We start with a list of the possible states of a particle - these are based on different configurations -- spin-up/down, charge+/-, and so on. I could reduce these to a list with only one entry on it -- a specific particle of a specific and charge.
lets call this list a 'set' and each member of the list an 'element' of the set. Thus the set can have from zero to as many elements as we choose.
Now lets create another list, or set, whose elements are all of the possible states of the universe. -- perhaps you ask 'How the @#$ do you do that!?'
Since we can only get 'good' data on our local system, to determine the possible states of the universe we use universal constants to tell us what the universe could NOT be like -- and so what is left is a list of what states it could be like. This is a common method used by humans to determine the nature of something that they cannot EXPLICITLY identify -- they do it by implication; applying known invarient values that help to get close to 'what is' in that they remove 'what is not'.
We now have two sets, lets call the 'particle' one set A and the 'universe' one set B. Our intention is to gain more details on the elements of A when combined with the elements of B, and we do this by creating what is called a Cartesian Product --- written as C = A x B, where the 'x' symbol means 'cross' and what this does is give us a set C where each element is made-up of each element of A together with each element of B -- thus forming what are called "ordered pairs" e.g. A1+B1, A1+B2, ...A2+B1,..etc etc (Note that the term 'ordered pairs' forces the concept of ordinality into the arguement and with it come properties that are distinctly different from those used in expressing cardinality - where cardinality is expressed through the association of probabilities with an ordered pair).
We in fact need to refine this so: C = A x (B-A) since ALL states of the universe must include the particle and we dont want to 'double-up'!
So..what exactly is set C? Set C contains all of the possible pairings of all of the possible states of a particle in all of the possible states of the universe. In fact, set C contains TWO lots of information - each ordered pair can be treated as a state and C contains the actual values for all states (objective 'precise' fact) as well as each state's probability of existing (approximations and so not so 'objective')
Each state's probability is made-up of combining the probability of the particle state with the probability of the universe state.
Set C therefore can be treated as a list of probabilities and in the context of Schrodinger's Wave Mechanics, set C IS the wave equation which is a probability list expressed in a wave format -- looks like a normal distribution curve (which it is) with the most likely states in the middle and the least likely at either end.
What happens in physics is that we can write-down the wave equation prior to an experiment and then when the experiement is run and completed this act of measurement should give a state that is on the list. - it always does, and the finding of the state is the point where the abstract wave 'collapses' -- where probable becomes actual; something I can point at and say "THATS IT!! 7.8912..etc etc".
Set C can in fact be expressed in one of FOUR possible ways in QM. Either as a form corresponding to P.A.M. Dirac's Transformative Mechanics, or as a form in Heisenberg's Matrix Mechanics (set A is column data and Set B is row data and a cell in the matrix will correspond to the measured result.) or as a form in Schrodinger's Wave Mechanics, or as a form on Feynman's Sum of Histories.
Of all these forms, only Heisenberg and Schrodinger work dualistically in that they link objective, 'hard' 'what is' with subjective, 'soft', 'what could be'. Thus in the translation process, to get from Dirac to Feynman you MUST go through either of the methods of Heisenberg or Schrodinger
The 'interesting' one is Feynman's Sum of Histories where EVERYTHING is treated as a wave and it is the constructive and destructive interferences created by mixing waves that 'causes' reality. Many people seem to see this as a form of confirming 'objective non-locality' - it isnt; it demonstrates one of the ways in which we can map reality and is in fact a function of how our brain works in that one aspect is to work on probabilities -- a right brain function which is very 'wave' oriented in that the right brain deals with creating illusions by emphasising a continuum rather than discreteness; it creates 'interference' so that the reductionist bias left brain of others has to work at trying to identify the real from the illusional. (Thus the right brain is 'rooted' in the functions of the amygdala -- part of the mid-brain that controls fight/flight behaviour which is in fact BLUFF. The left is more 'rooted' in the behaviour of the mid brain's hippocampus that functions more on establishing EXPLICIT locations and relations.)
In physics we have all of the four above mentioned methods of representation and so can choose which one is the more useful in a given context; usually it is the middle two - Schrodinger/Heisenberg with a bias to Schrodinger. The 'reason' for Schrodinger is that we retain a degree of 'hardness' (left brain bias) and remove some of the complexities in rigid matrix formats, replacing them with wave formats (right brain).
It is important to recognise that all of these systems are created by 'in here' and serve to help us describe 'out there'; they are mathematical constructs that go towards describing whole/aspects interactions -- wholes and aspects being the way the brain categorises and processes data both 'in here' and from 'out there'.
At the brain level, the left brain is biased to reductionism. precision, and an emphasis on EXPLICIT identification -- "YES THATS IT!". It is thus result oriented with a single context and a strong text.
The right brain is biased to dealing with aspects, approximations, pattern matching and IMPLICIT identification -- it can point but it cant 'say'. It is cause oriented with a multi-context bias and a weak text. Note that being contextual it contains both the aspects that support a text as well as the NEGATION of the text. (recall how we try to identify implicity by stressing what something is NOT).
Combining left and right gives 'us' -- ranging from extreme left - location bias, fundamentalist, conservative, literal interpretations, through the more analogy prone middle to the more symbol/metaphor, non-locality prone, changing right.
All of our disciplines, when refined, are symbols/metaphors for describing whole/aspects behaviour. the problem with this is that, since there is a degree of hierarchy 'in here', so symbols/metaphors can be expressed at higher levels as if 'fact' and so we must be vary wary and selective in what we consider to be 'real' and what is 'illusion'. As an example of this see the section on wave interference and recursive dichotomisations.