and it was happening in Gottingen, Germany. In early summer, 1922, Professor Niels Bohr, who then headed a brand-new institute of physics in Denmark called the Copenhagen School, had come to give a lecture.
Among the students who had gathered to hear Bohr was twenty-year-old Werner Heisenberg. This occasion would be the first of many meetings between Heisenberg and Bohr. Together these two would change the meaning of physics. Eager to rid physics of mechanical models, they would herald a new school, a school of discontinuists. Their interpretations would lead to a revolution of thought.
Heisenberg wrote of this first meeting with Bohr in his book, Physics and Beyond. After some remarks concerning Bohr’s atomic theory, he wrote:
Bohr must have gathered that my remarks sprang from profound interest in his atomic theory…. He replied hesitantly … and asked me to join him that afternoon on a walk over the Hain Mountain…. This walk was to have profound repercussions on my scientific career, or perhaps it is more correct to say that my real scientific career only began that afternoon…. Bohr’s remark [that afternoon] reminded me that atoms were not things….3
But if atoms were not “things,” then what were they? Heisenberg’s answer was that all classical ideas about the world had to be abandoned. Motion could no longer be described in terms of the classical concept of a thing moving continuously from one place to another. This idea only made sense for large objects; it did not make sense if the “thing” was atom-sized. In other words, concepts are reasonable only when they describe our actual observations rather than our ideas about what we think is happening. Since an atom was not seen, it was not a meaningful concept.
Young Heisenberg: A vison of uncertainty?
Heisenberg’s thoughts had been influenced by Einstein. In 1905, Einstein had carefully laid out the steps to relativity. He had recognized that, in order to speak about such notions as space and time, one must provide operational definitions—definitions that detailed how these things were measured. For example, space is what a ruler measures and time is what a clock measures. For anyone armed with these empirical and objective forms, space and time lose their mystery. Everyone holding rulers and clocks can agree on the definitions, because they can agree on which operations to do with these instruments.
A concept is useful when we all know how its measurement is to be accomplished. This viewpoint led Heisenberg to question any concept that had no operational definition. Atoms were not observable, but the light coming from them was observable. Thus, Heisenberg developed a new form of mathematical tools based upon the frequencies of the light that was seen, rather than the position and momentum of an unobservable electron within an unseen atom. These new mathematical tools were developed from the mathematics of operators and not the mathematics of numbers.
An operator in mathematics performs a duty. It changes or modifies a mathematical function in a defined way. For example, the operator called “square” will multiply any mathematical function by itself. (Thus when “square” operates on “x,” it makes “x2.” When it operates on 5, it makes 25, etc. Operators are also capable of being operated upon. Thus “square” can be multiplied by the number 3, which can be an operator as well as a simple number. This makes “3*square,” a new operator. When “three*square” operates upon 5 it makes 75 instead of 25. Two or more operators can also be multiplied together.) Heisenberg discovered, with the help of Max Born, that his mathematical operators, which corresponded to the observed frequencies and intensities of the light from atoms, obeyed a strange law of multiplication. The order in which you multiplied the operators was important. If the operators were, for example, A and B, then AB did not equal BA. (If we use the previous example of “three”square,”we see that “three*square” is not the same as “square*three.” For when “three*square” operates on 5 it gives 75, but “square “three” operating on 5 gives the result of multiplying 3 times 5 and then squaring. This gives 225, instead. Thus “three*square does not equal “square*three.”) Did that mean that the physical world depended as well upon the order in which you observed things?
Later, Born and Pascual Jordan carried Heisenberg’s mathematics a step further. They were guided by Bohr’s Principle of Correspondence that showed that the classical mechanical viewpoint would correspond with the quantum mechanical viewpoint whenever the quantum numbers describing the old Bohr orbits were quite large compared with 1. By following this principle, they were able to find mathematical operators for the position and the momentum of the electron instead of the frequencies and intensities used by Heisenberg. The surprising fact was that these operators, too, depended on the order in which they were carried out. A new and previously unsuspected picture of the universe was emerging.
The new tools of operator algebra were later found to be related to the mathematics of matrices. A matrix, an array of numbers, must be handled in a careful, well-defined way. The rules governing the use of matrices were found to be identical to the mathematical rules used to handle operators. Consequently, Heisenberg’s development of quantum mechanics came to be called matrix mechanics. The wave mechanics of de Broglie and Schroedinger was still being investigated, however, and eventually it became apparent that the two forms of mathematical expression were simply disguised versions of the same thing. Schroedinger discovered this4 and offered formal mathematical proof of their equivalence. For a while, interest in the purely operational matrix mechanics dropped.
Yet Heisenberg wasn’t ready to dismiss the insights he had garnered from his matrix mechanics. He began to explore his observational basis for reality using the Schroedinger wave. Born’s probability interpretation suggested how he should proceed, and, following the tradition of Einstein, Heisenberg attempted to describe the method by which the position and momentum of an atom-sized object could be measured.
To see something, we must shine light on it. Determining the location of an electron would require our sense of sight. But for so tiny an object as an electron, Heisenberg knew a special kind of microscope would be needed. A microscope magnifies images by catching light rays, which were originally moving in different directions, and forcing them all to move in much the same direction toward the awaiting open eye. The larger the aperture or lens opening, the more rays of light there are to catch. In this way, a better image is obtained, but the viewer pays a price for that better image.
The price is that we don’t know the precise path taken by the light ray after it leaves the object we were trying to view in the first place. Oh, we will see it all right—a fraction after its collision with the little light photon that was gathered up by the microscope. But was that photon heading north before it was corralled by the lens, was it heading south, or southwest? Once the photon is gathered in, that information gets lost.
But so what? We do get an exact measurement of the position of the electron. We can point out just where it was. Well, not exactly. We still must worry about the kind of light we use. Try to imagine painting a fine portrait the size of a Lincoln penny. What kind of brush would you use? The finer the hairs of your brush, the better your ability to create the miniature. If you were to reduce the size of the portrait even further, you would need an even finer brush.
