A brief History of Time pdfdrive com

FIGURE 4.1

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FIGURE 4.2
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 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!

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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 the 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.

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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.
However, 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

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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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