Mathematics and the Real World (27 page)

30. THE WONDER EQUATION

It was Erwin Schrödinger (1887–1961), an Austrian physicist and mathematician who studied and lived in Germany, who offered a mathematical explanation of the quantum effect. He presented what became known as Schrödinger's equation in a paper he wrote in 1926, when he was a professor at the University of Zurich, an equation that earned him the Nobel Prize in 1933 jointly with the British physicist Paul Dirac (1902–1984). Later he moved to Berlin but left Germany in 1933 as an anti-Nazi gesture and moved to England. From there he went on to the United States, but after a short stay in Princeton he moved to England, then to Scotland and finally, in 1936, decided to return to Austria, to the University of Graz. There he had to issue a retraction of the censure of the Nazis he had published three years earlier. Yet, his position in Graz was canceled and he moved in 1940 to the Republic of Ireland, which was neutral during World War II, where he founded the Institute of Advance Studies in Dublin. He later apologized to Einstein for his apparent support of Nazism, for Einstein was and remained his good friend.

The explanation, if it can be called that, suggested by Schrödinger for the various effects was a differential equation of the wave, or string-type, equation (see section 22) and as such was similar to Maxwell's equation. Schrödinger's equation was more complex and contained new mathematical elements in its description of nature (the details of which are not relevant here). Just as the wave equation has solutions with “clean,” natural, characteristic frequencies, Schrödinger's equation also has such solutions. The characteristic solutions of the original string equation relate to effects that we feel, for example the “clean” vibrations of the strings of a musical instrument. Erwin Schrödinger proposed that the characteristic solutions to his equation describe the electrons of the atom, with the characteristic frequency describing the energy, while the length of the wave is related to the momentum of the electron, that is, the multiple of the speed and the mass. Just as the different elements in the string equation apply to different strings, here too different elements in Schrödinger's equation apply to different particles. A mathematical analysis of the equation that Schrödinger proposed for the known atoms showed a perfect match with all the results and experimental measurements obtained until then.

Again the information is in the form, the foundation of which was set by Maxwell: all the available information is explained using an equation whose relevance is not clear, but whose solutions enable new effects to be predicted. If the electron is the solution to Schrödinger's equation, then it is a wave. Yet no one had seen the wave of an electron. What does “the electron is a wave” mean? What is the medium in which the wave propagates? The answer is that the electron is a wave only insofar as it solves the Schrödinger equation, and even that solution we cannot perceive with our senses but can only measure certain quantities related to the frequency of the wave and its momentum or to the fact that a wave undulates. However, even in Schrödinger's days, science had already become accustomed to the fact that physics was mathematics that describes phenomena that cannot be perceived directly by the senses, and it can be understood only via its indirect effects, and therefore the equation was accepted as an equation that describes physics, meaning, it was accepted as physics.

The human brain, however, cannot absorb and deal with abstract quantities
without an intuitive picture, and such intuition can be founded only upon known concepts. This limitation brought about attempts to clarify the nature of the solutions to Schrödinger's equation. Schrödinger himself interpreted the fact that the electron is described by a wavelike function to mean that the charge of the electron is in effect spread around the nucleus, and the wave describes how the charge is spread. Another interpretation, innovative and original, was given by the German scientist Max Born (1882–1970), earning him the Nobel Prize in Physics in 1954. He suggested that the wave describes the probability of finding the electron in a particular spot. The probability itself is given by the square of the height of the wave at every location. Thus Born introduced a completely new element into the description of nature: randomness, lack of determinism. It is not lack of knowledge that brings about the effect that appears random, nor a statistical approximation, such as mechanical statistics uses statistics because of the inability to analyze every single particle, but randomness that is inherent in nature itself. Moreover, according to Born, when a force is exerted on an electron, such as when measuring its location, the randomness disappears. The electron behaves like a particle, and the wave focuses, or in the language of physicists, collapses, in the exact location of that particle.

One of the conclusions drawn was that an electron can pass simultaneously through two holes, but when its location is measured it loses that property and will “decide” through which hole it has passed. It is difficult to absorb these properties because they are formulated in terms of day-to-day objects, and in our daily lives we do not encounter such properties. The reason is that electrons are not objects about which evolution has taught us how to develop intuition.

Born's interpretation was not accepted unquestioningly. Einstein, for instance, opposed it vigorously, stating that “God does not play with dice.” After a while the idea spread in popular literature that Einstein was opposed to the whole quantum theory. That is incorrect. On the contrary, as we saw in the previous section, he was one of the founding fathers of the theory and certainly fully accepted Schrödinger's equation as describing physics as it is. Einstein was just opposed to the interpretation proposed by Born, which assumed a law of nature that permitted randomness. Nonetheless, Born's
interpretation proved to be a reliable instrument for analyzing physics and as a source of proven hypotheses. Today Born's interpretation is accepted as a correct description of nature, of course at the subatomic level.

A mathematical analysis carried out by a student of Bohr and Born, the German scientist Werner Heisenberg (1901–1976), which earned him the Nobel Prize in Physics in 1932, yielded the
uncertainty principle
. This principle stated that it was impossible to determine accurately either the location or the momentum of an electron. Mountains of interpretations and implications, even including analysis of our daily lives, have been derived from this principle, most of them with no logical basis. The principle is a mathematical one. To formulate it mathematically requires some concepts to be developed that are beyond the scope of this book (for the mathematicians among the readers, the principle drives from the fact that the location operator and the momentum operator do not commute), but its implications for physics can be described.

The electron has both wave and particle properties. As a particle it is located at a defined spot, but to calculate its momentum we need to consider the wave as a whole. As soon as the location of the electron is measured, it stops being a wave and its momentum cannot be determined exactly. When calculating the momentum, the wave aspect of the electron is used, and its exact location is not known. This applies to electrons and to other particles. It is not relevant to situations and objects that are not simultaneously waves and particles. The principle is relevant to other situations in which the effect has two complementary aspects, such as signal processing. The signal can be described both by its frequency and by its progress in time. Both are parallel to the location and the momentum of the particle, and the uncertainty principle states that it is impossible to simultaneously describe both exactly (for the mathematicians, the product of the second moment of the function and its Fourier transform is bounded from below). These are mathematical principles that can be interpreted in mathematical uses, but caution must be exercised in trying to apply them in other situations.

It is worth repeating and emphasizing the new system of relations that appeared and developed at that time between nature, mathematics, and
the description of nature. The basis consists of nature itself. We describe it with the help of mathematical equations. The equations can describe quantities whose properties are not directly accessible to us and of which we have no perception or feeling. The justification for the correctness of the equations is purely and simply that there is a match between the effects they predict and experimental results, and they constitute a source for additional discoveries. In order to analyze the behavior of the solutions to the equations, our brains need an interpretation in terms that they can imagine. The interpretation is “correct” only to the extent that with its help we can perform an analysis of the solutions to the equations. We have no alternative but must learn about nature in terms of the interpretation that we have developed. We must always bear in mind, however, that it is merely an interpretation of the mathematics that describes nature; nature itself can produce surprises.

31. GROUPS OF PARTICLES

At the beginning of the 1930s the subatomic situation was relatively simple: the atom was made up of a nucleus, inside which were neutrons and protons, and around which revolved things that were waves and also particles, that is, electrons. Light particles were also known, that is, photons. But it soon became apparent that subatomic reality was far more complex. First, following precise experiments that analyzed the frequency of the radiation it was found that there was not just one type of electron. In effect there are two types. The difference between the two types of radiation that they created was explained by the suggestion that the electron, while revolving around the nucleus, also revolves on its own axis, and the direction of this movement, to the left or to the right, is what gives the different types of radiation. The physicists called this turning on its axis “spin.” Again, there is no guarantee that the electron particle, which is also a wave, does actually spin on its own axis. This property, however, would provide a good explanation for the difference between the two types of electrons.

Then the positron was discovered, a particle similar to an electron but
with a positive charge. The positron was discovered as a result of a mathematical analysis of Dirac's equation, which is a version of Schrödinger's equation adapted for the electron, yet with an additional solution with a positive charge. This solution indicated the possibility that an “antimatter” particle existed. If this particle meets the “matter” particle, they will both disappear and become energy. Within a short time it was indeed found that the mathematical solution, somewhat exotic we might add, was realized by a real particle in nature, the positron. Within a few years other “matter” (and “antimatter”) particles were discovered, which were then given the collective name of elementary particles. Initially these particles were studied by examining cosmic radiation and the results of the cosmic particles striking the earthly atoms. Cosmic radiation has great energy, but a large part of it is absorbed in the atmosphere. Research into the elementary particles was therefore then carried out by raising photographic plates to a great height and documenting the impact between the cosmic radiation and the earthly particles. Later, other means were developed, such as bubble chambers and later still particle accelerators, that recorded the collision between the accelerated particles and other particles and the changes that the collisions caused, including the creation of new particles. These were characterized by their energy, the frequency of their wave, their mass, and their spin, the level of which increased and now not only reflected direction but was also given values of a half, a third, and so on. The episode of creating a picture of the subatomic world is fascinating but falls outside the scope of this book. Suffice it to say here that a list was formed of the elementary particles, but to understand the order underlying it required mathematics. Schrödinger's equations, other equations that developed from them, Born's interpretation, and what was derived from them were sufficient to describe properties of the particles but did not explain their allocation according to the various properties. For that, a mathematical element that had not been used before in this area was incorporated, namely, groups.

The classification of the various particles by their properties led to their being arranged in tables according to their type. A particular type of particle is called a hadron. Two scientists, Murray Gell-Mann of Caltech in California and the Israeli Yuval Ne'eman (1925–2006), who was then (in 1961)
at Imperial College, London, and later at Tel Aviv University, both noticed in that year, independently, that hadrons can be placed in several tables according to the characteristics of their spins so that each table consisted of exactly eight particles. Moreover, the relation between the particles in every table matched what the mathematicians had for a long time called the SU(3) group.

It is not necessary to get involved deeply in group theory to understand its role in describing the physical situation. A group is a collection of mathematical elements and the relations between them, for example, revolving the plane through either 90, 180, 270, or 360 degrees, with the last of those alternatives bringing us back to the starting position. This is a group whose elements are the turns, and the action between every two elements is the turn obtained after two consecutive revolutions. That is to say the relationships are: a turn of 180 degrees followed by a turn of 270 degrees is equal to a turn of 90 degrees, and so on. This is a group with four elements. To describe such turns, there is clearly no need to call them a group and to use sophisticated mathematics. What is special about mathematical terminology is that it enables us to describe more-complex systems. For instance, turning a die through 90 degrees in one direction around one of the three axes will define a more complex group. Mathematics studies even more complex groups and groups in which the relations between the elements are more complicated.