Unclear explanation: HUP in Griffiths’ 3rd Edition

Question

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.

Answer 1

The original arguments provided by Heisenberg to stress some consequences of the non-commuting algebra of observables (Heisenberg's microscope) very often cause confusion when people meet the statistical uncertainty relations which are not a principle but a theorem which is a direct consequence of the formalism of QM and its statistical interpretation.

Just looking at the way the so-called HUP is derived, for example in Griffith's book, it is clear that in no place there is mention of two measurements on the same physical system. The reason is that two measurements of one component of the position, say $x$, and the corresponding component of the momentum, $p_x$, taken at different times, would implies that the second measurement is not on the same state as the first one, at variance with the request that the two uncertainties, $\Delta x$ and $\Delta p_x$ should be relative to the same quantum state. Moreover, a really simultaneous measurement would be incompatible with the formalism and its well established agreement with experiments.

What the statistical uncertainty relations say is that the statistics of two independent set of measurements on an ensemble of similarly prepared quantum systems all in the same state, each measurement performed on a different member of the ensemble, will be distributed according to two distributions, concentrated in such a way that the resulting statistical spreads will satisfy the uncertainty relations.

Possible limits to the product of uncertainties of non-commuting variables for measurements on the same systems have been analyzed in recent years both on the experimental and on the theoretical side.

Answer 2

GiorgioP's answer is great. I want to address the final part of your quote though.

the spread here refers to the fact that measurements made of identically prepared systems do not yield identical results.

I don't think this part is specifically about the HUP. Even for just single measurements of indentically prepared states $|\psi\rangle$ we will not always get identical results. For any observable $A$ with eigenvalues $a$ there is always a probability of measuring a value of $a$ of $$P(a)= |\langle a|\psi\rangle|^2$$ If for more than one $a$ value $\langle a|\psi\rangle\neq0$, then you cannot expect the same result for each measurement. This has nothing to do with an uncertainty principle; it's just a consequence of the probabilistic nature of QM measurements.

Contrary to what many people think, the HUP doesn't show that spreads in measurements exist, rather it just shows how for certain pairs of measurements, their spreads (more specifically the product of the standard deviations) are constrained to be no smaller than a certain value. In other words, the HUP doesn't explain uncertainty, it just characterizes it.