Wave Function

The wave function of quantum mechanics can be regarded as a function of the experimental setup. The function represents the mathematical relationship between a setup described in terms of classical spacetime descriptors and what amounts to a Fourier transform of those descriptors, which we could call a "wave". Or a wave description of the setup.

If the following were the wave function

y = F(x)

then y would be the wave, and x the setup. The function is (with some caveats) invertible:

x = invF(y)

Now the wave function is defined mathematically, meaning that it does not depend on anything else to be correct. It is a definition. And it is global.

Now a mathematical function is very different from something that samples a mathematical function. For example a computer function can sample a mathematical function. Somewhat unfortunately the way you write a sampling function for a computer is often the same as the way you might write the mathematical function that was being sampled:

y = F(x)

The difference is that a single invocation of the sampling function results in just one of the infinite number of pairs identified by the mathematical function. The sampling function, in a sense, represents the relationship between one x and one y, whereas the mathematical function represents all of the pairs. Only if the sampling function was invoked for all possible x's, would the result become equivalent to a representation of the mathematical function. On the other hand the computer function's definition (it's code) could be regarded as equivalent to a representation of the mathematical function.

Now a sampling function is really interesting in relation to the problem of "wave function collapse". The function y = F(x) represents all x's and y's for which y = F(x) is true whereas a sampling version of the same function need only represent a single case (a state), as in the following computer program:

var x = 9
var y = F(x)

While the relationship between x and y is computed according to F(x) the resulting value of y could have been computed by any number of other functions. For example, if y turns out to be 3, then F could have been:

y = sqrt(x)
y = x/3
y =x - 6

It is only when one computes for all possible values of x, that the actual function used (F) becomes distinguishable from any other functions that might have also computed y. Of course it can also be indentified (distinguishable) by reading the code.

The point here is that while the rule can be regarded as determining the value of y (ie. mathematically - equivalent to reading the code), the relationship between any single value of y and it's corresponding x is not given by that rule (since any number of other rules could also express that relationship). Rules can be expressed mathematically but their indentity in the context of a sampling is statistical.

Here is a really rough sampling function that simulates/represents a particle detection in terms of a sampling function (one that samples the wave function).

x = Math.random() * detectorWidth;
y = Math.random() * detectorHeight;
r = Math.random();
a = WaveFunction(x,y);
b = WaveFunction(x,y);
if(r > (a * b))
displayParticleAt(x,y)


Apart from the QM formalism defining the probability of a detection as the square of the wave function the components a and b can be regarded as the two components of a joint probability, where one represents the particle while the other represents the detector. The joint probability represents the detector and particle being in the same place at the same time (ie. in the same frame of reference or the same classical universe).

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