> Regarding the similarity to a double slit interference pattern, what I found in my investigation (see my previous comment) is it was solely the result of the sorting algorithm used for the X coordinates of each possible outcome/weight. Ordering by the average X location results in a binomial distribution, but there are still ties to deal with. Applying secondary sorting to those based on the position of the left or rightmost X creates secondary binomial distributions within the main binomial distribution, similar to what Jonathan showed, except without all the spikiness because his example wasn’t consistently sorted in this manner.
> You can see the sorting I’m talking about by looking at the sorting of Xs in the output of weights in Jonathan’s double-slit examples. The bottom half is sorted as I stated, but the top half is somewhat out of order, corresponding to the spikes in the tallest curve in the center of the graphs.
> Maxime C, I had the same question as you – Why choose this sorting method for encoding the photon positions? My conclusion is that it doesn’t work however. Each additional step run with a large string produces an additional secondary binomial distribution. To get a smooth curve, we’d need to run billions of steps on a large string, but then we we’d also see billions of peaks, which no longer compares well with the expected diffraction pattern.
For example, Werner Heisenberg's doctoral thesis[1] arose from a contract of his doctor father Arnold Sommerfeld from a company that dealt with the channelling of the Isar river through the city of Munich. Very practical problems involving differential equations - kind of the bread and butter of physicists and engineers at the time.
What if quantum mechanics was found today in a world where the bread and butter has shifted to computer science, linear algebra and discrete math? Would we still end up with waves and differential equations, or would another formulation arise more naturally?
EDIT: I think a beautiful (but imperfect) example to illustrate this dichotomy in the ways of thinking is how the Bell inequality can be approached with photons and polarization or as a game. Thinking about Alice and Bob or polarized light, which would you prefer?
There was considerable disagreement between the factions of physicists who favoured the different versions which essentially ended when after some considerable theoretical effort (mostly by Dirac) it was shown that the two pictures are exactly equivalent.
Physicists still use whichever formulation is most suitable for whatever problem they're trying to solve, for example if you're analysing the something where you care about a bunch of bound states like the simple harmonic oscillator or the hydrogen atom then the matrix picture tends to be easier to work with.
You are right that wave mechanics was more popular than matrix mechanics because physicists were already very familiar with wave methods.
Heisenberg's matrix mechanics was the first, and formulated entirely in terms of linear algebra. See for example the introduction of basic linear algebra techniques in the famous Bohr-Jordan paper from 1925:
For me it's a reminder that physics describes how quantum systems evolve, but it doesn't (in a sense) tell us what those systems are. Are particles waves? Matrices? Excitations of a field?
Each of these descriptions works, so I can't exclusively say any one of them is what particles really are. Bring a macroscopic being is philosophically frustrating.
"Quantum mechanics is what you would inevitably come up with if you started with probability theory, and then said, let's try to generalize it so that the numbers we used to call 'probabilities' can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn't, but it could have been."
Well worth a read. In fact, I'd say it's worth buying the book just for this one chapter.
I discussed the difference earlier: https://news.ycombinator.com/item?id=38255476
Before this I didn’t know Stephen hosted “Live CEOing” sessions and now I wish this was the norm!
0: https://www.twitch.tv/videos/2083073452 (timestamp around 50:00)
https://www.preposterousuniverse.com/podcast/2021/07/12/155-...
Looking at the numbers on the graphs for single-slit diffraction, they are just binomial coefficient, at least mostly, not sure why there are pieces missing in the last rows. That is also what you expect when you repeatedly make binary decisions to go left or right. The article does not mention the binomial distributions once, it only appears in a comment.
And then they claim that it converge to the actual single-slit diffraction distribution, something with a Chebyshev polynomial and the sinc function, according to the article. Seemingly without justification besides looking at graphs and noting that they are both bell shaped. As said, not sure what is going on in the last rows of the graphs, but I would almost bet that the two functions are not the same, even in the limit as it becomes a Poisson distribution plus whatever the last rows do.
Why do they not just proof that the two are the same? The entire article seems to be about getting numbers out of their multiway system and then concluding that - if you squint hard enough - they look somewhat like diffraction patterns.
A Gaussian distribution, I think. But they're certaintly not the same function, and it should be immediately obvious to a math grad with experience in physics. The sinc function, for one, has secondary maxima (its plot in the article is very convenienty cropped to allow pretending those don't exist). Just put a hair in the path of a laser beam and you will see the local maxima in light intensity! Their "single-slit" string procedure, on the other hand, can only generate a single central peak. This really makes no sense at all.
I mean we obviously understand how they work at a pure mechanical level, and we have this analogy with lookup (keys, queries, values) and "attention," but do we really get it? Can someone explain to me why that design works so much better than lots of other things like RNNs?
Or did we just tinker a lot (a method known as "graduate student descent") guided by mathematical hunches and loose analogies with biological brains until we found something that kinda worked?
It wouldn't be the first time. AFAIK we got the idea of wings from birds and figured out how to fly with them before we had a really solid fluid mechanical understanding of how and why wings work the way they do. We just thought "hmm so birds fly, so lets try stuff that looks a bit like that..."
Quantum mechanics is used in very different ways by people in quantum information, condensed matter, many body physics, quantum field theory, nuclear physics and may other (sub)fields.
Of course it will be difficult for a quantum information theorist if they try to apply what they know directly to a hydrogen atom, but (speaking from experience) it will also be quite difficult for someone trained in what you call Hermitian quantum mechanics to directly apply what they know to quantum field theory, or quantum information or any other subfield that uses different language.
I strongly disagree with your summary "if you know Hermitian quantum mechanics then unitary quantum mechanics is conceptually straightforward. If you know unitary quantum mechanics then you will have a lot of new concepts and mathematics to learn before you understand hermitian quantum mechanics".
I challenge anyone trained in Hermitian quantum mechanics to make progress on (for example) proving or disproving the generalised quantum Stein's lemma, or any of the unsolved problems here https://arxiv.org/abs/2002.03233 using those methods.
For example, the paper you cite is entirely understandable for anyone with some training in hermitian QM. In contrast, good luck trying to understand elementary concepts like the spectrum of the hydrogen atom or interference of matter waves from unitary QM.
Of course the field of quantum info has progressed enormously and has its own interesting challenges, for which hermitian QM is all but useless.
Concepts like the spectrum of the hydrogen atom or interference phenomena aren't particularly difficult to understand conceptually: the Hamiltonian has some eigenvectors and eigenvalues, you use the Dirac equation and work them out. The "matter waves" interfere essentially in the same way that waves on the surface of a pond do.
The things that you're calling conceptual understanding I guess must be different to this: maybe something like detailed calculations of the structure of the spectrum?
For a pure mathematician, quantum mechanics is a lovely introduction to Hilbert Spaces.
I know there are what are called C*-algebras, which help solve some of these issues, but I don't know anything about them. I do know that the Hilbert Space approach is not sufficient.
Aaronson claims that QM "can be derived" (quote from the original comment) are followed by an introduction to some aspects of QM that leads to strictly none of these things. That is why I am unhappy with it, and I still do not see why I am "somewhat mistaken" (quote from you).
In fact, I can go even further and say that (from a quick glance at least) in his whole book positions and momenta make no appearance, let alone the correspondence principle. (I do not even see hermitian operators!) Without them I just do not see any reasonable "derivation" of (or, more properly, argumentation for) what I call QM.
I completely agree with you that from the perspective of a quantum information theorist computing the spectrum of the hydrogen atom is a rather complicated thing. I disagree wholeheartedly that this is part of the essence of quantum mechanics.
The hydrogen atom is one system, understanding conceptually that its behavior is governed by a self-adjoint operator and its spectrum is very relevant to the whole of quantum physics. Understanding exactly the details of the calculation I think are not. Especially because if you do the calculation within quantum mechanics without quantum field theory you will obtain a somewhat incorrect result anyway (you will miss interesting phenomena like Lamb shift).
Similarly interference is an interesting phenomenon that one needs to understand to understand quantum mechanics, but understanding the specific calculation of how interference makes nice patterns in some example setup isn't particularly enlightening.
I agree that the Heisenberg uncertainty principle is important, but it certainly be derived from Aaronson's point of view (e.g. the standard Robertson-Schrödinger inequality is easily obtained).
As an aside, I also think that self-adjoint operators and the correspondence principle are a fairly terrible way to think about observables in quantum mechanics. An obvious fact is that every measurement of (for example) the position of a particle has some unavoidable experimental error (our apparatus only has finite resolution) so the thing we actually measure in reality is some fuzzy observable which can not be represented as a self-adjoint operator. A POVM is a much more natural candidate (as a physicist it is natural to assume that the thing you get by adding some classical noise to your measurement is still a measurement).
No, or at least I did not mean to: I said "know how to compute", not "compute". One typically uses the Schrodinger equation to do so (although Pauli did not need it), but this starting point is nowhere to be found here.
> I agree that the Heisenberg uncertainty principle is important, but it certainly be derived from Aaronson's point of view (e.g. the standard Robertson-Schrödinger inequality is easily obtained).
Robertson-Schrödinger is fairly trivial, at least in Aaronson's finite-dimensional world. But you conveniently forgot how to "derive" the part of QM that actually gives you the value of the commutator sitting on the right-hand side. So will you just postulate it? That sounds pretty terrible pedagogically, and it might be better to provide at least some general discussion. And that discussion is exactly what I am advocating as a necessary ingredient in any self-respecting introduction to (let alone derivation of) QM.
> I also think that self-adjoint operators and the correspondence principle are a fairly terrible way to think about observables in quantum mechanics.
No teacher of QM should introduce POVMs before talking about positions and momenta.
In my opinion knowing how to use the Schrodinger equation to get the "spectrum of the hydrogen atom" is essentially a matter of historical interest but really not relevant to understanding things. Its quite cool you can do these tricks to derive a nice analytical form for the spectrum, but this approach emphatically does not generalise to more complicated systems (any non-trivial molecule) and even for the hydrogen atom the spectrum you get will be wrong anyway because of relativistic corrections and QFT-corrections.
> But you conveniently forgot how to "derive" the part of QM that actually gives you the value of the commutator sitting on the right-hand side.
I'm not sure what you're arguing is missing here? Once you've derived Robertson-Schrödinger you've just got a commutator there, for whatever observables you want to apply it to you just plug in the value.
>No teacher of QM should introduce POVMs before talking about positions and momenta.
I'm not talking about teaching here but thinking. You are probably right that most physics undergrads would not cope well with learning about POVMs. On the other hand I am tempted to argue for not teaching about the position operator and position in Schrödinger-style QM at all, or at least leaving it until quite late on. The way people teach QM has this weird thing where its pretty obviously wrong, because every physics undergrad knows we have special relativity, so there should be some nice symmetry between space and time which is completely missing in the Schrödinger equation. Time in the Schrödinger equation is a coordinate, and space (position) is a self-adjoint operator, which is just manifestly weird. Once you get to quantum field theory this gets fixed and position isn't an operator/observable anymore, it gets demoted back to a coordinate exactly the same as time.
FWIW, as someone who is interested in science pedagogy, and specifically as someone who actively engages with anti-science propaganda like young-earth creationism, I want to contribute this:
> In my opinion knowing how to use the Schrodinger equation to get the "spectrum of the hydrogen atom" is essentially a matter of historical interest but really not relevant to understanding things.
IMHO this is more than historical interest. It's a dramatic illustration of how science actually works, and specifically, that it does not rely on any appeal to authority, despite the superficial appearance of occasionally hearing people say things like, "Einstein teaches us that X" with the implication that X is therefore unquestionable gospel because Einstein said it. Here is an example of a calculation that anyone can do (with enough effort) and compare to the results of experiments that they can likewise do themselves (with enough effort). Of course, most people won't bother to put in this effort, but just knowing that they could if they wanted to is very powerful because it provides an actual reason why other people's results are generally trustworthy: even if you don't do the experiment, someone else might, and if the result turns out to be wrong then it will eventually be called out.
Also...
> this approach emphatically does not generalise to more complicated systems
This is spot on. Speaking from first-hand experience of my own intellectual journey into QM, focusing on single-particle systems and slogans like "any attempt to measure the position of the particle destroys the interference in the two-slit experiment" is extremely misleading. It leads to conceptual dead-ends that make it much harder to wrap your brain around entanglement than it should be. IMHO, QM pedagogy should start with entanglement and decoherence. In this respect, I think Aaronson gets it right.
But mainly I just wanted to thank you both for the privilege of being a fly on the wall while you discuss these things. It has generated a long reading list for me.
The gap in understanding between the students in the two courses is humongous. Both sets of students would need to essentially sit through half a semester of the classes of the other course to understand what they are saying.
The QC students don't know Schrodinger's equation at all, let alone how to solve it for the quantum harmonic oscillator or for the hydrogen atom (without which I agree you don't know QM). And the QM students know what Hamiltonian dynamics look like, but little about unitary dynamics.