Page 34 - A Brief History of Time - Stephen Hawking

A Brief History of Time - Stephen Hawking... Chapter 4
passing through both slits at the same time!

The phenomenon of interference between particles has been crucial to our understanding of the structure of atoms, the
basic units of chemistry and biology and the building blocks out of which we, and everything around us, are made. At the
beginning of this century it was thought that atoms were rather like the planets orbiting the sun, with electrons (particles
of negative electricity) orbiting around a central nucleus, which carried positive electricity. The attraction between the
positive and negative electricity was supposed to keep the electrons in their orbits in the same way that the gravitational
attraction between the sun and the planets keeps the planets in their orbits. The trouble with this was that the laws of
mechanics and electricity, before quantum mechanics, predicted that the electrons would lose energy and so spiral
inward until they collided with the nucleus. This would mean that the atom, and indeed all matter, should rapidly collapse
to a state of very high density. A partial solution to this problem was found by the Danish scientist Niels Bohr in 1913. He
suggested that maybe the electrons were not able to orbit at just any distance from the central nucleus but only at certain
specified distances. If one also supposed that only one or two electrons could orbit at any one of these distances, this
would solve the problem of the collapse of the atom, because the electrons could not spiral in any farther than to fill up
the orbits with e least distances and energies.

This model explained quite well the structure of the simplest atom, hydrogen, which has only one electron orbiting around
the nucleus. But it was not clear how one ought to extend it to more complicated atoms. Moreover, the idea of a limited
set of allowed orbits seemed very arbitrary. The new theory of quantum mechanics resolved this difficulty. It revealed that
an electron orbiting around the nucleus could be thought of as a wave, with a wavelength that depended on its velocity.
For certain orbits, the length of the orbit would correspond to a whole number (as opposed to a fractional number) of
wavelengths of the electron. For these orbits the wave crest would be in the same position each time round, so the
waves would add up: these orbits would correspond to Bohr’s allowed orbits. However, for orbits whose lengths were not
a whole number of wavelengths, each wave crest would eventually be canceled out by a trough as the electrons went
round; these orbits would not be allowed.

A nice way of visualizing the wave/particle duality is the so-called sum over histories introduced by the American scientist
Richard Feynman. In this approach the particle is not supposed to have a single history or path in space-time, as it would
in a classical, nonquantum theory. Instead it is supposed to go from A to B by every possible path. With each path there
are associated a couple of numbers: one represents the size of a wave and the other represents the position in the cycle
(i.e., whether it is at a crest or a trough). The probability of going from A to B is found by adding up the waves for all the
paths. In general, if one compares a set of neighboring paths, the phases or positions in the cycle will differ greatly. This
means that the waves associated with these paths will almost exactly cancel each other out. However, for some sets of
neighboring paths the phase will not vary much between paths. The waves for these paths will not cancel out Such paths
correspond to Bohr’s allowed orbits.
With these ideas, in concrete mathematical form, it was relatively straightforward to calculate the allowed orbits in more
complicated atoms and even in molecules, which are made up of a number of atoms held together by electrons in orbits
that go round more than one nucleus. Since the structure of molecules and their reactions with each other underlie all of
chemistry and biology, quantum mechanics allows us in principle to predict nearly everything we see around us, within
the limits set by the uncertainty principle. (In practice, however, the calculations required for systems containing more
than a few electrons are so complicated that we cannot do them.)

Einstein’s general theory of relativity seems to govern the large-scale structure of the universe. It is what is called a
classical theory; that is, it does not take account of the uncertainty principle of quantum mechanics, as it should for
consistency with other theories. The reason that this does not lead to any discrepancy with observation is that all the
gravitational fields that we normally experience are very weak. How-ever, the singularity theorems discussed earlier
indicate that the gravitational field should get very strong in at least two situations, black holes and the big bang. In such
strong fields the effects of quantum mechanics should be important. Thus, in a sense, classical general relativity, by
predicting points of infinite density, predicts its own downfall, just as classical (that is, nonquantum) mechanics predicted
its downfall by suggesting that atoms should collapse to infinite density. We do not yet have a complete consistent theory
that unifies general relativity and quantum mechanics, but we do know a number of the features it should have. The
consequences that these would have for black holes and the big bang will be described in later chapters. For the
moment, however, we shall turn to the recent attempts to bring together our understanding of the other forces of nature
into a single, unified quantum theory.

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