TITLE: Distribution of Quantum Measurement Uncertainties (what Exactly does the Heisenberg Uncertainty Principle tell us)?
QUESTION [0 upvotes]: In quantum mechanics, one of the first results students (and random people with an interest in physics) are exposed to is the Heisenberg Incertainty Principle. Informally, this principle expresses the product of the uncertainties of certain pairs of properties must be greater than some fixed value. Essentially, the "knowabilities" of certain pairs of properties of a quantum particle are linked such that certainty in one gives rise to uncertainty in another. Typically, the properties position and momentum are given as an example, so I will use those for the remainder of this question.
I will use $x$ and $\Delta x$ for position and uncertainty in position and $p$ and $\Delta p$ for momentum and uncertainty in momentum, all respectively.
My question is this:
If I make a measurement, and say I know the measurement has uncertainty $\Delta x$, what distribution should I expect "actual" position to follow. I know $\Delta x$ and let's say I know $\Delta p$ as well. These are numbers, they don't describe a physical state without some context. Are $\Delta x$ and $\Delta p$ the bounds of a uniform distribution describing the possible values of $x$ and $p$? Are they the variance of a normal distribution? Are they the variance (or some other property) of a distribution whose shape is determined by the quantum state of the particles interacting during the observation?
I have found some resources (like this one https://opentextbc.ca/universityphysicsv3openstax/chapter/the-heisenberg-uncertainty-principle/) that reference this but don't ever give an explicit answer. I also know that the Heisenberg Uncertainty Principle is a special case of the Cauchy-Schwarz inequality, which I plan to explore as an avenue to answering this question for myself.

REPLY [0 votes]: You appear to assume that a particle being measured has a definite position and momentum and that Δx and Δp represent an uncertainty in the result of the measurement. That is a misleading view. The uncertainty relation expresses the fact that a particle cannot posses a definite momentum and position simultaneously. This follows directly from the properties of waves. A pure sine wave (which is a mathematical abstraction) has a perfectly defined frequency but no location- it is spread throughout space.  A localised wave (ie one whose amplitude tails off to zero over some finite space) does not have a single frequency (although it might have a dominant one)- instead it is a superposition of many overtones. The more localised the wave, the more the overtones are present in the mix. The uncertainty relation simply represents that fact- that the narrower the spatial spread of the wave, the more the overtones are present in it, while conversely the more the overtones drop out to reveal a dominant fundamental frequency the more the wave must be spread through space.
So, a wave cannot have a well-defined position without having a wide spread of frequencies (ie momentum values), and it cannot have a well-defined frequency (ie momentum) without being widely spread through space.
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