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So you have this Equation of State Psi for a quantum system, and you want to figure out more than averages.

If you want to figure out the value of a set of parameters (position, momentum, spin, etc.), you can plot (Psi's magnitude squared) versus the values of those parameters, having integrated over all the possibilities of all other parameters that Psi is a function of.

This may require restating Psi into a form which allows this. One example: Even if Psi is given as a function of position, one can use the fourier transform of Psi to find the function's value as a function of momentum. Then you can plot it in reciprocal space.

Once you have (Psi magnitude squared) as a function of the chosen parameters, the height of the function at any given point on this plot is going to be the magnitude of the probability density of that region of parameter space. In other words, that is how likely it is to be there, moving that fast, have that direction of magnetic spin, be that kind of particle, or whatever parameter you chose.

The width of the function in your parameter space is a measure of how poorly defined its value is (if the value is largely the same over a wide range of values, it is just as likely to be there as the other end... not a lot of information!). On the other hand, the probability could be clumped together tightly into a bunch of points, or even exactly one. In this case, the value is very well defined.

If you only pay attention to one parameter, there is in theory no limit to how closely a parameter can be confined, even down to an absolute certainty. However, certain pairs of parameters prevent each other from being well-defined, because a function that produces a well-defined value in one of the parameters MUST have a poorly defined value in the other. This is known as the Heisenberg Uncertainty Principle.

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