This might be a bit of a reflected version of a post from 2016, triggered by some popular science videos. I have the vague impression that there’s something that I don’t like about many popular science pieces on quantum physics and I haven’t quite generalized it until now: “They” are not distinguishing between the uncertainty properties of quantum-stuff and the idealizations that every model makes in modelling the world.

A look back at Newtonian Physics

Let’s discuss a billiard ball on a billiard table. In Newtonian Physics one could describe that as an extended body with perfectly known position, shape, momentum, angular momentum together with other perfectly known positions \(\mathbf{x}_0\), shapes, momenta \(\mathbf{p}_0\) of other balls and table edges and perfectly known properties of the table (rolling resistance, brush direction of the hair, bounciness of the edges, friction when bouncing with a spin), etc, and one could perfectly predict how the balls will move. (Newton’s laws could be spelled out in an incredibly complicated differential equation¹, which might have one unique solution, which could maybe be found, …)

\[\mathbf{x}(t) = \mathrm{math\ here, depending\ on\ }\mathbf{x}_0,\mathbf{p}_0,t\mathrm{\ and\ other\ things}\]

A nice model with nothing too astonishing.

And anybody with a sense for the real world or modelling would remark that the limits of the model are clearly that no position or momentum would ever be perfectly known. But - imo - that’s something outside of the model. This might be a nice segue² to describe the initial parameters as probability distributions, maybe describe them with moments and cut off after the first few terms³.

\[\rho(\mathbf{x}_0) = \mathrm{multivariate\ normal\ distribution\ around\ }\bar{\mathbf{x}}_0\mathrm{\ with\ covariance\ }\mathbf{\Sigma}\]

We could also look at the solution of the equation of motion to parametrize the position of the white ball \(\mathbf{x}(t)\) at time \(t\) as a function of the initial condition, make a Taylor series,

\[ \mathbf{x}_0 = \bar{\mathbf{x}_0} + \mathbf{\Delta_x}\\ \mathbf{x}(t) = \bar{\mathbf{x}(t)} + \mathbf{J} \mathbf{\Delta_x} \]

and if we’re lucky the convergence radius is larger than the width of the probability distribution of the initial positions⁴, cut off the Taylor series after the linear term and now \(\mathbf{x}(t)\) also has a normal probability distribution.

\[\rho(\mathbf{x}(t)) = \mathrm{multivariate\ normal\ distribution\ around\ }\bar{\mathbf{x}(t)}\mathrm{\ with\ covariance\ }\mathbf{J^\star\Sigma J}^{\star\star}\]

(\(\star\) / \(\star\star\): I am too lazy to figure out if these are \(\mathbf{J}^{-1}\) or \(\mathbf{J}^T\). The point is: the covariance is proportional to \(\mathbf{\Sigma}\), unique, computable).

We’ve just invented first year physics error propagation! But that’s exactly not what I wanted to write about.

When the Taylor series doesn’t converge⁵ we (can) have very different situations for \(x(t)\) for very similar initial conditions. That can be called non-perturbative and is a typical introduction to chaos theory.⁶ (I never got past the introduction)

Still, within the model of Newtonian Physics we usually deal with perfectly known parameters and get perfectly accurate solutions. Everything unknown is loss in the translation from real world to model.

Now quantum

Instead of a billiard ball, let’s shoot an electron or photon or anything quantum-y onto another such thing. Here, the boundary condition of the model is not a given point in phase space, described by position and momentum, but rather a wave function⁷. The practitioner from the previous section could (and rightfully should!) still argue that the wave function of our photon is not perfectly known, but that’s what our model works with, so in the model we have a perfectly known wave function. And one “feature” of quantum physics is that this wave function may have a well defined position, it may have a well defined momentum, but it cannot have both. If we had a magical “prepare my experiment like I modelled it” machine, we could’ve asked it for a billiard ball with no angular momentum at precise lab coordinates and with a given momentum, but we can’t ask it for a photon beam with exact position and momentum.

\[\Phi(x, 0) = \mathrm{all\ perfectly\ known.\ No\ covariance}\]

And now our electron⁸ with perfectly known wave function as initial condition flies towards another electron on our quantum billiard table, and we cannot say with certainty what’s going to happen. We can compute a wave function for the final state⁹ and that then can try to translate the model’s language into “real world” words, which are: a probability distribution for possible positions of the electron after the collision.

\[\Phi(x, t) = \mathrm{again,\ all\ known.\ No\ covariance}\\ \rho(\mathbf{x}(t)) = \Phi(x, t)^\dagger\Phi(x, t) = \mathrm{non-trivial}\]

Seemingly as in the Newtonian case, there’s a probability distribution for the final state, but that is not the propagation of our ignorance of the initial parameters, but actually an inherent property of the model. (And there’s abundant experimental evidence that that’s not a shortcoming of the model but rather the model correctly modelling the randomness of the microscopic real world¹⁰.)

better examples?

Actually, I think particle decays are nicer examples: When does a tauon¹¹ decay? Into what? There, I wouldn’t even bother discussion the initial condition and we still have nice probabilistic outcomes. But in the spirit of xkcd 2501, maybe I’m already going too far in my assumption that the typical reader can imagine electron collisions and should talk about particles they never heard of. (Actually, science communicators have told me, one shouldn’t assume the layperson would understand what a particle physicist is talking about when mentioning “particle decays”. I would hope discussions of radioactive decays in high school math - as example for exponential functions - would form a basis. All you need to know is “now the thing is there, and now it isn’t anymore and there’s only splinters and trash”. So even taking into account that I also forgot gazillion of things from school, a bare minium - small thing undergoes rapid disassembly event - shouldn’t be too hard. … But I guess I shouldn’t ignore professionals’ advice, especially without doing any research into science communication myself).

Further reading / watching

• The remark about the uniqueness of the solutions to Newton’s equations of motion is motivated by this up and atom video that recently appeared in my timeline (youtube link).

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