Quantum Theory

The description of a particle in
quantum theory
The familiar, classical, description of a particle requires that, at all
times, it exists at a particular position. Indeed, the rules of classical
mechanics involve this position and allow us to calculate how it
varies with time. According to quantum mechanics, however, these
rules are only an approximation to the truth and are replaced by
rules that do not refer explicitly to this position but, instead,
predict the time variation of a quantity from which it is possible to
calculate the probability of the particle being in a particular place.
We shall indicate below the circumstances in which the classical
approximation is likely to be valid.
The probability will be a positive number (any probability has to
be positive) which, in general, will vary with time and with the
spatial point considered. As an example, figure 5 is a graph of such
a probability, and shows how it varies with the distance, denoted
by x, along a straight line from some fixed point 0. This graph
represents a particle which is close to the point labelled P. The
width of the distribution, shown in the figure as U,, gives some idea
of the uncertainty in the true position of the particle. There are
precise methods of defining this uncertainty but these are not
important for our purpose. Clearly a very narrow peak corresponds
to accurate knowledge of the position of the particle and, conversely,
a wide peak to inaccurate knowledge.

At this stage it might be thought that we can always use the
classical approximation, where particles have exact positions, by
working with sufficiently narrow peaks. However, if we do this we
lose something else. It turns out that the width of the peak is also
related to the uncertainty in the velocity of the particle, more
precisely the velocity in the direction of the line between the points
0 and P, only here the relation is the opposite way round: the
narrower the peak, the larger the uncertainty. In consequence,
although there is no limit to the accuracy with which either the
position or the velocity can be fixed, the price we have to pay for
making one more definite is loss of information on the other. This
faa is known as the Heisenberg uncertainty principle.
Quantitatively, this principle states that the product of the
position uncertainty and the velocity uncertainty is at least as large
as a certain fixed number divided by the mass of the particle being
considered. The fixed number is, in fact, the constant +z introduced
earlier. We can then write the uncertainty principle in the form
U,U, > &/m (2.1)
where U, is the uncertainty in the velocity and m is the mass of the
particle.
The quantity +I is Planck’s constant. We quote again its value,
this time in SI units:
4 = 1.05 x kgm2s-’.
This is a very small number! We can now see why quantum effects
are hard to see in the world of normal sized, i.e. ‘macroscopic’,
objects. For example, we consider a particle with a mass of one
gram (about the mass of a paper clip). Suppose we locate this to
an accuracy such that U, is equal to one hundredth of a centimetre
(10-4m). Then, according to equation (2.1), the error in velocity
will be about 10-”m per year. Thus we see that the uncertainty
principle does not put any significant constraint on the position and
velocity determinations of macroscopic objects. This is why
classical mechanics is such a good approximation to the macroscopic
world.
We contrast this situation with that which applies for an electron
inside an atom. The uncertainty in position cannot be larger than
the size of the atom, which is about 10-”m. Since the electron
mass is approximately kg, equation (2.1) then yields a
velocity uncertainty of around lo6 ms-’. This is a very large
velocity, as can be seen, for example, by the fact that it corresponds
to passage across the atom once every 10-l6s. Thus we guess,
correctly, that quantum effects are very important inside atoms.
Nevertheless, readers may be objecting on the grounds that, even
in the microscopic world, it is surely possible to devise experiments
that will measure the position and velocity of a particle to a higher
accuracy than that allowed by equation (2.1), and thereby
demonstrate that the uncertainty principle is not correct. Such
objections were made in the early days of quantum theory and were
shown to be invalid. The crucial reason for this is that the
measuring apparatus is also subject to the limitations of quantum
theory. In consequence we find that measurement of one of the
quantities to a particular accuracy automatically disturbs the other
and so induces an error that satisfies equation (2.1). As a simple
example of this, let us suppose that we wish to use a microscope to measure the position of a particle, as illustrated in figure 6. The
microscope detects light which is reflected from the particle. This
light, however, consists of photons, each of which carries momentum.
Thus the velocity of the particle is continuously being altered
by the light that is used to measure its position. It is not possible
to calculate these changes since they depend on the directions of the
photons after collision. The resulting uncertainty can be shown to
be that given by the uncertainty relation. The caption to figure 6
explains this more fully. Most textbooks of quantum theory, e.g.
those mentioned in the bibliography ($6.5), include a detailed
analysis of this experiment and of other similar ‘thought’
experiments.