Other applications of quantum theory

In this section we shall outline some of the most important applications
of quantum theory to various areas of physics, applications which ensured that, in spite of its problems, it rapidly gained acceptance.
Nothing in the remainder of our discussion will depend on
this section, so it may be omitted by readers who are in a hurry.
The section is also somewhat more demanding with regard to
background knowledge of physics than most.
The understanding of electricity and magnetism, besides being
the prerequisite for the scientific and technological revolutions of
this century, was the great culminating triumph of nineteenth century,
classical, physics. By combining simple experimental laws,
deduced from laboratory experiments, into a mathematically consistent
scheme, Maxwell unified electric and magnetic phenomena
in his equations of electromagnetism. These equations predicted
the existence of electromagnetic waves capable of travelling
through space with a calculable velocity. Visible light, radio waves,
ultraviolet light, heat radiation, x-rays, etc, are all examples, differing
only in frequency and wavelength, of such waves.
The first hint of any inadequacy within this scheme of classical
physics came with the calculation of the way in which the intensity
of electromagnetic radiation emitted by a ‘black body’ (i.e. a body
that absorbs all the radiation falling upon it at a particular
temperature) varies with the frequency of the radiation. The
assumptions which went into the calculation were of a very general
nature and were part of the accepted wisdom of classical physics;
the results, however, were clearly incompatible with experiment. In
particular, although there was agreement at low frequency, the
calculated distribution increased continuously at high frequency
rather than decreasing to zero as required.
Max Planck, in 1900, realised that one simple modification to the
assumptions would put everything right, namely, that emission and
absorption of radiation by a body can only occur in finite sized
‘packets’ of energy equal to h times the frequency. The constant of
proportionality introduced here, and denoted by h, is the original
Planck’s constant. For various reasons it is usual now to work
instead with the quantity h, which we quoted in equation (2.2), and
which is equal to h divided by 27r.
The packets of energy, introduced by Planck, are the ‘quanta’
which gave rise to the name quantum theory. Each such quantum
is now known to be a photon, i.e. a particle of electromagnetic
radiation, but such a concept was a heresy at the time of Planck’s
original suggestion; electromagnetic radiation (e.g. light, radio waves, etc) was known to be waves! The quantisation was therefore
assumed to be simply something to do with the processes of
emission and absorption.
Such a view was shown to be untenable by the observation of
the photoelectric effect, in which electrons are knocked out of
atoms by electromagnetic radiation. If we assume that the energy
in a uniform beam of light, incident upon a plate, is distributed
uniformly across the plate, then it is possible to calculate the time
required for sufficient energy to fall on one atom to knock out an
electron. This is normally of the order of several seconds, in
contrast to the observation that the effect starts immediately.
Further, the energy of the emitted electrons is, apart from a
constant, proportional to the frequency of the radiation. Einstein,
in 1905, showed that all the observations were in perfect agreement
with the assumption that the radiation travelled as photons, each
carrying the energy E appropriate to its frequency according to the
relation previously used by Planck:
E = hf (2.3)
where f is the frequency.
The final confirmation of the idea of photons came from the
observation, in 1922, of the Compton effect, in which radiation was
seen to decrease in frequency when it was scattered by electrons.
This can be explained very simply as being due to the loss of energy
in the photon-electron collision, a loss that can be exactly calculated
from the laws of conservation of energy and momentum.
Although quantum theory began with its application to radiation,
the ideas were soon applied to particles. In 191 l , de Broglie
suggested that, if waves can have particle properties, then it is
reasonable to expect particles to have wave properties. He
introduced the relation:
I = h/mv (2.4)
between the wavelength I , the velocity v, and the mass m of a
particle. The major achievements of quantum mechanics have
been, following this relation, in its application to matter, in
particular to the structure of atoms.
The experimental work of Rutherford, early this century,
showed that an atom consists of a small, positively charged,
nucleus, which contains most of the mass of the atom, surrounded by a number of negatively charged electrons which are bound to the
nucleus by the attractive electric force. Each atom was therefore
like a miniature solar system, with the electrons playing the role of
planets, orbiting the nuclear ‘sun’. Prior to the advent of quantum
theory there were, however, serious problems with this picture: why
did the orbiting electrons not radiate electromagnetic waves,
thereby losing energy so that they would fall into the nucleus? Why
were the energies available to a given atom only a set of discrete
numbers, rather than a continuum as would be expected from
classical mechanics?
Quantum theory provides a complete answer to these questions.
All the energy levels of atoms can be calculated from. the
Schrodinger equation, in perfect agreement with experiment. The
interactions between atoms, as observed in molecules, chemical
processes and atomic scattering experiments can also be understood
from this equation. As we mentioned in 81.1, quantum theory
successfully brought a whole new range of phenomena into the
domain of calculable physics.
The details of all this are outside the scope of this particular
book, but it is worthwhile to give a simple picture of why the wave
nature of the electron helps us to understand the quantum answers
to the problems mentioned above with the classical picture of the
atom. If we consider a wave on a string with fixed end points,
then only certain wavelengths are allowed, because an integral
multiple of the wavelength must fit exactly into the string. A consequence
is that the string can only vibrate with a particular set of
frequencies; a fact which is crucial to many musical instruments.
The frequencies which occur can be altered by changing either the
length or the tension of the string. In an atom the situation is
similar, except that, instead of having a wave on a string with fixed
end points, we have a wave on a circle (the orbit), which must join
smoothly on to itself. Thus the circumference of the circle has to be
an exact integral multiple of the wavelength. As we show in
Appendix 5 , this condition yields the energy levels of the simplest
atom.
The transition from one energy level in an atom to another, by
the emission of a photon, i.e. by electromagnetic radiation, is an
example of an important class of very typically quantum
phenomena, in which one particle spontaneously ‘decays’ into (say)
two others. Calling the first particle A and the others B and C, we can write this as
A+B+C.
If we start with a large number of A particles then, after a given
time, some of them will have decayed. It is usual to define a
‘half-life’ as the time taken for half of the particles in a large initial
sample to have decayed. The half-life depends on the process considered
and values ranging from tiny fractions of a second to times
beyond the age of the universe are known.
Even though the half-life for the decay of a certain type of
particle, e.g. the A particle above, might be known, it will not be
possible to say when a particular A particle will decay. This is
random; like, for example, the choice of transmission or reflection
in the potential barrier experiment. Indeed, one can think of some
types of decays as being rather like a particle bouncing backwards
and forwards between high potential barriers; eventually the
particle passes through a barrier and decay occurs. In general, if we
start with a wavefunction describing only identical A particles, then
it will change into a sum of a wavefunction describing A particles, which will have a magnitude decreasing with time, and one describing
(B + C), which will have an increasing magnitude.
Finally, we mention the recent, very accurate, experiments which
show that neutrons passing through a double slit, as in figure 13,
interfere exactly as predicted by quantum theory. An example is
shown in figure 14. These experiments were carried out in response
to the recent upsurge of interest in checking carefully the validity
of quantum theoretical predictions in as many circumstances as
possible. We shall later mention other such tests. In all cases so far
the theory is completely satisfactory.