Math II (Несколько текстов для зачёта), страница 25

By Max Tegmark and John Archibald Wheeler

Inset Article

QUANTUM CARDS

A SIMPLE FALLING CARD IN PRINCIPLE LEADS TO A QUANTUM MYSTERY

According to quantum physics, an ideal card perfectly balanced on its edge will fall down in both directions at once, in what is known as a superposition. The card's quantum wave function (blue) changes smoothly and continuously from the balanced state (left) to the mysterious final state (right) that seems to have the card in two places at once. In practice, this experiment is impossible with a real card, but the analogous situation has been demonstrated innumerable times with electrons, atoms and larger objects. Understanding the meaning of such superpositions, and why we never see them in the everyday world around us, has been an enduring mystery at the very heart of quantum mechanics. Over the decades, physicists have developed several ideas to resolve the mystery, including the competing Copenhagen and many-worlds interpretations of the wave function and the theory of decoherence.

Inset Article

COPENHAGEN INTERPRETATION

IDEA: Observers see a random outcome; probability given by the wave function.

ADVANTAGE: A single outcome occurs, matching what we observe.

PROBLEM: Requires wave functions to "collapse,"but no equation specifies when.

When a quantum superposition is observed or measured, we see one or the other of the alternatives at random, with probabilities controlled by the wave function. If a person has bet that the card will fall face up, when she first looks at the card she has a 50 percent chance of happily seeing that she has won her bet. This interpretation has long been pragmatically accepted by physicists even though it requires the wave function to change abruptly, or collapse, in violation of the Schrodinger equation.

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MANY-WORLDS INTERPRETATION

IDEA: Superpositions will seem like alternative parallel worlds to their inhabitants.

ADVANTAGE: The Schr6dinger equation always works: wave functions never collapse. ..BI.-PROBLEMS:

The bizarreness of the idea. Some technical puzzles remain.

If wave functions never collapse, the Schrbdinger equation predicts that the person looking at the card's superposition will herself enter a superposition of two possible outcomes: happily winning the bet or sadly losing. These two parts of the total wave function (of person plus card) carry on completely independently, like two parallel worlds. If the experiment is repeated many times, people in most of the parallel worlds will see the card falling face up about half the time. Stacked cards (right) show 16 worlds that result when a card is dropped four times.

Inset Article

DECOHERENCE: HOW THE QUANTUM GETS CLASSICAL

IDEA: Tiny interactions with the surrounding environment rapidly dissipate the peculiar quantumness of superpositions.

ADVANTAGES: Experimentally testable. Explains why the everyday world looks "classical" instead of quantum.

CAVEAT: Decoherence does not completely eliminate the need for an interpretation such as many-worlds or Copenhagen.

The uncertainty of a quantum' superposition (left) is different from the uncertainty of classical probability, as occurs after a coin toss (right).A mathematical object called a density matrix illustrates the distinction. The wave function of the quantum card corresponds to a density matrix with four peaks. Two of these peaks represent the 50 percent probability of each outcome, face up or face down. The other two indicate that these two outcomes can still, in principle, interfere with each other. The quantum state is still "coherent." The density matrix of a coin toss has only the first two peaks, which conventionally means that the coin is really either face up or face down but that we just haven't looked at it yet.

Decoherence theory reveals that the tiniest interaction with the environment, such as a single photon or gas molecule bouncing off the fallen card, transforms a coherent density matrix very rapidly into one that, for all practical purposes, represents classical probabilities such as those in a coin toss. The Schrbdinger equation controls the entire process.

SPLITTING REALITY

It is instructive to split the universe into three parts: the object under consideration, the environment, and the quantum state of the observer, or subject. The Schrodinger equation that governs the universe as a whole can be divided into terms that describe the internal dynamics of each of these three subsystems and terms that describe interactions among them. These terms have qualitatively very different effects.

The term giving the object's dynamics is typically the most important one, so to figure out what the object will do, theorists can usually begin by ignoring all the other terms. For our quantum card, its dynamics predict that it will fall both left and right in superposition. When our observer looks at the card, the subject-object interaction extends the superposition to her mental state, producing a superposition of joy and disappointment over winning and losing her bet. She can never perceive this superposition, however, because the interaction between the object and the environment (such as air molecules and photons bouncing off the card) causes rapid decoherence that makes this superposition unobservable.

Even if she could completely isolate the card from the environment (for example, by doing the experiment in a dark vacuum chamber at absolute zero), it would not make any difference. At least one neuron in her optical nerves would enter a superposition of firing and not firing when she looked at the card, and this superposition would decohere in about 10-20 second, according to recent calculations. If the complex patterns of neuron firing in our brains have anything to do with consciousness and how we form our thoughts and perceptions, then decoherence of our neurons ensures that we never perceive quantum superpositions of mental states. In essence, our brains inextricably interweave the subject and the environment, forcing decoherence on us.

M. T. and J.A.W.

The Authors

MAX TEGMARK and JOHN ARCHIBALD WHEELER discussed quantum mechanics extensively during Tegmark's three and a half years as a postdoc at the Institute for Advanced Studies in Princeton, N.J. Tegmark is now an assistant professor of physics at the University of Pennsylvania. Wheeler is professor emeritus of physics at Princeton, where his graduate students included Richard Feynman and Hugh Everett III (inventor of the many-worlds interpretation). He received the 1997 Wolf Prize in physics for his work on nuclear reactions, quantum mechanics and black holes. In 1934 and 1935 Wheeler had the privilege of working on nuclear physics in Niels Bohr's group in Copenhagen. On arrival at the institute he asked a workman who was trimming vines running up a wall where he could find Bohr. "I'm Niels Bohr," the man replied. The authors wish to thank Emily Bennett and Ken Ford for their help with an earlier manuscript on this topic and Jeff Klein, Dieter Zeh and Wojciech H. Zurek for their helpful comments.

REASSESSING AN ANCIENT ARTIFACT

Source: Science News, 01/27/2001, Vol. 159 Issue 4, p56, 1/3p

Author(s): Peterson, Ivars

From New Orleans at the Joint Mathematics Meetings

The famous Mesopotamian clay tablet known as Plimpton 322 has tantalized historians of mathematics ever since its discovery more than 60 years ago. Scholars have considered the tablet to be an anomalous mathematical exercise well in advance of its time. They have variously interpreted the cryptic columns of numbers, written in the wedge-shaped script called cuneiform, as a trigonometric table or a sophisticated scheme for generating Pythagorean triples. A Pythagorean triple is a set of three whole numbers, a, b, and c, such that a[2] + b[2] = c[2].

Now, Eleanor Robson of the Oriental Institute at the University of Oxford in England offers an alternative explanation of the tablet's purpose. The tablet served as a guide for a teacher preparing exercises involving squares and reciprocals, she suggests. Robson also pinpoints the tablet's date to within 40 years of 1800 B.C. and says that it probably came from Larsa, a Mesopotamian city about 100 miles southeast of Babylon.

Previous historians had typically failed to consider the tablet's cultural context and relied on later mathematical developments to infer its purpose. For example, the concept of angle measurement, which is essential for a trigonometric table, was not developed until nearly 2,000 years after the tablet was made. New scholarly approaches to Mesopotamian mathematics, however, combine historical, linguistic, and mathematical techniques to address questions such as, How did Mesopotamians approach mathematical problems, and what role did these problems play in their society? "We need to understand the document in its historical and cultural context," Robson says. "Neglecting these factors can hinder our interpretations."

By comparing Plimpton 322 with other ancient tablets, Robson established that its style is consistent with temple records and documents of about 1800 B.C. in Larsa. Scrutiny of various mathematical tablets revealed the importance of computational methods based on reciprocals (1/x) and squares (chi square) of numbers. Robson also found examples of student exercises that consisted of problem lists, each one registering essentially the same problem with slightly different numbers.

Such evidence enables modern mathematicians to view Plimpton 322 "not as a freakish anomaly in the history of early mathematics but as the epitome of Mesopotamian mathematical culture at its best," Robson says. "It's a well-organized, well-executed, beautiful piece of mathematics." Robson describes her findings in a report scheduled for publication in HISTORIA MATHEMATICA.

PRIME PROOF ZEROS IN ON CRUCIAL NUMBERS

Source: Science News, 12/02/2000, Vol. 158 Issue 23, p357, 1/2p

Author(s): Peterson, I.

Fermat's last theorem is just one of many examples of innocent-looking problems that can long stymie even the most astute mathematicians. It took about 350 years to prove Fermat's tantalizing conjecture.

Now, Preda Mihailescu of the Swiss Federal Institute of Technology in Zurich has proved a theorem that is likely to lead to a solution of Catalan's conjecture, another venerable problem involving relationships among whole numbers. He describes his result in a paper to be published in the Journal of Number Theory.

"This is a very important contribution," says mathematician Andrew Granville of the University of Georgia in Athens. Mihailescu's work probably puts the resolution of Catalan's problem into the foreseeable future, he notes.

Named for Belgian mathematician Eugone Charles Catalan, the conjecture concerns powers of whole numbers. For example, the sequence of all squares and cubes of whole numbers greater than 1 begins with the integers 4, 8, 9, 16, 25, 27, and 36. In this sequence, 8 (the cube of 2) and 9 (the square of 3) are not only powers but also consecutive whole numbers.

In 1844, Catalan asserted that among powers of whole numbers, the only pair of consecutive numbers that arises is 8 and 9. Since then, Catalan's conjecture has posed a challenge to number theorists akin to that provided by Fermat's last theorem (SN: 11/5/94, p. 295).

Solving Catalan's problem amounts to a search for whole number solutions to the equation x[superscript]p - y[superscript]q = 1, where x, y, p, and q are all greater than 1. The conjecture suggests that there is only one such solution: 3[superscript]2 - 2[superscript]3 = 1.

In a major step toward resolving Catalan's conjecture, Robert Tijdeman of the University of Leiden in the Netherlands showed in 1976 that even if it is not true, there is a finite rather than an infinite number of solutions to the equation. In effect, each of the exponents p and q must be less than a certain value.

Last year, Maurice Mignotte of the Universite Louis Pasteur in Strasbourg, France, demonstrated that p had to be less than 7.15 5 1011 and q less than 7.78 5 10[superscript]16. Meanwhile, computations showed that no consecutive powers other than 8 and 9 occur below 10[superscript]7.

In the latest advance, Mihailescu proved that, if additional solutions to the equation exist, the exponents p and q are a pair of what are known as double Wieferich primes. These pairs obey the following relationship: p[superscript](q - 1) must leave a remainder of 1 when divided by q[superscript]2, and q[superscript](p - 1) must leave a remainder of 1 when divided by p[superscript]2. The pair of prime numbers 2 and 1,093 fits this relationship.

Only six examples of double Wieferich primes have been identified so far. All of these pairs are below the range specified by the computations addressing Catalan's conjecture. A major collaborative computational effort (http://www.ensor.org) has now been mounted to find additional double Wieferich primes, but mathematicians are betting that a theoretical approach to proving Catalan's conjecture will beat out the computers.

MATH & MUSIC: THE MAGICAL CONNECTION

Source: Scholastic Parent & Child, Dec2000/Jan2001, Vol. 8 Issue 3, p50, 5p, 3c

Author(s): Church, Ellen Booth

Math and music unite the two hemispheres of the brain--a powerful force for learning.

did you ever consider the skills your child uses when she sings a song such as "This Old Man"? She is matching and comparing (through pitch, volume, and rhythm), patterning and sequencing (through melody, rhythm, and lyrics), and counting numbers and adding. Add dramatic hand movements or clapping to the beat, and you have created an entire package of learning rolled into one song!

In recent years, there has been a considerable amount of research on the effect of music on brain development and thinking. Neurological research has found that the higher brain functions of abstract reasoning as well as spatial and temporal conceptualization are enhanced by music activities. Activities with music can generate the neural connections necessary for using important math skills.