What's the relationship between the definition of the uncertainty principle using standard deviations vs using $\Delta x$ and $\Delta p$?

Question

So I've heard two different explanations of the uncertainty principle, both of which make sense on their own, but I'm having a hard time figuring out how they're connected. The first is that the uncertainty principle is really a statistical principle. Basically, we can measure the position of a particle any individual time however precisely our equipment allows. Same with the momentum. However, if we take multiple measurements, there will always be some variation in both data sets and the product of the standard deviations will always be greater than or equal to $h/2$.

The other explanation I've heard explains it in terms of properties of waves. Essentially, a particle is really just the superposition of many different waves in the relevant field. A period wave doesn't HAVE a well-defined position or momentum, so we have to add together many different waves of varying momentums in order to create enough destructive interference that most of the peaks cancel out and we get a wave-packet with a fairly well-defined position, though there will always be SOME variation. A similar process is required for momentum. This is the most intuitive explanation of the uncertainty principle I've heard, but how is it related to the statistical definition? I know it has something to do with the wave-function being a probability function (or, more specifically, its squared magnitude is the probability density function) but I'm not sure exactly what.

I think part of what's got me confused is that I don't understand how it's possible to add up a bunch of waves to (for lack of a better term) "generate" a particle with a better defined position/momentum. The math makes sense, but how does it match up to how we actually perform measurements? I don't need the details of how specific measurement devices work but more the concept of measurement in quantum mechanics and how it corresponds to the idea of taking the superposition of many waves to create a wave-packet and what that has to do with taking multiple measurements and having variation in the data.

I feel like I'm starting to actually get at least a bit of a grip on how quantum mechanics works (I've been fascinated by it for years), but I'm quite confused about this. Any help is greatly appreciated.

Answer 1

You more or less answered your own question. The wave properties are described by a wave function that is governed by the dynamical equations of the system under investigation and the statistical properties appear when you make measurements of this wave function. Such measurements will sample the wave function in a random fashion so that the squared modulus of the wave function gives the probability density for the measurement results. In that way, the uncertainty principle is described by both.

The wave function can either be represented in "position" space, as a function of position coordinates, or in the Fourier domain as a function of the wave vectors. One can move from one to the other via (inverse) Fourier transforms. Even in classical optics (Fourier optics), one can already see how an optical field can be represented as a superposition of plane waves. That then leads to the understand of the uncertainty relationship for the wave function.

Measurements of the wave functions always involve interactions between the wave function and the measurement device. Such an interaction involves a transfer of energy and momentum governed by Planck's (and de Broglie's) relation between frequency and energy (wavenumber and momentum), which brings in Plank's constant. Therefore, we find Planck's constant in the uncertainty relationship.

Answer 2

A preliminar point should be made clear, before discussing Uncertainty Relations. According to Quantum Mechanics a particle is never a wave or a superposition of waves. Classical waves have nothing to do with individual quantum particles. A relation with waves does exist, but it is much more subtle: waves can used to obtain the probability distribution of observable quantities. There is a big difference with the the statement that a particle is a wave. If this would be true, it would be possible to detect arbitrarily small amount of any observable like charge, spin etc., while experimentally we know that that quantities can only have well definite values.

The content of the Uncertainty Principle (UP) is apparently simple. Nevertheless, after its statement by Werner Heisenberg, strongly influenced by Bohr's complementary principle, it underwent an important mutation after Robertson's derivation of a general inequality for the product of the variances of the statistical distributions of the values of two non-commuting operators. Coexistence of the original Heisenberg's and statistical interpretations still causes conceptual problems and misconceptions in understanding UP.

After Robertson's work, UP relations are presented as a statement about the distribution of the possible values of two no-commuting observables, when measured with arbitrary high accuracy in a given quantum state. As such, there is no reference to experimental disturbs and even the observer effect plays a minor role. Moreover, there is not any reference to wave-like properties of individual particles. Actually, the statistical intepretation of UP implies only that, if an ensemble of quantum systems has been prepared in a given state $\left| 0 \right>$, independent measurements of observable represented by operators $A$ and $B$ would imply that the distribution of measurements of $A$ and $B$ should be subject to the inequality $$ \sigma^2_A \sigma^2_B \geq \big| \dfrac{1}{2i} \langle 0[A,B] 0\rangle \big|^2. $$

As such, the UP would say something quite different from the original ideas contained in the analysis of Heisenberg's microscope. Unfortunately, this point is often not made clear in discussions on UP.

The present point of view on Heisenberg's formulation is that it was an attempt to address a different problem. which has a common origin in the non-commutation of some pairs of operators representing observables, but does not coincides with Robertson's statistical result.

This last point has emerged quite clear by a revival of interest, in the last couples of decades, for the physical content of UP with respect to the problem of (almost) simultaneous measurements of non-commuting observables.

Indeed, the non commuting of two operators, according to the basic postulates of QM implies that it is not possible to measure at the same time the two quantities (thus something quite different from the unrelated set of measurements of the statistical analysis of UP). The reason is that one of the basic postulates of QM says that the effect of a measurement of a quantity A is to bring the quantum system into one of the eigenstates of the corresponding operator. However two non-commuting operators do not have a set of common eigenvectors, then the theoretical impossibility of a simultaneous measurement.

In recent years, people have started to analyze quantitatively such impossibility, asking questions about how good could be, on theoretical basis, a joint measurement of two non-commuting observables. See for instance the paper by Cyril Branciard on PNAS and references therein contained.

Under such new viewpoint, it is possible to recover in a semi-quantitative form the original Heisenberg's formulation, although the exact value of the "uncertainty" may be slightly different.