A brief History of Time

partition, but a small amount will go through the slits. Now suppose one
places a screen on the far side of the partition from the light. Any point on
the screen will receive waves from the two slits. However, in general, the
distance the light has to travel from the source to the screen via the two slits
will be different. This will mean that the waves from the slits will not be in
phase with each other when they arrive at the screen: in some places the
waves will cancel each other out, and in others they will reinforce each
other. The result is a characteristic pattern of light and dark fringes.
The remarkable thing is that one gets exactly the same kind of fringes if
one replaces the source of light by a source of particles such as electrons
with a definite speed (this means that the corresponding waves have a
definite length). It seems the more peculiar because if one only has one slit,
one does not get any fringes, just a uniform distribution of electrons across
the screen. One might therefore think that opening another slit would just

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increase the number of electrons hitting each point of the screen, but,
because of interference, it actually decreases it in some places. If electrons
are sent through the slits one at a time, one would expect each to pass
through one slit or the other, and so behave just as if the slit it passed
through were the only one there - giving a uniform distribution on the
screen. In reality, however, even when the electrons are sent one at a time,
the fringes still appear. Each electron, therefore, must be 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

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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

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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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