What is the evidence for the claim that the Uncertainty Principle is not a result of inadequacies in the measuring instruments/technique?

Question

If we accept that quantum mechanics is a good theory for describing microscopic systems (for, there is plenty of evidence that suggests quantum mechanics is a good theory), then the property of non-commuting operators results in a general uncertainty principle of which the Heisenberg Uncertainty Principle is a special case. And that is fundamental to the theory. Non-commuting quantities happily account for things like the two-slits experiment and the Uncertainty Principle pops out of the theory as natural consequence.

However, is the claim "Uncertainty Principle is NOT a result of inadequacies in the measuring instruments/technique" merely a conjecture which is well supported by fact that nowadays quantum mechanics is probably the most successful theory in physics?

Answer 1

As you wrote, the uncertainty relations (UR) are a direct consequence of the theory. They can be proved just by evaluating the expected dispersion around the average of the observable values of two non-commuting observables.

Theoretical expectation values and then variances do not contain information about possible experimental uncertainty. They are exact mathematical consequences of the theory. When QM axioms say that the possible outcome of a measurement of an observable $A$ belongs to the spectrum of the corresponding operator on a Hilbert space, this should be intended as an exact statement (without reference to any inadequacies in the measuring instruments/technique).

Treatment of the consequences of limited-precision experiments is a separate story. It is not different from the situation in classical Physics, where we have as well exact formal results, and the analysis of measurement limitations is a kind of post-processing of the theory.

Answer 2

That quantum uncertainty is not the same as classical uncertainty (characterizing an,y measurement device) as can be shown in an interference experiment (two-slit experiment is a model one, but there are real-world experiments based on it). While the classical uncertainty is fully characterized by a probability distribution, $p(x)$, the quantum uncertainty is characterized by probability amplitude: $\psi(x)$, so that the corresponding probability is $p(x)=|\psi(x)|^2$.

E.g., in the two slit experiment, the classical uncertainty from the electrons passing through the two slits would result in the addition of probabilities: $$p(x)=p(x|1)p_1 + p(x|2)p_2,$$ where $p_j$ is the probability to pass through slit $j$, whereas $p(x|j)$ is the uncertainty of electrons passing through slit $j$. The two are uncorrelated.

In quantum case the probability distribution on screen is $$ p(x)=|a_1\psi_1(x)+a_2\psi_2(x)|^2=|a_1\psi_1(x)|^2+|a_2\psi_2(x)|^2+2\Re\left[a_1a_2^*\psi_1(x)\psi_2^*(x)\right]= p(x|1)p_1+p(x|2)p_2+2\Re\left[a_1a_2^*\psi_1(x)\psi_2^*(x)\right], $$ where $p_j$|a_j|^2$, $p(x|j)=|\psi_j(x)|^2$.

Thus, quantum and classical treatment produce different results, which can be tested experimentally.

Answer 3

The uncertainty principle is nothing but a mathematical conclusion. This conclusion emerges from the physical hypothesis of Hilbert space and operators on it.

A system is fully described by a vector in Hilebert space, where the Schwartz inequality can be applied. Variables are represented by hermitian operators, namely q-number, leading to non-trival commutation relations.

No measurement is taken in this procedure, so the uncertainty principle has nothing to do with measurements. Nevertheless uncertainty does exist during a measurement, often results of inadequacies in the measuring instruments/technique, which can be reduced by improving device porformance or experimental techniques. The total uncertaity is combination of these two parts.

That' why uncertainty due to the uncertianty principle is called the quantum limit.

Answer 4

Quantum mechanics is an extremely well-validated theory that has withstood all attempts to explain it away with reference to limitations in measurements/experimental technique.

If you are for some reason fundamentally opposed to believing this then you unlikely to be convinced otherwise by a post in this forum. It takes some years of study to internalise the intuition that the classical picture is wrong. I will try to point you in the correct direction.

One of the postulates of quantum mechanics is that observables correspond to mathematical operators, and that the possible measured values of these observables correspond to the operators spectrum.

In that context the Heisenberg uncertainty principle is just one of many uncertainty relations that occur. There is in fact an uncertainty principle for each pair of non-commuting observables. In that sense there is nothing 'special' about the the Heisenberg uncertainty principle. It does however present an obvious challenge to our classical intuition , and is often presented first for historical and pedagogical reasons.

The important point is that the various uncertainty principles follow inevitably if we accept that observables are represented by (potentially non-commuting) operators.

If you are happy to settle for experimental evidence for the fundamental truth of a different uncertainty relation (for spin components), I suggest you read about the sequential Stern-Gerlach experiment. This is explained well in the first chapter of Sakurai's 'Modern Quantum Mechanics'. It provides, in a conceptually clear way, strong evidence that we simply cannot, even in principle simultaneously measure the different spin components of a particle.

To quote Sakurai:

It is to be clearly understood that the limitation we have encountered in determining $S_z$ and $S_x$ is not due to the incompetence of the experimentalist. By improving the experimental techniques we cannot make the $S_z$ - component out of the third apparatus ... disappear. The peculiarities of quantum mechanics is imposed upon us by the experiment itself [emphasis added].

It is important to understood that the mathematical theory was created to make sense of these kinds of experiments - someone didn't arbitrarily decide to represent observables by operators because they felt like it, it was necessary to make sense of the experimental data.