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Why is quantum mechanics based on probability theory? [duplicate]

What makes us formulate quantum mechanics based on probability theory?

Isn't the real quantum world based on unknown laws to us?

Is it possible that results of an experiment will be measurable in another way but not expected value?

Answer

I'll have a go to show that the concept of probability is a mathematical tool for formulating a  theory of the mechanics that governs  the microcosm, which ended into Quantum mechanics as we know it.

To start with, what is [probability theory][1] in mathematics ?

>Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted.

In contrast, Quantum Mechanics is a theory with dynamical solutions of specific differential equations with imposed physical boundary conditions. **There is nothing random about these solutions.** Thus QM is not based on probability *theory* as the events  are not random and are not from the [distributions appearing][2] in the studies of probability theory.

It uses the concept of probability: the outcomes of particle experiments  repeated many times are predicted by the  QM solutions, an individual experiment having a calculable  probability of appearing taken from those solutions.

Quantum Mechanics became necessary because there were experiments where classical mechanics was not able to predict or explain them. Particles, entities with specific (x,y,z,t) and (p_x,p_y,p_z,E), exhibited also a wave nature, but not of a classical wave i.e.  of a medium reacting to energy passing. A new type of wave that appears in the probability space .

[An example][3] is the double slit experiment with individual electrons measured over time.

![double slit][4]

A probability distribution can be defined by these data, that has nothing to do with the probability theory's distribution and everything to do with the QM mathematical solution giving a wave function, whose square  gives the probability of an electron appearing at a given space point.

[1]: http://en.wikipedia.org/wiki/Probability_theory
  [2]: http://en.wikipedia.org/wiki/Probability_theory#Probability_distributions
  [3]: http://en.wikipedia.org/wiki/Double_slit
  [4]: http://i.stack.imgur.com/ZVdhv.jpg

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Why is quantum mechanics based on probability theory? [duplicate]

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