[eigenket]

I think this is a very common view, and a somewhat mistaken one.

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.

[prof-dr-ir]

With conceptually straightforward I meant that the concepts are easy to pick up.

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.

[eigenket]

I think maybe we have some difference in how we're talking about things.

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?

[prof-dr-ir]

In my definition, someone cannot claim to know QM if they do not know (a) Heisenberg's uncertainty principle, (b) how to compute interference patterns or (c) how to compute the spectrum of hydrogen.

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.

[eigenket]

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).

[prof-dr-ir]

> 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.

[eigenket]

> 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.