Upon rereading all the above, seems like briefly discussing HUP would be of interest to you.
Regard 'uncertain' as 'non deterministic' in descriptions below. [I bet someone will object to that!]
These are from my notes, so, thankfully, I can just copy and paste:
Synopsis: If you search HUP in these forums you can rummage through many pages of disagreements and clarifications. The quotes below are slightly edited posts [for brevity, clarity] from those discussions. [I had little idea myself what HUP REALLY meant until arguments/discussions/and some research papers were dissected in these forums.]My own single sentence explanations :
A] Get a better instrument and you'll get better measurement results to any accuracy.
B] Quantum theory does not predict the outcomes of single measurements, it only predicts the ensemble [statistical] properties of multiple measurements.
C] In classical mechanics we can predict with absolute precision, to arbitrary accuracy, the future position and momentum [for example] of a single particle; The HUP says no you can't: you can only make a statistically based prediction! Summary Details:
It IS possible to simultaneously measure the position and momentum of a single particle. The HUP doesn't say anything about whether you can measure both in a single measurement at the same time. That is a separate issue.
It is possible to measure position and momentum simultaneously…a single measurement of a particle. What we can't do is to prepare an identical set of states… such that we would be able to make an accurate prediction about what the result of a position measurement would be and an accurate prediction about what the result of a momentum measurement would be….for an ensemble of future measurements.
What we call "uncertainty" is a property of a statistical distribution. The HUP isn't about a single measurement and what can be obtained out of that single measurement. It is about how well we can predict subsequent measurements given the ‘identical’ initial conditions. The commutativity and non commutivity of operators applies to the distribution of results, not an individual measurement. This "inability to repeat identical measurement results" is in my opinion better described as an inability to prepare a state which results in identical observables.
The uncertainty principle results from uncertainties which arise when attempting to prepare a set of identically prepared states…from identical initial conditions. The wave function is not associated with an individual particle but rather with the probability for finding particles at a particular position.
What we can't do is to prepare an identical set of states [that yields identical measurements]. NO STATE PREPARATION PROCEDURE IS POSSIBLE WHICH WOULD YIELD AN ENSEMBLE OF SYSTEMS IDENTICAL IN ALL OF THEIR OBSERVABLE PROPERTIES. [instead, identical’ state preparation procedures yield a statistical distribution of observables [measurements].]
Fredrik:
To prepare a state is to bring a particle on which we intend to do a measurement to the measuring device. Different ways of doing that may give us different average results. Two ways of doing it (two preparation procedures) are considered equivalent if no series of measurements can distinguish between them (i.e. if they give us the same wavefunction, or more generally, the same state operator/density matrix). These equivalence classes are often called "states".
The uncertainty principle restricts the degree of statistical homogeneity which it is possible to achieve in an ensemble of similarly prepared systems. A non-destructive position measurement is a state preparation that localizes the particle in the sense that it makes its wavefunction sharply peaked. This of course "flattens" its Fourier transform, so if the Fourier transform was sharply peaked before the position measurement, it isn't anymore.
The Uncertainty Principle finds its natural interpretation as a lower bound on the statistical dispersion among similarly prepared systems resulting from identical state preparation procedures and is not in any real sense related to the possible disturbance of a system by a measurement. The distinction between measurement and state preparation is essential for clarity.
A quantum state (pure or otherwise) represents an ensemble of similarly prepared systems. For example, the system may be a single electron. The ensemble will be the conceptual (infinite) set of all single electrons which have been subjected to some state preparation technique (to be specified for each state), generally by interaction with a suitable apparatus.
