Wave functions give results that are quite different from measurements we obtain for the location of more familiar, macroscopic objects. We can never be sure about where exactly an electron is at a given instant of time; rather we can only predict the probability of finding the electron in a given region of space at a given instant of time. The wave function of an electron enables us to determine that probability. To obtain the probability density we calculate the square of the wave function.
Probabilities of finding the object within a certain region are determined from the probability densities. An important conclusion is that we cannot state with certainty the location of an electron, only the probability of finding it at each of many locations.
The following essay describes some of the differences between our knowledge of large objects and our knowledge of the very small.
Interlude: From Newtonian to Quantum Views of Nature
Adapted from The Fascination of Physics by
Jacqueline D. Spears and Dean Zollman © 1986,1996. Reprinted
with
permission of the authors.
More than 50 years have passed since the wave and particle models
merged to become a new
model of the physical world. In the early days of this century,
physicists voiced strong
arguments for and against the wave function and its interpretations.
Now, the arguments have
become less emotional; the concepts less unsettling. Passing years
and new generations of
physicists have a way of turning a revolutionary thought into
a tradition; the new physics into
the old physics. In the midst of this settled acceptance of modern
physics, we must realize the
enormous impact quantum mechanics and wave functions have upon
a physicist’s view of
“reality.” We pause briefly to examine the remarkable
transformation from the physics of
Newton to that of the modern quantum physicist.
When Isaac Newton introduced his three laws of motion, he provided
a structure within which
we could understand all motion – from the falling apple to
the orbiting planet. Once we knew
all the forces acting on an object, we could predict all future
motions with complete accuracy.
By placing certainty squarely within the grasp of human intelligence,
Newton created an
enormously comforting view of our universe. This feeling of certainty
was stated well by the
French mathematician Pierre LaPlace:
An intelligence which at a given instant
knew all the forces acting in nature and the
position of every object in the universe – if endowed
with a brain sufficiently vast
to make all necessary calculations – could describe with
a single formula the
motions of the largest astronomical bodies and those of the
smallest atoms. To such
an intelligence, nothing would be uncertain; the future, like
the past, would be an
open book.
Newton’s model created an image of a rational
world proceeding in a rational way – a world
view eagerly embraced by philosophers, theologians, and physicists
alike.
Beneath this world view lie two very important assumptions. The
first is that all events are
ordered, not random. To Newton and his contemporaries, all motion
was completely
determined by whomever or whatever started the universe. These
motions obeyed and would
continue to obey a series of orderly rules that could be discovered
by the careful observer.
The second assumption was that the physicist acts as an objective
observer of events. Newton
and his contemporaries believed that Beneath this world view lie
two very important
assumptions. The first is that all events are ordered, not random.
To Newton and his
contemporaries, all motion was completely determined by whomever
or whatever started the
universe. These motions obeyed and would continue to obey a series
of orderly rules that
could be discovered by the careful observer. The second assumption
was that the physicist
acts as an objective observer of events. Newton and his contemporaries
believed that while
the measurer does have some impact on the events he or she measures,
this impact is minimal
and predictable. Events continue, according to a system of ordered
rules, with an existence
independent of the observer. All that remained was for science
to discover the rules.
During the eighteenth and nineteenth centuries, when Newton’s
laws were applied to objects
as small as molecules, this world view prevailed. In principle,
physicists believed, once they
knew the momentum and position of each molecule, they could predict
all future motions of all
molecules. Completing these measurements and calculations for
a gram of water, let alone the
entirety of the universe, was not humanly possible, so statistical
or probabilistic descriptions
were adopted. Consistent with Newton’s world view, probabilities
were needed only to
compensate for an information overload, not because of the inherent
unknowability of nature.
What does the new world view have to say to us about our knowledge?
Implicit in the
probabilistic interpretation now given to matter waves is the
assumption that, on the
microscopic level, events are random. Wave descriptions provide
us with information about
the probabilities associated with this random behavior; particle
measurements convert these
probabilities into brief certainties. Further, objective observers
have become active
participants in the world that they are trying to describe. Physicists
now acknowledge that the
types of measurements they undertake affect the observations and
models they subsequently
construct. Words like particle, position, and path have no meaning
apart from the way in
which the experimenter measures them. These words describe our
way of ordering the events
we see, not a true underlying structure of nature. Newton’s
view of an orderly nature that
exists independent of how we observe it exists no more.
For many physicists the radical departure from more traditional
ideas was difficult to accept.
Erwin Schrödinger, whose equations were the Newton’s
laws of quantum mechanics,
remained uncomfortable with the probabilistic interpretation given
to matter waves. Albert
Einstein, whose quantum explanation of the photoelectric effect
won a Nobel Prize, also
remained unconvinced. He felt that quantum theory was only a stepping
stone to a more
complete understanding of matter. In this view, probabilities
do not represent nature but
rather, people’s limited ability to comprehend nature. In
a letter to Max Born in 1926,
Einstein summarized his and perhaps many others’ feelings:
Quantum theory is certainly imposing. But
an inner voice tells me that it is not yet
the real thing. The theory says a lot, but does not really
bring us closer to the secrets
of the “old” one. I, at any rate, am convinced He
is not playing at dice.
Only time will tell whether Einstein’s
inner voice was the voice of wisdom or the voice of a
past, unwilling to give way to the future.