Paul (P. A. M.) Dirac formulated the most elegant version of the mathematical principles of quantum mechanics after hearing a lecture by
Werner Heisenberg on his new ideas of "matrix mechanics." Shortly after matrix mechanics,
Erwin Schrödinger developed his "wave mechanics" and showed it was equivalent to the Heisenberg picture.
Dirac then combined the matrix and wave formulations using abstract symbolic methods from classical mechanics called Poisson brackets and canonical transformations.
In his textbook The Principles of Quantum Mechanics, Paul Dirac introduced the concepts of superposition, the projection postulate, the axiom of measurement, and indeterminacy using simple examples with polarized photons.
Dirac's examples suggest a very simple and inexpensive experiment that we call the Dirac 3polarizers experiment to demonstrate the notions of quantum states, the preparation of quantum systems in states with known properties, the superposition of states, the measurement of various properties, the projection or representation of a state vector in another basis set of vectors, and the infamous "collapse" or "reduction" of the wave function and the resulting indeterminacy.
In their Copenhagen interpretation of quantum mechanics, Niels Bohr and Heisenberg said that the results of quantum measurements must be expressible in classical concepts because it is the language that humans can understand. By contrast, Dirac argued that the nonintuitive concepts of quantum mechanics, though impossible to understand in terms of classical concepts, could be mastered through long familiarity with them.
The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g. proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses.
Information physics attempts to articulate some new concepts, albeit slightly modified versions of intuitive classical concepts. We associate quantum waves with possibilities and a quantum particle with actualization of a possibility. Quantum physics lets us calculate the probabilities for each possibility, to an extraordinary degree of accuracy. Although the calculation involves abstract complex quantities and the motion through space of immaterial information about those possibilities, the result is both understandable (if nonintuitive because never experienced) and visualizable.
The Information Interpretation of quantum mechanics is based on three simple premises:
When you hear or read that electrons are both waves and particles, think "eitheror" 
first a wave of possibilities, then an actual particle.
 Quantum systems evolve in two ways:
 the first is the wave function deterministically exploring all the possibilities for interaction, interfering with itself as it travels,
 the second is the particle randomly choosing one of those possibilities to become actual.
 No knowledge can be gained by a "conscious observer" unless new information has already been irreversibly recorded in the universe. That information can be created and recorded in three places:
 in the target quantum system,
 in the combined target system and measuring apparatus,
 it can then become knowledge in the observer's mind.

The measuring apparatus is quantal, not deterministic or "classical." It need only be statistically determined and capable of recording the irreversible information about an interaction. The human mind is similarly only statistically determined.
We even try to visualize some of these concepts, including Dirac's three polarizers, the twoslit experiment, and the EinsteinPodolskyRosen thought experiment.
Excerpts
From the Preface to The Principles of Quantum Mechanics, First Edition
Chapter 1 of The Principles of Quantum Mechanics, Fourth Edition