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This equation is a statement of the He¿senberg uncertaintụ principle that we
mentioned earlier.
Since the right-hand side of Eq. (6.22) is a constant, this equation says that
1Í we try to “pin down” a particle by forcing it to be at a particular place, 1%
ends up by having a high speed. Or if we try to fÍorce it to go very slowly, or at a
precise velocity, it “spreads out” so that we do not know very well just where ït
1s. Particles behave in a funny wayl
'The uncertainty principle describes an inherent fuzziness that must exist in
any attempt to describe nature. Ôur most precise description of nature rwusf
ben terms of probabilöties. There are some people who do not like this way of
describing nature. They feel somehow that if they could only tell what is reallu
going on with a particle, they could know its speed and position simultaneously.
In the early days of the development of quantum mechanics, Einstein was quite
worried about this problem. He used to shake his head and say, “But, surely God
--- Trang 137 ---
Fig. 6-11. A way of visualizing a hydrogen atom. The density (white-
ness) of the cloud represents the probability density for observing the
electron.
does not throw dice in determining how electrons should go!” He worried about
that problem for a long time and he probably never really reconciled himself to
the fact that this is the best description of nature that one can give. There are
still one or two physicists who are working on the problem who have an intuitive
conviction that it is possible somehow to describe the world in a different way
and that all of this uncertainty about the way things are can be removed. No
one has yet been successful.
The necessary uncertainty in our specifcation of the position of a particle
becomes most important when we wish to describe the structure of atoms. In the
hydrogen atom, which has a nucleus of one proton with one electron outside of
the nucleus, the uncertainty in the position of the electron is as large as the atom
itselft We cannot, therefore, properly speak of the electron moving in some “orbit”
around the proton. The most we can say is that there is a certain chanee p(r) AV,
of observing the electron in an element of volume AV at the distance r from
the proton. The probability density p(z) is given by quantum mechanics. For
an undisturbed hydrogen atom p(z) = Ae~?*/*, The number ø is the “typical”
radius, where the function is decreasing rapidly. 5ince there is a small probability
of fñnding the electron at distances from the nucleus mụuch greater than ø, we
may think of ø as “the radius of the atom,” about 10—†19 meter.
W© can form an image of the hydrogen atom by imagining a “cloud” whose
density is proportional to the probability density for observing the electron. A
--- Trang 138 ---
sample of such a cloud is shown in EFig. 6-11. “Thus our best “picture” of a
hydrogen atom is a nucleus surrounded by an “electron cloud” (although we
reall mean a “probability cloud”). The electron is there somewhere, but nature
permits us to know only the chance of fñnding it at any particular place.
In its eforts to learn as much as possible about nature, modern physics has
found that certain things can never be “known” with certainty. Much of our
knowledge must always remain uncertain. "The mos we can know is in terms of
probabilities.
--- Trang 139 ---
Tho Thoortg ©Ÿ Ấnrcrtff(rffOre
7-1 Planetary motions
In this chapter we shall diseuss one of the most far-reaching generalizations oŸ
the human mỉnd. While we are admiring the human mind, we should take some
time of to stand in awe of a na#ure that could follow with such completeness
and generality such an elegantly simple principle as the law of gravitation. What
1s this law of gravitation? It is that every object in the universe attracts every
other obJect with a force which for any two bodies is proportional to the mass
of each and varies inversely as the square of the distance between them. “This
statement can be expressed mathematically by the equation
F=G——..
T to this we add the fact that an object responds to a force by accelerating in
the direction of the force by an amount that is inversely proportional to the mass
of the object, we shall have said everything required, for a sufficiently talented
mathematician could then deduce all the consequences of these two principles.
However, since you are not assumed to be sufficiently talented yet, we shall
discuss the consequences in more detail, and not just leave you with only these
two bare principles. We shall brieRy relate the story of the discovery of the
law of gravitation and discuss some of its consequences, its efects on history,
the mysteries that such a law entails, and some reñnements of the law made
by Einstein; we shall also discuss the relationships of the law to the other laws
of physics. All this cannot be done in one chapter, but these subjects will be
treated in due time in subsequent chapters.
The story begins with the ancients observing the motions of planets among
the stars, and finally deducing that they went around the sun, a fact that was
rediscovered later by Copernicus. Exactly ho the planets went around the sun,
--- Trang 140 ---
with exactly t0høt motion, took a littÌe more work to discover. In the beginning
of the fñfteenth century there were great debates as to whether they really went
around the sun or not. 'Eycho Brahe had an idea that was diferent from anything
proposed by the ancients: his idea was that these debates about the nature of the
motions of the planets would best be resolved if the actual positions of the planets
in the sky were measured sufficiently accurately. IÝ measurement showed exactly
how the planets moved, then perhaps it would be possible to establish one or
another viewpoint. 'This was a tremendous idea—that to fñnd something out, it is
better to perform some careful experiments than to carry on deep philosophical
areuments. Pursuing this idea, Tycho Brahe studied the positions of the planets
for many years in his observatory on the island of Hven, near Copenhagen. He
made voluminous tables, which were then studied by the mathematician Kepler,
after Iycho's death. Kepler discovered from the data some very beautiful and
remarkable, but simple, laws regarding planetary motion.
7-2 Kepler?s laws
First of all, Kepler found that each planet goes around the sun in a curve
called an ellpse, with the sun at a focus of the ellipse. An ellipse is not just an
oval, but is a very specifc and precise curve that can be obtained by using two
tacks, one at each focus, a loop oŸ string, and a pencil; more mathematically, it
1s the locus oŸ all points the sum oŸ whose distances from two fixed points (the
foci) is a constant. Ôr, iŸ you will, it is a foreshortened circle (Fig. 7-1).
“EN |
rị + ra = 2a
Fig. 7-1. An ellipse.