Experimental status of Heisenbergs uncertainty relations for position and momentum with regard to different interpretations

Question

I am reading "The philosophy of quantum mechanics" by Max Jammer. In the chapter about the uncertainty relations he writes:

Turning now to the question of the empirical support, we unhesitatingly declare that rarely in the history of physics has there been a principle of such universal importance which so few credentials of experimental tests. In fact, as far as $I_1$ is concerned, no methods seem presently to be available to measure, for example, the position and the momentum of an individual electron simultaneously with sufficient precision to evaluate the errors involved. [...] $I_2$ obviously enjoys a far better empirical backing. In fact, many measurements often referred as vindications of Heisenberg's relations are experimental corroborations of precisely of $I_2$.

$I_1$ and $I_2$ above revers to two interpretations of uncertainty relations, where:

1. The *nonstatistical interpretation* $I_1$ according to which it is impossible, in principle, to specify precisely the simultaneous values of canonically conjugate variables that describe the behaviour of single (individual) physical systems.

2. The *statistical interpretation* $I_2$ according to which the product of the standard deviations of two canonically conjugate variables has a lower bound given by $h/4\pi$.

Now I am wondering what's the experimental status of the uncertainty relations today regarding the position momentum uncertainty. Which are the important papers discussing (direct) experimental tests for $I_1$, which ones for $I_2$?

Has been any of the many Gedanken-experiments about the uncertainty relations implemented as a serious experimental test for the uncertainty relations?

Comments

This is an excellent question, I'd like to know the answer, too. There are POVMs for measuring position and momentum simultaneously, so the question makes sense is some sense, but it is not so easy to figure out what precisely this sense is. One of the difficulties is that there are no established rules for deciding what precisely constitutes an approximate but high precision position measurement is, or the same for a momentum measurement.

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A double slit experiment, as well as diffraction experiments with electrons (photons, etc.), are direct experiments where this uncertainty is manifested.

Answer 1

1. The *nonstatistical interpretation* I1

according to which it is impossible, in principle, to specify precisely the simultaneous values of canonically conjugate variables that describe the behaviour of single (individual) physical systems.

This is what experimental physicists would consider the standard interpretation: the HUP is inherent in the quantum nature of the particle under study. The HUP is binding not on any two measurable variables of a particle, but on the ones that are conjugate to each other due to the operator structure of quantum mechanics.

2. The *statistical interpretation* I2

according to which the product of the standard deviations of two canonically conjugate variables has a lower bound given by h/4π.

This is not what the quantum state is about, not about measurement uncertainty which is given as a standard deviation on any measurable variables of a system . The  Quantum mechanical uncertainty is not given by the width of some statistical distribution ( a Poisson or a Gausian), but from the probability distributions calculated from the quantum mechanical wave functions for the given boundary conditions.

A recent experimental  publication makes the difference between statistical measurement error and the HUP  clear.

Abstract:

While there is a rigorously proven relationship about uncertainties intrinsic to any quantum system, often referred to as "Heisenberg's Uncertainty Principle," Heisenberg originally formulated his ideas in terms of a relationship between the precision of a measurement and the disturbance it must create. Although this latter relationship is not rigorously proven, it is commonly believed (and taught) as an aspect of the broader uncertainty principle. Here, we experimentally observe a violation of Heisenberg's "measurement-disturbance relationship", using weak measurements to characterize a quantum system before and after it interacts with a measurement apparatus. Our experiment implements a 2010 proposal of Lund and Wiseman to confirm a revised measurement-disturbance relationship derived by Ozawa in 2003. Its results have broad implications for the foundations of quantum mechanics and for practical issues in quantum mechanics

In this experimental work, the difference between measurement error and inherent quantum mechanical uncertainty is made clear . They conclude"

In conclusion, using weak measurements to experimentally characterize a system before and after it interacts with a measurement apparatus, we have directly measured its precision and the disturbance. This has allowed us to measure a violation of Heisenberg's hypothesized MDR. Our work conclusively shows that, although correct for uncertainties in states, the form of Heisenberg's precision limit is incorrect if naively applied to measurement. Our work highlights an important fundamental difference between uncertainties in states and the limitations of measurement in quantum mechanics.

A search on the net gives many examples of consistency with the HUP, here is a fairly recent one.

The Heisenberg uncertainty principle for material objects is an essential corner stone of quantum mechanics and clearly visualizes the wave nature of matter. Here, we report a demonstration of the Heisenberg uncertainty principle for the fullerene molecule C70 at a temperature of 900 K. We do this by showing the increase in molecular momentum spread after passage through a narrow slit with a variable width down to 70 nm. We find good quantitative agreement with the theoretical expectation.

The HUP is important in quantum computing ( a subject I am not familiar with) and a number of recent publications exist with qubits in mind.