Wave Mechanics 195
matter how accurately they are measured. The uncertainty principle
follows from Heisenberg’s equations. Once formulated, it is easy to
demonstrate that it arises from general considerations of measuring
atomic phenomena and indeed, explains the necessity of a probabilistic
description of atomic processes. We shall return to this question later,
but let us continue our description of the uncertainty principle.
The uncertainty principle states that there is an intrinsic built-in
theoretical limitation to how precisely one may measure both the
position and the momentum of a particle. It is possible to measure either
the momentum or the position of the particle as accurately as one cares
to, however, one’s measurement of the other variable suffers as a
consequence. For instance, it one reduces the uncertainty in one’s
measurement of the particle’s momentum one automatically increases the
uncertainty in the measurement of its position. Labeling the uncertainty
in the measurement of the momentum, p, and the position, x, by ∆p and
∆x respectively, the mathematical expression of the uncertainty principle
takes the following form: the product of ∆p times ∆x is always greater
than or equal to h, Planck’s constant, i.e., ∆p ∆x > h.
It is obvious from this formulation that if ∆p = 0 then ∆x becomes
very large or vice versa. In fact, if one knows the momentum precisely
such that ∆p = 0 then ∆x becomes infinite, which means one loses all
information about the position.
Heisenberg showed that the uncertainty principle also applies to the
measurements of the energy, E, and the lifetime, t, of a system. The
measurement of one interferes with one’s knowledge of the other. If ∆E
and ∆t are the respective uncertainties of the energy and time
measurements then the uncertainty principle states that the product of
these uncertainties will always be greater than or equal to h or ∆E ∆t > h.
Many physicists and lay thinkers found the uncertainty principle a
complete anathema. They could not conceive how theoretical limitation
to measurements could possibly be imposed upon physics. They were
also offended by the probabilistic nature of the quantum mechanics,
which the uncertainty principle seems to epitomize. The uncertainty
principle, to my way of thinking, on the other hand, represents a natural
limitation to the study of microscopic quantities whose energy is
quantized. The uncertainty principle helps one understand why a
probabilistic description of atomic processes is necessary.
In order to describe a physics system, which, after all, is the object of
physics, it is first necessary to observe or know the system, i.e. to be able