In chapter 1, part 6 of the 3rd Ed. Of Introduction to Quantum Mechanics, Griffiths’ says
“Please understand what uncertainty principle means:Like position measurements, momentum measurements yield precise answers — the spread here refers to the fact that measurements made of identically prepared systems do not yield identical results. “
To me this explanation means: if we grab an ensemble of particles with the prepared with same wave function, and we take the $x$ and $p$ measurements on each particle (somehow simultaneously?, does it matter?), the product of the standard deviations of the collection of this measurements is always greater or equal to some value. Is this correct? Additionally, this quote invokes that the uncertainty principle does not apply on the measurement uncertainty of a single particle’s x and p measurements. Is this correct? Does it apply to a single wave functions standard deviations in position and momentum? Is this the same standard deviation as that obtained when actually doing the ensemble measurements?
For context: I am a 3rd year physics undergrad prereading the text for next quarters intro quantum class. I already covered all the other Griffiths book.
I also read this post (Is the uncertainty principle a statement about limits on our predictive rather than our measurement abilities?) but it does not answer my question about the simultaneity of the measurements or the applicability pf the principle to one particle’s uncertainty in measurement.