Math (Несколько текстов для зачёта), страница 7

where P is its period of revolution. Substituting for v in the force formula gives

F = m[sub p] v[sup 2]/r

= m[sub p] 4 Pi[sup 2] r[sup 2]/r P[sup 2]

= M[sub p] 4 Pi[sup 2] r/P[sup 2]

Kepler's third law says that P[sup 2] = kr[sup 3], where k is a proportionality constant. Substituting further then gives

F = m[sub p] 4 Pi[sup 2] r/kr[sup 3]

= 4 Pi[sup 2]/k x m[sub p]/r[sub r],

that is,

F proportional to m[sup p]/r[sup 2].

The sun exerts that force, F, on the planet. At a given distance r, it is proportional to the mass of the planet.

By Newton's third law of motion, the planet exerts the same force on the Sun. Since the Sun's force on the planet depends on the mass of the planet, it seems reasonable to suppose that the planet's force on the Sun depends on the mass of the Sun, which means that the mutual force depends on both masses. The mutual force cannot depend on the sum of the masses, since doubling a mass doubles the force but doubling one term of a sum does not double the sum; that is, a + 2b is not twice a + b. Newton assumed that the force depended on the product of the masses, an assumption that agrees with the result that doubling either factor in a product doubles the product, that is, (2a) x b = a x (2b) = 2(ab). So for the Sun and a planet, the mutual force of attraction, F, is

F proportional to m[sub s] m[sub p]/r[sup 2]

or

F proportional to m[sub s] m[sub p]/r[sup 2]

where G is a constant that needs to be determined by experiment and r is the distance between the centers of the two objects. Newton spent many years investigating this phenomenon and it took the invention of a little thing called calculus to prove that r is the distance between the centers of the objects.

The preceding result is Newton's law of universal gravitation. By universal, Newton meant that it applies equally to all objects, both terrestrial and celestial. To test his law, Newton compared the falling of an object at the surface of Earth (the famous apple?) to the falling of the Moon. Figure 10 (Feynman 1995) shows what is meant by a "falling" Moon. In one second, the Moon travels from A to B in its orbit. If Earth did not attract the Moon, it would travel along the tangent instead. Thus the distance s is the distance it has "fallen." In right triangle ABC,

s/x = x/2r - s approximately equal to x/2r,

since s < x < r; or

s approximately equal to x[sup 2]/2r.

The quantity x is the distance that the Moon travels in one second. Since the moon's average distance from the center of Earth is about 385 000 km,

x = 1 second/1 month x 2 Pi r approximately equal to 4.24 x 10[sup -7] x 2 Pi x 3.85 x 10[sup 8] m approximately equal to 1026 m.

Substituting this result into the previous formula gives

s approximately equal to x[sup 2]/2r

approximately equal to (1026m)[sup 2]/2 x 3.85 x 10[sup 8] m

approximately equal to 0.0014 m.

So 0.0014 m is the distance that the Moon falls in one second. At the surface of Earth, which is 6400 km from its center, an object falls about 5 m in one second. If Earth's gravitational pull varies inversely as the square of the distance from its center, Newton reasoned, then the distance that the Moon falls in one second should be (6400/385000)[sup 2], or approximately 0.00028 times the distance that an object on Earth falls in one second. Our figures agree with Newton's reasoning, since (0.00028)(5) approximately equal to 0.0014.

DETERMINING THE ASTRONOMICAL UNIT

The traditional way to determine the distance to inaccessible objects is by triangulation. Triangulation to very distant objects is usually done using the concept of parallax. Parallax describes the phenomenon that occurs when you hold your finger in front of your face and alternately close your left eye and your right eye. Your finger appears to shift its position with respect to the background. Figure 11 illustrates how this idea can be used to measure the distance to object O. First stand at a position A so that object O is aligned with some other object, much farther away than O. Then move to the side to a new position B. Object O and the more distant object are no longer aligned but rather subtend some angle q at your eye. This angle can be measured. If the more distant object is sufficiently far away, the lines to it from A and B are nearly parallel and <p approximately equal to <q. Angle p is called the parallax angle. As long as angle p is small, the baseline AB can be taken as an arc of a circle with center at O and radius x. Since AB and the parallax angle can be measured, distance x can be calculated using the arc-length formula from geometry, giving

AB = p/360 2 Pi x arrow right = 180 x AB/Pi p

Since distances to astronomical objects are so enormous, angle p is always very small, sometimes only a fraction of a second of arc; and so approximating a segment with an arc does not make any measurable difference. The smallness of angle p also explains why measuring the base angles at A and B, as would be done in solving a triangle that was less "long and skinny," is impractical.

Students cannot collect their own data for most of the problems in this article, but they can get hands-on experience using parallax to measure distances, as my geometry classes have done. In addition to a tape measure, the only equipment needed is a device that measures small angles with some accuracy. Working in groups of four, my students made their own parallax-measuring devices using a piece of Styrofoam about 60 cm by 15 cm. See figure 12. About 50 cm from one end, they placed a row of pins spaced so that adjacent pins would subtend angles of 0.5 degree when viewed from that end. Of course, they had to use the previously mentioned arc-length formula to calculate how far apart to place the pins. After making their measuring devices, they practiced measuring small angles and distances in the classroom, where I could be certain that they knew what they needed to do and what quantities they needed to measure, that is, angle q and baseline AB. I then sent the groups outside after giving each group a description of some specific object on campus, such as a water tower or telephone pole; a specific place to stand to measure its distance; and a specific object to use as the distant background object. I also gave each group a photograph with these objects marked, to help them orient themselves.

To find the length of the astronomical unit, astronomers could in theory measure the parallax of the Sun from two points on Earth's surface. Obtaining this measurement is nearly impossible in practice, though, because of the Sun's angular size, brightness, and distance. However astronomers can triangulate the distance to some planet, say, Mars, to obtain its distance from Earth in miles or kilometers. Its distance is already known in AU, and from this result, the size of an AU can be calculated. This result then gives the scale of the entire Solar System.

The story told here of necessity is incomplete, but I hope that it is tantalizing enough with all its connections to history and science to encourage interested teachers and students to explore the topic further.

The author thanks Craig Merow for his assistance in preparing this manuscript for publication.

TABLE 1 Orbital Radii of Planets, in AU

Legend for Chart:

A - Planet

B - Copernicus's Value

C - Modern Value

A B C

Mercury 0.36 0.387

Venus 0.72 0.723

Earth 1.00 1.00

Mars 1.5 1.52

Jupiter 5 5.20

Saturn 9 9.54

TABLE 2 Illustration of Kepler's Third Law

Legend for Chart:

A - Planet

B - Semimajor Axis a (AU)

C - Sidereal Period P (yrs.)

D - A[sup 3]

E - p[sup 2]

A B C D E

Mercury 0.387 0.241 0.058 0.058

Venus 0.723 0.615 0.378 0.378

Earth 1 1 1 1

Mars 1.524 1.881 3.54 3.54

Jupiter 5.203 11.86 141 141

Saturn 9.539 29.46 868 868

CONSISTENT HISTORIES AND QUANTUM MEASUREMENTS

Source: Physics Today, Aug99, Vol. 52 Issue 8, p26, 6p, 2 diagrams, 1bw

Author(s): Griffiths, Robert B.; Omnes, Roland

The traditional Copenhagen orthodoxy saddles quantum theory with embarrassments like Schrodinger's cat and the claim that properties don't exist until you measure them. The consistent-histories approach seeks a sensible remedy.

Students of quantum theory always find it a very difficult subject. To begin with, it involves unfamiliar mathematics: partial differential equations, functional analysis, and probability theory. But the main difficulty, both for students and their teachers, is relating the mathematical structure of the theory to physical reality. What is it in the laboratory that corresponds to a wavefunction, or to an angular momentum operator? Or, to use the picturesque term introduced by John Bell,(n1) what are the "beables" (pronounced BE-uh-bulls) of quantum theory--that is to say, the physical referents of the mathematical terms?

In most textbooks, the mathematical structures of quantum theory are connected to physical reality through the concept of measurement. Quantum theory allows us to predict the results of measurements--for example, the probability that this counter rather than that one will detect a scattered particle. That the concept of measurement played an important role in the early development of quantum theory is evident from Niels Bohr's account of his discussions with Albert Einstein at the 1927 and 1930 Solvay conferences.(n2) And it soon became part of the official "Copenhagen" interpretation of the theory.

But what may well have been necessary for the understanding of quantum theory at the outset has not turned out to provide a satisfactory permanent foundation for the subject. Later generations of physicists who have tried to make a measurement concept a fundamental axiom for the theory have discovered that this raises more problems than it solves. The basic difficulty is that any real apparatus in the laboratory is composed of particles that are presumably subject to the same quantum laws as the phenomenon being measured. So, what is special about the measuring process? Is not the entire universe quantum mechanical?

When quantum theory is applied to astrophysics and cosmology, the whole idea of using measurements to interpret its predictions seems ludicrous. Thus, many physicists nowadays regard what has come to be called "the measurement problem" as one of the most intractable difficulties standing in the way of understanding quantum mechanics.

Two measurement problems

There are actually two measurement problems that conventional textbook quantum theory cannot deal with. The first is the appearance, as a result of the measurement process, of macroscopic quantum superposition states such as Erwin Schrodinger's hapless cat. The second problem is to show that the results of a measurement are suitably correlated with the properties the measured system had before the measurement took place--in other words, that the measurement has actually measured something.

The macroscopic-superposition problem is so difficult that it has provoked serious proposals to modify quantum theory, despite the fact that all experiments carried out to date have confirmed the theory's validity. Such proposals have either added new, "hidden" variables to supplement the usual Hilbert space of quantum wavefunctions, or they have modified the Schrodinger equation so as to make macroscopic superposition states disappear. (For a discussion of two such proposals, see the two-part article by Sheldon Goldstein in PHYSICS TODAY, March 1998, page 42, and April 1998, page 38.) But even such radical changes do not resolve the second measurement problem.

Both problems can, however, be resolved without adding hidden variables to the Hilbert space and without modifying the Schrodinger equation. In a series of papers starting in 1984, an approach to quantum interpretation known as consistent histories, or decoherent histories, has been introduced by us and by Murray Gell-Mann and James Hartle.(n3) The central idea is that the rules that govern how quantum beables relate to each other, and how they can be combined to form sensible descriptions of the world, are rather different from what one finds in classical physics.

In the consistent-histories approach, the concept of measurement is not the basis for interpreting quantum theory. Instead, measurements can be analyzed, together with other quantum phenomena, in terms of physical processes. And there is no need to invoke mysterious long-range influences and similar ghostly effects that are sometimes claimed to be present in the quantum world.(n4)

Quantum histories