Planck's quantum of action is like an ideal indivisible atom.

There is no dispute over how these images or equations are derived. I like an elegant
derivation as much as any mathematician, but here I am going to spare you the derivations.
It is worth noting though that the "uncertainty relations" are very closely related to
formulae in a powerful branch of mathematics called Fourier Analysis which was
extensively developed and well known for generations before the discovery of the
physical quantum of action. The close tie that the "uncertainty relations" have to
classical Fourier (or harmonic) analysis has made their interpretation in light of
the quantum principle quite controversial.

In the formula, dx*dp > h, x stands for position, p stands for momentum and h is simply
a number, a very tiny one, although one could recalibrate one's instruments so that h is
equal to 1. The h is named after Max Planck (1858-1947) who used it in the year 1900
to describe something as simple as heat. The symbol h is called Planck's constant, or
Planck's quantum of action. It is a fundamental physical constant equal to the smallest
quantity of action that can occur in the universe. (In the Bohr atom, the action is the angular
momentum inherent in the orbital motion of electrons. Angular momentum has the same
physical units as action.) Using classical physics and the assumption that there is an
irreducible "atom of action" h, one can derive a huge amount of modern physics.

Planck's constant is like an ideal atom that cannot be split into smaller pieces. In geometric
terms, a single quantum of action can be thought of as a moldable "substance" which can be
stretched and twisted but whose overall "volume" is fixed. In classical physics, there is no
fundamental unit of length, no "atom" of time and no minimum increment of mass. But there
are fundamental combinations of length, time and mass. There is a fundamental constant for
the speed of light and there is a universal constant for measuring the attractive force between
any two masses. In quantum physics, there are a few other constants but none more basic than
Planck's constant for measuring "action". This action can be realized as the product of certain
pairs of variables as occurs in the "uncertainty relations". The quantum principle forces these
"conjugate variables" to be inversely related to each other. When one is big, the other one is
small so that their product is a shape whose "volume" is at least h. What does this mean?
What does it have to do with picturing fundamental reality?

In 1927, Heisenberg worked directly with Bohr at Bohr's fabulous institute in Copenhagen,
Denmark (supported in part by Carlsberg Beer!). Bohr's take on the "uncertainty relations"
was quite different from Heisenberg's. The inverse relationship between dx and dp implies
that if dx is small then dp is large. To Heisenberg this meant that if dx is small, then there
is little uncertainty in the position of a particle and a great amount of uncertainty in its
momentum. Bohr's take was more like this: If one is able to precisely measure a particle's
position, then in that experimental framework the particle does not have a well defined
momentum. A completely different experimental setup can give momentum measurements,
but because dx*dp >h, this new setup will forbid any well defined position measurements.
It is wrong to say that the position is now uncertain. Instead, the position attribute does not
exist in the momentum-measuring setup. It is not simply unknowable. Whether something
is "to be or not to be" is in part up to us.

Atomic properties are properly defined in the context of the experimental conditions in which
they can be measured. These properties are contingent upon the circumstances which bring them
into being. In the language of Scottish Law, the experimental apparatus is both "art and part"
in bringing about that which appears to happen. Individual properties are brought into being
in part by our choice of questions to ask, issues to ignore, and answers to heed. When a new
question is asked in conjunction with a new experimental arrangement, the previous properties
vanish and have no control over the new ones that appear. Bohr warned that the principle of
complementarity not only "set a limit to the extent of the information obtainable by measurement,
but also set a limit to the meaning which we may attribute to such information."

Quantum physics introduces an observer-dependency that the popular press has blown out 
of proportion. What really exists is not waves or particles or any other transitory appearance 
in the phenomenal realm. Quantum physics confirms the existence of harmonies that are
independent of us. They lie on the noumenal side of the map described earlier. The Periodic
Table of Elements is grounded on an a priori ontological realm of pre-established harmonies.
Here is what Max Planck had to say about this quantum business in 1933:

… sensory perceptions do not of themselves create the physical world around us …
they bring news of another world which lies outside of ours and is entirely independent of us.
. . . the external world forces itself upon our recognition with its own elemental power . . .
measurements . . . give no direct information about external reality.  They are only a register
or representation of reactions to physical phenomena.  As such they contain no explicit
information and have to be interpreted. As Helmholtz said, measurements furnish the
physicist with a sign which he must interpret . . . .

In a changing environment, position and momentum are fleeting phenomena. No atomic particle
can have both of these classical attributes at the same time.  We have the illusion that these
phenomena co-exist in large objects whose inherent action is huge compared to h. No one
knows exactly how the quantized behavior of atomic matter becomes smooth and regular
at our level.  It is a big question that we will bypass here, but see below for references.*

We will now use complementarity to interpret phenomena and signs far removed from physics.
From Bohr's "renunciation of the visualization of atomic phenomena," we will learn how to
coordinate and transcend a wide range of cherished and naive notions that are mistaken for
reality. The quantal substrate underlying phenomena cannot be forced into the terms of daylight
without compromising its indivisible nature. Just as Shakespeare does not appear in his scenes,
quanta are not directly visible in phenomena, yet are responsible for them. Between Shakespeare's
lines one can imagine him winking, laughing and sighing.  Behind the scenes of our daily life there
is an unseeable realm (a Dirac Sea) ruled in part by quantum physics. "The whole body like a
chaos capable of any form that the next daring spirit shall brood upon it."*  The classical
observables reign and show off, while the quantum physics does the actual governing.

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* For an indepth study of the unresolved issues, see 
Decoherence and the Appearance of a Classical World in Quantum Theory
by Giulini, Joos, Kiefer, Kupsch, Stamatescu & Zeh. This is a difficult book in 
mathematical physics published by Springer Verlag, 1996. It is as an example of 
what the scientific work in this area looks like. There is a huge amount of 
pseudoscience to avoid in learning the lessons of the quantum. Sir Roger Penrose
is also trying to figure how our ordinary world emerges from the quantum jungle; 
see Shadows of the Mind (1994)

* Year 1649 quote from The Oxford English Dictionary

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