Biomedical Engineering Reference
In-Depth Information
pattern of alternating strong and weak lines in the spectra of diatomic molecules, a
problem in which Bohr's theory had failed. He showed that two forms of molecular
hydrogen should exist, depending on the relative directions of the proton spin, and
that the form with spins aligned (orthohydrogen) should be three times as abun-
dant as the other with spins opposed (parahydrogen). This discovery was cited in
the award of the 1932 Nobel Prize in physics to Heisenberg.
The concept of building an atomic theory on observables and its astounding suc-
cesses let to a revolution in physics. In classical physics, objects move with certain
endowed properties, such as position and velocity at every moment in time. If one
knows these two quantities at any one instant and also the total force that an object
experiences, then its motion is determined completely for all times by Newton's
second law. Such concepts are applied in celestial mechanics, where the positions
of the planets can be computed backwards and forwards in time for centuries. The
same determinism holds for the motion of familiar objects in everyday life. How-
ever, on the atomic scale, things are inherently different. In an experiment that
would measure the position and velocity of an electron in orbit about a nucleus,
the act of measurement itself introduces uncontrollable perturbations that prevent
one's obtaining all the data precisely. For example, photons of very short wave-
length would be required to localize the position of an electron within an atomic
dimension. Such photons impart high momentum in scattering from an electron,
thereby making simultaneous knowledge of the electron's position and momen-
tum imprecise.
In 1927 Heisenberg enunciated the uncertainty principle, which sets the limits
within which certain pairs of quantities can be known simultaneously. For momen-
tum
p
and position
x
(in the direction of the momentum) the uncertainty relation
states that
p x
≥
.
(2.30)
Here
p
and
x
are the uncertainties (probable errors) in these quantities, deter-
mined simultaneously; the product of the two can never be smaller than
,which
it can approach under optimum conditions. Another pair of variables consists of
energy
E
and time
t
,forwhich
.
(2.31)
The energy of a system cannot be measured with arbitrary precision in a very short
time interval. These uncertainties are not due to any shortcomings in our measur-
ing ability. They are a result of the recognition that only observable quantities have
an objective meaning in physics and that there are limits to making measurements
on an atomic scale. The question of whether an electron “really” has a position and
velocity simultaneously—whether or not we try to look—is metaphysical. Schools
of philosophy differ on the fundamental nature of our universe and the role of the
observer.
Whereas observation is immaterial to the future course of a system in classical
physics, the observer's role is a basic feature of quantum mechanics, a formal-
ism based on observables. The uncertainty relations rule out classical determinism
E t
≥
