Ah, I see. You've changed from talking about concepts and conceptual understanding to talking about computations.

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).

> Ah, I see. You've changed from talking about concepts and conceptual understanding to talking about computations.

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.

> One typically uses the Schrodinger equation to do so

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.

I just wanted to chime back in here and say that I am finding this discussion absolutely fascinating and enlightening. This is HN at its best. Thank you.

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.

I will chime in to say that I have several times taught a course on Quantum Computing using an Scott Aaronson type approach and a course on Quantum Mechanics in the traditional way. With some overlap of students.

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.